Real Analysis

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Library of Congress Cataloging-in-Publication Data:

Folland, Gerald B.
    Real analysis : modern techniques and their applications / Gerald B. Folland. — 2nd ed.
        p. cm. — (Pure and applied mathematics)
      “A Wiley-Interscience publication.”
      Includes bibliographical references and index.
      ISBN 0-471-31716-0 (cloth : alk. paper)
      1. Mathematical analysis. 2. Functions of real variables.
  I. Title. II. Series: Pure and applied mathematics (John Wiley & Sons: Unnumbered)
  QA300.F67 1999
  515—dc21
  98–37260

To the memory of my mother and father
Helen B. Folland
and
Harold F. Folland

Preface

The name “real analysis” is something of an anachronism. Originally applied to the theory of functions of a real variable, it has come to encompass several subjects of a more general and abstract nature that underlie much of modern analysis. These general theories and their applications are the subject of this book, which is intended primarily as a text for a graduate-level analysis course. Chapters 1 through 7 are devoted to the core material from measure and integration theory, point set topology, and functional analysis that is a part of most graduate curricula in mathematics, together with a few related but less standard items with which I think all analysts should be acquainted. The last four chapters contain a variety of topics that are meant to introduce some of the other branches of analysis and to illustrate the uses of the preceding material. I believe these topics are all interesting and important, but their selection in preference to others is largely a matter of personal predilection.

The things one needs to know in order to read this book are as follows:

1. First and foremost, the classical theory of functions of a real variable: limits and continuity, differentiation and (Riemann) integration, infinite series, uniform convergence, and the notion of a metric space.

2. The arithmetic of complex numbers and the basic properties of the complex exponential function ex+iy = ex(cosy + isiny). (More advanced results from complex function theory are used only in the proof of the Riesz-Thorin theorem and in a few exercises and remarks.)

3. Some elementary set theory.

4. A bit of linear algebra — actually, not much beyond the definitions of vector spaces, linear mappings, and determinants.

All of the necessary material in (1) and (2) can be found in W. Rudin’s classic Principles of Mathematical Analysis (3rd ed., McGraw-Hill, 1976) or its descendants such as R. S. Strichatrz’s The Way of Analysis (Jones and Bartlett, 1995) or S. G. Krantz’s Real Analysis and Foundations (CRC Press, 1991). A summary of the relevant facts about sets and metric spaces is provided here in Chapter 0. The reader should begin this book by examining §0.1 and §0.5 to become familiar with my notation and terminology; the rest of Chapter 0 can then be referred to as needed.

Each chapter concludes with a section entitled “Notes and References.” These sections contain miscellaneous remarks, acknowledgments of sources, indications of results not discussed in the text, references for further reading, and historical notes. The latter are quite sketchy, although references to more detailed sources are provided; they are intended mainly to give an idea of how the subject grew out of its classical origins. I found it entertaining and instructive to read some of the original papers, and I hope to encourage others to do the same.

A sizable portion of this book is devoted to exercises. They are mostly in the form of assertions to be proved, and they range from trivial to difficult; hints and intermediate steps are provided for the more complicated ones. Every reader should peruse them, although only the most ambitious will try to work them all out. They serve several purposes: amplification of results and completion of proofs in the text, discussion of examples and counterexamples, applications of theorems, and development of further ideas. Instructors will probably wish to do some of the exercises in class; to maximize flexibility and minimize verbosity, I have followed the principle of “When in doubt, leave it as an exercise,” especially with regard to examples. Exercises occur at the end of each section, but they are numbered consecutively within each chapter. In referring to them, “Exercise n” means the nth exercise in the present chapter unless another section is explicitly mentioned.

The topics in the book are arranged so as to allow some flexibility of presentation. For example, Chapters 4 and 5 do not depend on Chapters 1–3 except for a few examples and exercises. On the other hand, if one wishes to proceed quickly to Lp theory, one can skip from §3.3 to §§5.1–2 and thence to Chapter 6. Chapters 10 and 11 are independent of Chapters 8 and 9 except that the ideas in §8.6 are used in Chapter 10.

The new features of this edition are as follows:

The material on the n-dimensional Lebesgue integral (§§2.6–7) has been rearranged and expanded.

Tychonoff’s theorem (§4.6) is proved by an elegant argument recently discovered by Paul Chernoff.

The chapter on Fourier analysis has been split into two chapters (8 and 9). The material on Fourier series and integrals (§§8.3–5) has been rearranged and now contains the Dirichlet-Jordan theorem on convergence of Fourier series. The material on distributions (§§9.1–2) has been extensively rewritten and expanded.

A section on self-similarity and Hausdorff dimension (§11.3) has been added, replacing the outdated calculation of the Hausdorff dimension of Cantor sets in the old §10.2.

Innumerable small changes have been made in the hope of improving the exposition.

The writer of a text on such a well-developed subject as real analysis must necessarily be indebted to his predecessors. I kept a large supply of books on hand while writing this one; they are too numerous to list here, but most of them can be found in the bibliography. I am also happy to acknowledge the influence of two of my teachers: the late Lynn Loomis, from whose lectures I first learned this subject, and Elias Stein, who has done much to shape my point of view. Finally, I am grateful to a number of people — especially Steven Krantz, Kenneth Ross, and William Faris — whose comments and corrigenda concerning the first edition have helped me to prepare the new one.

GERALD B. FOLLAND

Seattle, Washington

Chapter 0

Prologue

The purpose of this introductory chapter is to establish the notation and terminology that will be used throughout the book and to present a few diverse results from set theory and analysis that will be needed later. The style here is deliberately terse, since this chapter is intended as a reference rather than a systematic exposition.

0.1 THE LANGUAGE OF SET THEORY

It is assumed that the reader is familiar with the basic concepts of set theory; the following discussion is meant mainly to fix our terminology.

Number Systems. Our notation for the fundamental number systems is as follows:

= the set of positive integers (not including zero)

= the set of integers

= the set of rational numbers

= the set of real numbers

= the set of complex numbers

Logic. We shall avoid the use of special symbols from mathematical logic, preferring to remain reasonably close to standard English, We shall, however, use the abbreviation iff for “if and only if.”

One point of elementary logic that is often insufficiently appreciated by students is the following: If A and B are mathematical assertions and −A, −B are their negations, the statement “A implies B” is logically equivalent to the contrapositive statement “−B implies −A.” Thus one may prove that A implies B by assuming -B and deducing -A, and we shall frequently do so. This is not the same as reductio ad absurdum, which consists of assuming both A and −B and deriving a contradiction.

Sets. The words “family” and “collection” will be used synonymously with “set,” usually to avoid phrases like “set of sets.” The empty set is denoted by Ø, and the family of all subsets of a set X is denoted by (X):

Here and elsewhere, the inclusion sign ⊂ is interpreted in the weak sense; that is, the assertion “E ⊂ X” includes the possibility that E = X.

If is a family of sets, we can form the union and intersection of its members:

Usually it is more convenient to consider indexed families of sets:

in which case the union and intersection are denoted by

If Eα ∩ Eβ = Ø whenever α ≠ β, the sets Eα are called disjoint. The terms “disjoint collection of sets” and “collection of disjoint sets” are used interchangeably, as are “disjoint union of sets” and “union of disjoint sets.”

When considering families of sets indexed by , our usual notation will be

and likewise for unions and intersections. In this situation, the notions of limit superior and limit inferior are sometimes useful:

The reader may verify that

If E and F are sets, we denote their difference by E \ F:

and their symmetric difference by EΔF:

When it is clearly understood that all sets in question are subsets of a fixed set X, we define the complement Ec of a set E (in X):

In this situation we have deMorgan’s laws:

If X and Y are sets, their Cartesian product X x Y is the set of all ordered pairs (x, y) such that x X and y Y. A relation from X to Y is a subset of X x Y. (If Y = X, we speak of a relation on X.) If R is a relation from X to Y, we shall sometimes write xRy to mean that (x, y) R. The most important types of relations are the following:

Equivalence relations. An equivalence relation on X is a relation R on X such that

The equivalence class of an element x is {y X : xRy}. X is the disjoint union of these equivalence classes.

Orderings. See §0.2.

Mappings. A mapping f : X → Y is a relation R from X to Y with the property that for every x X there is a unique y Y such that xRy, in which case we write y = f(x). Mappings are sometimes called maps or functions; we shall generally reserve the latter name for the case when Y is or some subset thereof.

If f : X → Y and g : Y → Z are mappings, we denote by g o f their composition:

If D ⊂ X and E ⊂ Y, we define the image of D and the inverse image of E under a mapping f : X → Y by

It is easily verified that the map f⁻¹ : (Y) → (X) defined by the second formula commutes with union, intersections, and complements:

(The direct image mapping f : (X) → (Y) commutes with unions, but in general not with intersections or complements.)

If f : X → Y is a mapping, X is called the domain of f and f(X) is called the range of f. f is said to be injective if f(x₁) = f(x₂) only when x₁ = x₂, surjective if f(X) = Y, and bijective if it is both injective and surjective. If f is bijective, it has an inverse f⁻¹ : Y → X such that f⁻¹ o f and f o f⁻¹ are the identity mappings on X and Y, respectively. If A ⊂ X, we denote by f|A the restriction of f to A:

A sequence in a set X is a mapping from into X. (We also use the term finite sequence to mean a map from {1,…, n} into X where n .) If f : → X is a sequence and g : → satisfies g(n) < g(m) whenever n < m, the composition f o g is called a subsequence of f. It is common, and often convenient, to be careless about distinguishing between sequences and their ranges, which are subsets of X indexed by . Thus, if f(n) = xn, we speak of the sequence {xn}₁∞; whether we mean a mapping from to X or a subset of X will be clear from the context.

Earlier we defined the Cartesian product of two sets. Similarly one can define the Cartesian product of n sets in terms of ordered n-tuples. However, this definition becomes awkward for infinite families of sets, so the following approach is used instead. If {Xα}αA is an indexed family of sets, their Cartesian product ΠαA Xα is the set of all maps f : A → ∪αA Xα such that f(α) Xα for every α A. (It should be noted, and then promptly forgotten, that when A = {1, 2}, the previous definition of X₁ x X₂ is set-theoretically different from the present definition of Π²₁ Xj. Indeed, the latter concept depends on mappings, which are defined in terms of the former one.) If X = ΠαA Xα and α A, we define the αth projection or coordinate map πα : X → Xα by πα(f) = f(α). We also frequently write x and xα instead of f and f(α) and call xα the αth coordinate of x.

If the sets Xα are all equal to some fixed set Y, we denote ΠαA Xα by YA:

If A = {1,…, n}, YA is denoted by Yn and may be identified with the set of ordered n-tuples of elements of Y.

0.2 ORDERINGS

A partial ordering on a nonempty set X is a relation R on X with the following properties:

if xRy and yRz, then xRz;

if xRy and yRx, then x = y;

xRx for all x.

If R also satisfies

if x, y X, then either xRy or yRx,

then R is called a linear (or total) ordering. For example, if E is any set, then (E) is partially ordered by inclusion, and is linearly ordered by its usual ordering. Taking this last example as a model, we shall usually denote partial orderings by ≤, and we write x < y to mean that x ≤ y but x ≠ y. We observe that a partial ordering on X naturally induces a partial ordering on every nonempty subset of X. Two partially ordered sets X and Y are said to be order isomorphic if there is a bijection f : X → Y such that x₁ ≤ x₂ iff f(x₁) ≤ f(x₂).

If X is partially ordered by ≤, a maximal (resp. minimal) element of X is an element x X such that the only y X satisfying x ≤ y (resp. x ≥ y) is x itself. Maximal and minimal elements may or may not exist, and they need not be unique unless the ordering is linear. If E ⊂ X, an upper (resp. lower) bound for E is an element x X such that y ≤ x (resp. x ≤ y) for all y E. An upper bound for E need not be an element of E, and unless E is linearly ordered, a maximal element of E need not be an upper bound for E. (The reader should think up some examples.)

If X is linearly ordered by ≤ and every nonempty subset of X has a (necessarily unique) minimal element, X is said to be well ordered by ≤, and (in defiance of the laws of grammar) ≤ is called a well ordering on X. For example, is well ordered by its natural ordering.

We now state a fundamental principle of set theory and derive some consequences of it.

0.1 The Hausdorff Maximal Principle. Every partially ordered set has a maximal linearly ordered subset.

In more detail, this means that if X is partially ordered by ≤, there is a set E ⊂ X that is linearly ordered by ≤, such that no subset of X that properly includes E is linearly ordered by ≤. Another version of this principle is the following:

0.2 Zorn’s Lemma. If X is a partially ordered set and every linearly ordered subset of X has an upper bound, then X has a maximal element.

Clearly the Hausdorff maximal principle implies Zorn’s lemma: An upper bound for a maximal linearly ordered subset of X is a maximal element of X. It is also not difficult to see that Zorn’s lemma implies the Hausdorff maximal principle. (Apply Zorn’s lemma to the collection of linearly ordered subsets of X, which is partially ordered by inclusion.)

0.3 The Well Ordering Principle. Every nonempty set X can be well ordered.

Proof. Let be the collection of well orderings of subsets of X, and define a partial ordering on as follows. If ≤₁ and ≤₂ are well orderings on the subsets E₁ and E₂, then ≤₁ precedes ≤₂ in the partial ordering if (i) ≤₂ extends ≤₁, i.e., E₁ ⊂ E₂ and ≤₁ and ≤₂ agree on E₁ and (ii) if x E₂ \ E₁ then y ≤₂ x for all y E₁. The reader may verify that the hypotheses of Zorn’s lemma are satisfied, so that has a maximal element. This must be a well ordering on X itself, for if ≤ is a well ordering on a proper subset E of X and x₀ X \ E, then ≤ can be extended to a well ordering on E ∪ {x₀} by declaring that x ≤ x₀ for all x E.

0.4 The Axiom of Choice. If {Xα}αA is a nonempty collection of nonempty sets, then ΠαA Xα is nonempty.

Proof. Let X = ∪αA Xα. Pick a well ordering on X and, for α A, let f(α) be the minimal element of Xα. Then f ΠαA Xα.

0.5 Corollary. If {Xα}αA is a disjoint collection of nonempty sets, there is a set Y ⊂ ∪αA Xα such that Y ∩ Xα contains precisely one element for each α A.

Proof. Take Y = f(A) where f ΠαA Xα.

We have deduced the axiom of choice from the Hausdorff maximal principle; in fact, it can be shown that the two are logically equivalent.

0.3 CARDINALITY

If X and Y are nonempty sets, we define the expressions

to mean that there exists f : X → Y which is injective, bijective, or surjective, respectively. We also define

to mean that there is an injection but no bijection, or a surjection but no bijection, from X to Y. Observe that we attach no meaning to the expression “card(X)” when it stands alone; there are various ways of doing so, but they are irrelevant for our purposes (except when X is finite—see below). These relationships can be extended to the empty set by declaring that

For the remainder of this section we assume implicitly that all sets in question are nonempty in order to avoid special arguments for Ø. Our first task is to prove that the relationships defined above enjoy the properties that the notation suggests.

0.6 Proposition. card(X) ≤ card(Y) iff card(Y) ≥ card(X).

Proof. If f : X → Y is injective, pick x₀ X and define g : Y → X by g(y) = f⁻¹(y) if y f(X), g(y) = x₀ otherwise. Then g is surjective. Conversely, if g : Y → X is surjective, the sets g⁻¹({x}) (x X) are nonempty and disjoint, so any f ΠxX g⁻¹ ({x}) is an injection from X to Y.

0.7 Proposition. For any sets X and Y, either card(X) ≤ card(Y) or card(Y) ≤ card(X).

Proof. Consider the set of all injections from subsets of X to Y. The members of can be regarded as subsets of X x Y, so is partially ordered by inclusion. It is easily verified that Zorn’s lemma applies, so has a maximal element f, with (say) domain A and range B. If x₀ X \ A and y₀ Y \ B, then f can be extended to an injection from A ∪ {x₀} to Y ∪ {y₀} by setting f(x₀) = y₀, contradicting maximality. Hence either A = X, in which case card(X) ≤ card(Y), or B = Y, in which case f⁻¹ is an injection from Y to X and card(Y) ≤ card(X).

0.8 The Schröder-Bernstein Theorem. If card(X) ≤ card(Y) and card(Y) ≤ card(X) then card(X) = card(Y).

Proof. Let f : X → Y and g : Y → X be injections. Consider a point x X: If x g(Y), we form g⁻¹(x) Y; if g⁻¹(x) f(X), we form f⁻¹(g⁻¹(x)); and so forth. Either this process can be continued indefinitely, or it terminates with an element of X \ g(Y) (perhaps x itself), or it terminates with an element of Y \ f(X). In these three cases we say that x is in X∞, XX, or XY; thus X is the disjoint union of X∞, XX, and XY. In the same way, Y is the disjoint union of three sets Y∞, YX, and YY. Clearly f maps X∞ onto Y∞ and XX onto YX, whereas g maps YY onto XY. Therefore, if we define h : X → Y by h(x) = f(x) if X X∞ ∪ XX and h(x) = g⁻¹ (x) if x XY, then h is bijective.

0.9 Proposition. For any set X, card(X) < card((X)).

Proof. On the one hand, the map f(x) = {x} is an injection from X to (X). On the other, if g : X → (X), let Y = {x X : x g(x)}. Then Y g(X), for if Y = g(x₀) for some x₀ X, any attempt to answer the question “Is x₀ Y?” quickly leads to an absurdity. Hence g cannot be surjective.

A set X is called countable (or denumerable) if card(X) ≤ card(). In particular, all finite sets are countable, and for these it is convenient to interpret “card(X)” as the number of elements in X:

If X is countable but not finite, we say that X is countably infinite.

0.10 Proposition.

a. If X and Y are countable, so is X x Y.

b. If A is countable and Xα is countable for every α A, then ∪αA Xα is countable.

c. If X is countably infinite, then card(X) = card().

Proof. To prove (a) it suffices to prove that ² is countable. But we can define a bijection from to ² by listing, for n successively equal to 2, 3, 4,…, those elements (j, k) ² such that j + k = n in order of increasing j, thus:

As for (b), for each α A there is a surjective fα : → Xα, and then the map f : x A → ∪αA Xα defined by f(n, α) = fα(n) is surjective; the result therefore follows from (a). Finally, for (c) it suffices to assume that X is an infinite subset of . Let f(1) be the smallest element of X, and define f(n) inductively to be the smallest element of E\ {f(1),…, f(n − 1)}. Then f is easily seen to be a bijection from to X.

0.11 Corollary. and are countable.

Proof. is the union of the countable sets , {−n : n }, and {0}, and one can define a surjection f : ² → by f(m, n) = m/n if n ≠ 0 and f(m, 0) = 0.

A set X is said to have the cardinality of the continuum if card(X) = card(). We shall use the letter as an abbreviation for card():

0.12 Proposition. card(()) = .

Proof. If A ⊂ , define f(A) to be ∑nA 2−n if \ A is infinite and 1 + ∑nA 2−n if \ A is finite. (In the two cases, f(A) is the number whose base-2 decimal expansion is 0.a₁a₂ · · · or 1.a₁a₂ · · ·, where an = 1 if n A and an = 0 otherwise.) Then f : () → is injective. On the other hand, define g : () → by g(A) = log(∑nA 2−n) if A is bounded below and g(A) = 0 otherwise. Then g is surjective since every positive real number has a base-2 decimal expansion. Since card(()) = card(()), the result follows from the Schröder-Bernstein theorem.

0.13 Corollary. If card(X) ≥ , then X is uncountable.

Proof. Apply Proposition 0.9.

The converse of this corollary is the so-called continuum hypothesis, whose validity is one of the famous undecidable problems of set theory; see §0.7.

0.14 Proposition.

a. If card(X) ≤ and card(Y) ≤ , then card(X x Y) ≤ .

b. If card(A) ≤ and card(Xα) ≤ for all α A, then card(∪αA Xα) ≤ .

Proof. For (a) it suffices to take X = Y = (). Define ϕ, : → by ϕ(n) = 2n and (n) = 2n − 1. It is then easy to check that the map f : ()² → () defined by f(A, B) = ϕ(A) ∪ (B) is bijective. (b) follows from (a) as in the proof of Proposition 0.10.

0.4 MORE ABOUT WELL ORDERED SETS

The material in this section is optional; it is used only in a few exercises and in some notes at the ends of chapters.

Let X be a well ordered set. If A ⊂ X is nonempty, A has a minimal element, which is its maximal lower bound or infimum; we shall denote it by inf A. If A is bounded above, it also has a minimal upper bound or supremum, denoted by sup A. If x X, we define the initial segment of x to be

The elements of Ix are called predecessors of x.

The principle of mathematical induction is equivalent to the fact that is well ordered. It can be extended to arbitrary well ordered sets as follows:

0.15 The Principle of Transfinite Induction. Let X be a well ordered set. If A is a subset of X such that x A whenever Ix ⊂ A, then A = X.

0.16 Proposition. If X is well ordered and A ⊂ X, then ∪xA Ix is either an initial segment or X itself.

Proof. Let J = ∪xA Ix. If J ≠ X, let b = inf(X \ J). If there existed y J with y > b, we would have y Ix for some x A and hence b Ix, contrary to construction. Hence J ⊂ Ib, and it is obvious that Ib ⊂ J.

0.17 Proposition. If X and Y are well ordered, then either X is order isomorphic to Y, or X is order isomorphic to an initial segment in Y, or Y is order isomorphic to an initial segment in X.

Proof. Consider the set of order isomorphisms whose domains are initial segments in X or X itself and whose ranges are initial segments in Y or Y itself. is nonempty since the unique f : {inf X} → {inf Y} belongs to , and is partially ordered by inclusion (its members being regarded as subsets of X x Y). An application of Zorn’s lemma shows that has a maximal element f, with (say) domain A and range B. If A = Ix and B = Iy, then A ∪ {x} and B ∪ {y} are again initial segments of X and Y, and f could be extended by setting f(x) = y, contradicting maximality. Hence either A = X or B = Y (or both), and the result follows.

0.18 Proposition. There is an uncountable well ordered set Ω such that Ix is countable for each x Ω. If Ω’ is another set with the same properties, then Ω and Ω’ are order isomorphic.

Proof. Uncountable well ordered sets exist by the well ordering principle; let X be one. Either X has the desired property or there is a minimal element x₀ such that Ix₀ is uncountable, in which case we can take Ω = Ix₀. If Ω’ is another such set, Ω’ cannot be order isomorphic to an initial segment of Ω or vice versa, because Ω and Ω’ are uncountable while their initial segments are countable, so Ω and Ω’ are order isomorphic by Proposition 0.17.

The set Ω in Proposition 0.18, which is essentially unique qua well ordered set, is called the set of countable ordinals. It has the following remarkable property:

0.19 Proposition. Every countable subset of Ω has an upper bound.

Proof. If A ⊂ Ω is countable, ∪xA Ix is countable and hence is not all of Ω. By Proposition 0.16, there exists y Ω such that ∪xA Ix = Iy, and y is thus an upper bound for A.

The set of positive integers may be identified with a subset of Ω as follows. Set f(1) = inf Ω, and proceeding inductively, set f(n) = inf(Ω\{f(1),…, f(n-1)}). The reader may verify that f is an order isomorphism from to Iω, where ω is the minimal element of Ω such that Iω is infinite.

It is sometimes convenient to add an extra element ω₁ to Ω to form a set Ω* = Ω ∪ {ω₁} and to extend the ordering on Ω to Ω* by declaring that x < ω₁ for all x Ω. ω₁ is called the first uncountable ordinal. (The usual notation for ω₁ is Ω, since ω₁ is generally taken to be the set of countable ordinals itself.)

0.5 THE EXTENDED REAL NUMBER SYSTEM

It is frequently useful to adjoin two extra points ∞ (= +∞) and −∞ to to form the extended real number system , and to extend the usual ordering on by declaring that −∞ < x < ∞ for all x . The completeness of can then be stated as follows: Every subset A of has a least upper bound, or supremum, and a greatest lower bound, or infimum, which are denoted by sup A and inf A. If A = {a₁,… an}, we also write

From completeness it follows that every sequence {xn} in has a limit superior and a limit inferior:

The sequence {xn} converges (in ) iff these two numbers are equal (and finite), in which case its limit is their common value. One can also define lim sup and lim inf for functions f : → , for instance:

The arithmetical operations on can be partially extended to :

We make no attempt to define ∞ − ∞, but we abide by the convention that, unless otherwise stated,

(The expression 0 · ∞ turns up now and then in measure theory, and for various reasons its proper interpretation is almost always 0.)

We employ the following notation for intervals in : if −∞ ≤ a < b ≤ ∞,

We shall occasionally encounter uncountable sums of nonnegative numbers. If X is an arbitrary set and f : X → [0, ∞], we define ∑xX f(x) to be the supremum of its finite partial sums:

(Later we shall recognize this as the integral of f with respect to counting measure on X.)

0.20 Proposition. Given f : X → [0, ∞], let A = {x : f(x) > 0}. If A is uncountable, then ∑xX f(x) = ∞. If A is countably infinite, then ∑xX f(x) = ∑∞₁ f(g(n)) where g : → A is any bijection and the sum on the right is an ordinary infinite series.

Proof. We have A = ∪∞₁ An where An = {x : f(x) > 1/n}. If A is uncountable, then some An must be uncountable, and ∑xF f(x) > card(F)/n for F a finite subset of An; it follows that ∑xX f(x) = ∞. If A is countably infinite, g : → A is a bijection, and BN = g({1,…, N}), then every finite subset F of A is contained in some BN. Hence

Taking the supremum over N, we find

and then taking the supremum over F, we obtain the desired result.

Some terminology concerning (extended) real-valued functions: A relation between numbers that is applied to functions is understood to hold pointwise. Thus f ≤ g means that f(x) ≤ g(x) for every x, and max(f, g) is the function whose value at x is max(f(x), g(x)). If X ⊂ and f : X → , f is called increasing if f(x) ≤ f(y) whenever x ≤ y and strictly increasing if f(x) < f(y) whenever x < y; similarly for decreasing. A function that is either increasing or decreasing is called monotone.

If f : → is an increasing function, then f has right- and left-hand limits at each point:

Moreover, the limiting values f(∞) = supa f(x) and f(−∞) = infa f(x) exist (possibly equal to ±∞). f is called right continuous if f(a) = f(a+) for all a and left continuous if f(a) = f(a−) for all a .

For points x in or , |x| denotes the ordinary absolute value or modulus of x, . For points x in n or n, |x| denotes the Euclidean norm:

We recall that a set U ⊂ is open if, for every x U, U includes an interval centered at x.

0.21 Proposition. Every open set in is a countable disjoint union of open intervals.

Proof. If U is open, for each x U consider the collection x of all open intervals I such that x I ⊂ U. It is easy to check that the union of any family of open intervals containing a point in common is again an open interval, and hence Jx = ∪Ix I is an open interval; it is the largest element of x. If x, y U then either Jx = Jy or Jx ∩ Jy = Ø, for otherwise Jx ∪ Jy would be a larger open interval than Jx in x. Thus if , the (distinct) members of are disjoint, and U = J. For each J , pick a rational number f(J) J. The map f : → thus defined is injective, for if J ≠ J′ then J ∩ J’ = Ø; therefore is countable.

0.6 METRIC SPACES

A metric on a set X is a function ρ : X x X → [0, ∞) such that

ρ(x, y) = 0 iff x = y;

ρ(x, y) = ρ(y, x) for all x, y X;

ρ(x, z) ≤ ρ(x, y) + ρ(y, z) for all x, y, z X.

(Intuitively, ρ(x, y) is to be interpreted as the distance from x to y.) A set equipped with a metric is called a metric space. Some examples:

i. The Euclidean distance ρ(x, y) = |x − y| is a metric on n.

ii. ρ₁(f, g) = ƒ₀¹ |f(x) − g(x)| dx and ρ∞(f, g) = sup0≤x≤1 |f(x) − g(x)| are metrics on the space of continuous functions on [0, 1].

iii. If ρ is a metric on X and A ⊂ X, then ρ|(A x A) is a metric on A.

iv. If (X₁, ρ₁) and (X₂, ρ₂) are metric spaces, the product metric ρ on X₁ x X₂ is given by

Other metrics are sometimes used on X₁ x X₂, for instance,

These, however, are equivalent to the product metric in the sense that we shall define at the end of this section.

Let (X, ρ) be a metric space. If x X and r > 0, the (open) ball of radius r about x is

A set E ⊂ X is open if for every x E there exists r > 0 such that B(r, x) ⊂ E, and closed if its complement is open. For example, every ball B(r, x) is open, for if y B(r, x) and ρ(x, y) = s then B(r − s, y) ⊂ B(r, x). Also, X and Ø are both open and closed. Clearly the union of any family of open sets is open, and hence the intersection of any family of closed sets is closed. Also, the intersection (resp. union) of any finite family of open (resp. closed) sets is open (resp. closed). Indeed, if U₁,… Un are open and x ∩n₁ Uj, for each j there exists rj > 0 such that B(rj, x) ⊂ Uj, and then B(r, x) ⊂ ∩n₁ Uj where r = min(r₁,…, rn), so ∩n₁ Uj is open.

If E ⊂ X, the union of all open sets U ⊂ E is the largest open set contained in E; it is called the interior of E and is denoted by E°. Likewise, the intersection of all closed sets F ⊃ E is the smallest closed set containing E; it is called the closure of E and is denoted by . E is said to be dense in X if = X, and nowhere dense if has empty interior. X is called separable if it has a countable dense subset. (For example, n is a countable dense subset of n.) A sequence {xn} in X converges to x X (symbolically: xn → x or lim xn = x) if limn→∞ ρ(xn, x) = 0.

0.22 Proposition. If X is a metric space, E ⊂ X, and x X, the following are equivalent:

a. x .

b. B(r, x) ∩ E ≠ Ø for all r > 0.

c. There is a sequence {xn} in E that converges to x.

Proof. If B(r, x) ∩ E = Ø, then B(r, x)c is a closed set containing E but not x, so x . Conversely, if x , since ()c is open there exists r > 0 such that B(r, x) ⊂ ()c ⊂ Ec. Thus (a) is equivalent to (b). If (b) holds, for each n there exists xn B(n⁻¹, x) ∩ E, so that xn → x. On the other hand, if B(r, x) ∩ E = Ø, then ρ(y, x) ≥ r for all y E, so no sequence of E can converge to x. Thus (b) is equivalent to (c).

If (X₁, ρ₁) and (X₂, ρ₂) are metric spaces, a map f : X₁ → X₂ is called continuous at x X if for every > 0 there exists δ > 0 such that ρ₂(f(y), f(x)) < whenver ρ₁(x, y) < δ — in other words, such that f⁻¹ (B(, f(x))) ⊃ B(δ, x). The map f is called continuous if it is continuous at each x X₁ and uniformly continuous if, in addition, the δ in the definition of continuity can be chosen independent of x.

0.23 Proposition. f : X₁ → X₂ is continuous iff f⁻¹(U) is open in X₁ for every open U ⊂ X₂.

Proof. If the latter condition holds, then for every x X₁ and > 0, the set f⁻¹(B(, f(x))) is open and contains x, so it contains some ball about x; this means that f is continuous at x. Conversely, suppose that f is continuous and U is open in X₂. For each y U there exists y > 0 such that B(y, y) ⊂ U, and for each x f⁻¹ ({y}) there exists δx > 0 such that B(δx, x) ⊂ f⁻¹(B(y, y)) ⊂ f⁻¹(U). Thus f⁻¹ (U) = ∪xf−1 (U) B (δx, x) is open.

A sequence {xn} in a metric space (X, ρ) is called Cauchy if ρ(xn, xm) → 0 as n, m → ∞. A subset E of X is called complete if every Cauchy sequence in E converges and its limit is in E. For example, n (with the Euclidean metric) is complete, whereas n is not.

0.24 Proposition. A closed subset of a complete metric space is complete, and a complete subset of an arbitrary metric space is closed.

Proof. If X is complete, E ⊂ X is closed, and {xn} is a Cauchy sequence in E, {xn} has a limit in X. By Proposition 0.22, x = E. If E ⊂ X is complete and x , by Proposition (0.22) there is a sequence {xn} in E converging to x. {xn} is Cauchy, so its limit lies in E; thus E = .

In a metric space (X, ρ) we can define the distance from a point to a set and the distance between two sets. Namely, if x X and E, F ⊂ X,

Observe that, by Proposition 0.22, ρ(x, E) = 0 iff x . We also define the diameter of E ⊂ X to be

E is called bounded if diam E < ∞.

If E ⊂ X and {Vα}αA is a family of sets such that E ⊂ ∪αA Vα, {Vα}αA is called a cover of E, and E is said to be covered by the Vα’s. E is called totally bounded if, for every > 0, E can be covered by finitely many balls of radius . Every totally bounded set is bounded, for if x, y ∪n₁ B(, zj), say x B(, z₁) and y B(, z₂), then

(The converse is false in general.) If E is totally bounded, so is , for it is easily seen that if E ⊂ ∪n₁ B(, zj), then ⊂ ∪n₁ B(2, zj).

0.25 Theorem. If E is a subset of the metric space (X, ρ), the following are equivalent:

a. E is complete and totally bounded.

b. (The Bolzano-Weierstrass Property) Every sequence in E has a subsequence that converges to a point of E.

c. (The Heine-Borel Property) If {Vα}αA is a cover of E by open sets, there is a finite set F ⊂ A such that {Vα}αF covers E.

Proof. We shall show that (a) and (b) are equivalent, that (a) and (b) together imply (c), and finally that (c) implies (b).

(a) implies (b): Suppose that (a) holds and {xn} is a sequence in E. E can be covered by finitely many balls of radius 2⁻¹, and at least one of them must contain xn for infinitely many n: say, xn B₁ for n N₁. E ∩ B₁ can be covered by finitely many balls of radius 2⁻², and at least one of them must contain xn for infinitely many n N₁: say, xn B₂ for n N₂. Continuing inductively, we obtain a sequence of balls Bj of radius 2-j and a decreasing sequence of subsets Nj of such that xn Bj for n Nj. Pick n₁ N₁, n₂ N₂,… such that n₁ < n₂ < · · ·. Then {xnJ} is a Cauchy sequence, for ρ(xnJ, xnk) < 21−j if k > j, and since E is complete, it has a limit in E.

(b) implies (a): We show that if either condition in (a) fails, then so does (b). If E is not complete, there is a Cauchy sequence {xn} in E with no limit in E. No subsequence of {xn} can converge in E, for otherwise the whole squence would converge to the same limit. On the other hand, if E is not totally bounded, let > 0 be such that E cannot be covered by finitely many balls of radius . Choose xn E inductively as follows. Begin with any x₁ E, and having chosen x₁,…, xn, pick xn+1 E \ ∪n₁ B(, xj). Then ρ(xn, xm) > for all n, m, so {xn} has no convergent subsequence.

(a) and (b) imply (c): It suffices to show that if (b) holds and {Vα}αA is a cover of E by open sets, there exists > 0 such that every ball of radius that intersects E is contained in some Vα, for E can be covered by finitely many such balls by (a). Suppose to the contrary that for each n there is a ball Bn of radius 2−n such that Bn ∩ E ≠ Ø and Bn is contained in no Vα. Pick xn Bn ∩ E; by passing to a subsequence we may assume that {xn} converges to some x E. We have x Vα for some α, and since Vα is open, there exists > 0 such that B(, x) ⊂ Vα. But if n is large enough so that ρ(xn, x) < /3 and 2−n < /3, then Bn ⊂ B(, x) ⊂ Vα, contradicting the assumption on Bn.

(c) implies (b): If {xn} is a sequence in E with no convergent subsequence, for each x E there is a ball Bx centered at x that contains xn for only finitely many n (otherwise some subsequence would converge to x). Then {Bx}xE is a cover of E by open sets with no finite subcover.

A set E that possesses the properties (a)–(c) of Theorem 0.25 is called compact. Every compact set is closed (by Proposition 0.24) and bounded; the converse is false in general but true in n.

0.26 Proposition. Every closed and bounded subset of n is compact.

Proof. Since closed subsets of n are complete, it suffices to show that bounded subsets of n are totally bounded. Since every bounded set is contained in some cube

it is enough to show that Q is totally bounded. Given > 0, pick an integer k > R√n/, and express Q as the union of kn congruent subcubes by dividing the interval [−R, R] into k equal pieces. The side length of these subcubes is 2R/k and hence their diameter is < 2, so they are contained in the balls of radius about their centers.

Two metrics ρ₁ and ρ₂ on a set X are called equivalent if

It is easily verified that equivalent metrics define the same open, closed, and compact sets, the same convergent and Cauchy sequences, and the same continuous and uniformly continuous mappings. Consequently, most results concerning metric spaces depend not on the particular metric chosen but only on its equivalence class.

0.7 NOTES AND REFERENCES

§§0.1–0.4: The best exposition of set theory for beginners is Halmos [62], and Smullyan and Fitting [135] is a good text on a more advanced level. Kelley [83] also contains a concise account of of basic axiomatic set theory. All of these books present a deduction of the Hausdorff maximal principle from the axiom of choice, as does Hewitt and Stromberg [76].

The axiom of choice (or one of the propositions equivalent to it) is generally taken as one of the basic postulates in the axiomatic formulations of set theory. Some mathematicians of the intuitionist or constructivist persuasion reject it on the grounds that one has not proved the existence of a mathematical object until one has shown how to construct it in some reasonably explicit fashion, whereas the whole point of the axiom of choice is to provide existence theorems when constructive methods fail (or are too cumbersome for comfort). People who are seriously bothered by such objections belong to a minority that does not include the present writer; in this book the axiom of choice is used sparingly but freely.

The continuum hypothesis is the assertion that if card(X) < , then X is countable. (Since it follows easily from the construction of Ω, the set of countable ordinals, that card(Ω) ≤ card(X) for any uncountable X, an equivalent assertion is that card(Ω) = .) It is known, thanks to Gödel and Cohen, that the continuum hypothesis and its negation are both consistent with the standard axioms of set theory including the axiom of choice, assuming that those axioms are themselves consistent. (An exposition of the consistency and independence theorems for the axiom of choice and the continuum hypothesis can be found in Smullyan and Fitting [135].) Some mathematicians are willing to accept the continuum hypothesis as true, seemingly as a matter of convenience, but Gödel [56] and Cohen [26, p. 151] have both expressed suspicions that it should be false, and as of this writing no one has found any really compelling evidence on one side or the other. My own feeling, subject to revision in the event of a major breakthrough in set theory, is that if the answer to one’s question turns out to depend on the continuum hypothesis, one should give up and ask a different question.

§0.6: A more detailed discussion of metric spaces can be found in Loomis and Sternberg [95] and DePree and Swartz [32].

Chapter 1

Measures

In this chapter we set forth the basic concepts of measure theory, develop a general procedure for constructing nontrivial examples of measures, and apply this procedure to construct measures on the real line.

1.1 INTRODUCTION

One of the most venerable problems in geometry is to determine the area or volume of a region in the plane or in 3-space. The techniques of integral calculus provide a satisfactory solution to this problem for regions that are bounded by “nice” curves or surfaces but are inadequate to handle more complicated sets, even in dimension one. Ideally, for n we would like to have a function μ that assigns to each E ⊂ n a number μ(E) [0, ∞], the n-dimensional measure of E, such that μ(E) is given by the usual integral formulas when the latter apply. Such a function μ should surely possess the following properties:

i. If E₁, E₂,… is a finite or infinite sequence of disjoint sets, then

ii. If E is congruent to F (that is, if E can be transformed into F by translations, rotations, and reflections), then μ(E) = μ(F).

iii. μ(Q) = 1, where Q is the unit cube

Unfortunately, these conditions are mutually inconsistent. Let us see why this is true for n = 1. (The argument can easily be adapted to higher dimensions.) To begin with, we define an equivalence relation on [0, 1) by declaring that x ~ y iff x – y is rational. Let N be a subset of [0, 1) that contains precisely one member of each equivalence class. (To find such an N, one must invoke the axiom of choice.) Next, let R = ∩ [0, 1), and for each r R let

That is, to obtain Nr, shift N to the right by r units and then shift the part that sticks out beyond [0, 1) one unit to the left. Then Nr ⊂ [0, 1), and every x [0, 1) belongs to precisely one Nr. Indeed, if y is the element of N that belongs to the equivalence class of x, then x Nr where r = x – y if x ≥ y or r = x − y + 1 if x < y; on the other hand, if x Nr ∩ Ns, then x − r (or x − r + 1) and x − s (or x − s + 1) would be distinct elements of N belonging to the same equivalence class, which is impossible.

Suppose now that μ : () → [0, ∞] satisfies (i), (ii), and (iii). By (i) and (ii),

for any r R. Also, since R is countable and [0, 1) is the disjoint union of the Nr’s,

by (i) again. But μ([0, 1)) = 1 by (iii), and since μ(Nr) = μ(N), the sum on the right is either 0 (if μ(N) = 0) or ∞ (if μ(N) > 0). Hence no such μ can exist.

Faced with this discouraging situation, one might consider weakening (i) so that additivity is required to hold only for finite sequences. This is not a very good idea, as we shall see: The additivity for countable sequences is what makes all the limit and continuity results of the theory work smoothly. Moreover, in dimensions n ≥ 3, even this weak form of (i) is inconsistent with (ii) and (iii). Indeed, in 1924 Banach and Tarski proved the following amazing result:

Let U and V be arbitrary bounded open sets in n, n ≥ 3. There exist k and subsets E₁,…, Ek, F₁,…, Fk of n such that

– the Ej’s are disjoint and their union is U;

– the Fj’s are disjoint and their union is V;

– Ej is congruent to Fj for j = 1,…, k.

Thus one can cut up a ball the size of a pea into a finite number of pieces and rearrange them to form a ball the size of the earth! Needless to say, the sets Ej and Fj are very bizarre. They cannot be visualized accurately, and their construction depends on the axiom of choice. But their existence clearly precludes the construction of any μ : (n) → [0, ∞] that assigns positive, finite values to bounded open sets and satisfies (i) for finite sequences as well as (ii).

The moral of these examples is that n contains subsets which are so strangely put together that it is impossible to define a geometrically reasonable notion of measure for them, and the remedy for the situation is to discard the requirement that μ should be defined on all subsets of n. Rather, we shall content ourselves with constructing μ on a class of subsets of n that includes all the sets one is likely to meet in practice unless one is deliberately searching for pathological examples. This construction will be carried out for n = 1 in §1.5 and for n > 1 in §2.6.

It is worthwhile, and not much extra work, to develop the theory in much greater generality. The conditions (ii) and (iii) are directly related to Euclidean geometry, but set functions satisfying (i), called measures, arise also in a great many other situations. For example, in a physics problem involving mass distributions, μ(E) could represent the total mass in the region E. For another example, in probability theory one considers a set X that represents the possible outcomes of an experiment, and for E ⊂ X, μ(E) is the probability that the outcome lies in E. We therefore begin by studying the theory of measures on abstract sets.

1.2 σ-ALGEBRAS

In this section we discuss the families of sets that serve as the domains of measures.

Let X be a nonempty set. An algebra of sets on X is a nonempty collection of subsets of X that is closed under finite unions and complements; in other words, if E₁,…, En , then n₁ Ej ; and if E , then Ec . A σ-algebra is an algebra that is closed under countable unions. (Some authors use the terms field and σ-field instead of algebra and σ-algebra.)

We observe that since ∩j Ej = (j Ecj)c, algebras (resp. σ-algebras) are also closed under finite (resp. countable) intersections. Moreover, if is an algebra, then Ø and X , for if E we have Ø = E ∩ Ec and X = E Ec.

It is worth noting that an algebra is a σ-algebra provided that it is closed under countable disjoint unions. Indeed, suppose {Ej}∞₁ ⊂ . Set

Then the Fk’s belong to and are disjoint, and ∞₁ Ej = ∞₁ Fk. This device of replacing a sequence of sets by a disjoint sequence is worth remembering; it will be used a number of times below.

Some examples: If X is any set, (X) and {Ø, X} are σ-algebras. If X is uncountable, then

is a σ-algebra, called the σ-algebra of countable or co-countable sets. (The point here is that if {Ej}∞₁ ⊂ , then ∞₁ Ej is countable if all Ej are countable and is co-countable otherwise.)

It is trivial to verify that the intersection of any family of σ-algebras on X is again a σ-algebra. It follows that if ε is any susbset of (X), there is a unique smallest σ-algebra (ε) containing ε, namely, the intersection of all σ-algebras containing ε. (There is always at least one such, namely, (X).) (ε) is called the σ-algebra generated by ε. The following observation is often useful:

1.1 Lemma. If ε ⊂ () then (ε) ⊂ ().

Proof. () is a σ-algebra containing ε; it therefore contains (ε).

If X is any metric space, or more generally any topological space (see Chapter 4), the σ-algebra generated by the family of open sets in X (or, equivalently, by the family of closed sets in X) is called the Borel σ-algebra on X and is denoted by X. Its members are called Borel sets. X thus includes open sets, closed sets, countable intersections of open sets, countable unions of closed sets, and so forth.

There is a standard terminology for the levels in this hierarchy. A countable intersection of open sets is called a Gδ set; a countable union of closed sets is called an Fσ set; a countable union of Gδ sets is called a Gδσ set; a countable intersection of Fσ sets is called an Fσδ set; and so forth. (δ and σ stand for the German Durchschnitt and Summe, that is, intersection and union.)

The Borel σ-algebra on will play a fundamental role in what follows. For future reference we note that it can be generated in a number of different ways:

1.2 Proposition. is generated by each of the following:

a. the open intervals: ε₁ = {(a, b) : a < b},

b. the closed intervals: ε₂ = {[a, b] : a < b},

c. the half-open intervals: ε₃ = {(a, b] : a < b} or ε₄ = {[a, b) : a < b},

d. the open rays: ε₅ = {(a, ∞) : a } or ε₆ = {(−∞, a) : a },

e. the closed rays: ε₇ = {[a, ∞) : a } or ε₈ = {(−∞, a] : a }.

Proof. The elements of εj for j ≠ 3, 4 are open or closed, and the elements of ε₃ and ε₄ are Gδ sets — for example, . All of these are Borel sets, so by Lemma 1.1, for all j. On the other hand, every open set in is a countable union of open intervals, so by Lemma 1.1 again, . That for j ≥ 2 can now be established by showing that all open intervals lie in (εj) and applying Lemma 1.1. For example, . Verification of the other cases is left to the reader (Exercise 2).

Let be an indexed collection of nonempty sets, , and πα : X → Xα the coordinate maps. If α is a σ-algebra on Xα for each α, the product σ-algebra on X is the σ-algebra generated by

We denote this σ-algebra by we also write The significance of this definition will become clearer in §2.1; for the moment we give an alternative, and perhaps more intuitive, characterization of product σ-algebras in the case of countably many factors.

1.3 Proposition. If A is countable, then is the σ-algebra generated by .

Proof. If Eα α, then where Eβ = Xβ for β ≠ α; on the other hand, . The result therefore follows from Lemma 1.1.

1.4 Proposition. Suppose that α is generated by A. Then is generated by . If A is countable and Xα εα for all α, is generated by e.

Proof. Obviously . On the other hand, for each α, the collection is easily seen to be a σ-algebra on Xα that contains εα and hence α. In other words, for all E α, α A, and hence . The second assertion follows from the first as in the proof of Proposition 1.3.

1.5 Proposition. Let X₁,…, Xn be metric spaces and let X = Πn₁ Xj, equipped with the product metric. Then . If the Xj’s are separable, then .

Proof. By Proposition 1.4, is generated by the sets π⁻¹j (Uj), 1 ≤ j ≤ n, where Uj is open in Xj. Since these sets are open in X, Lemma 1.1 implies that . Suppose now that Cj is a countable dense set in Xj, and let εj be the collection of balls in Xj with rational radius and center in Cj. Then every open set in Xj is a union of members of εj — in fact, a countable union since εj itself is countable. Moreover, the set of points in X whose jth coordinate is in Cj for all j is a countable dense subset of X, and the balls of radius r in X are merely products of balls of radius r in the Xj’s. It follows that XJ is generated by . Therefore , by Proposition 1.4.

1.6 Corollary. .

We conclude this section with a technical result that will be needed later. We define an elementary family to be a collection ε of subsets of X such that

Ø ε,

if E, F ε then E ∩ F ε,

if E ε then Ec is a finite disjoint union of members of ε.

1.7 Proposition. If ε is an elementary family, the collection of finite disjoint unions of members of ε is an algebra.

Proof. If A, B ε and Bc = J₁ Cj (Cj ε, disjoint), then A \ B = J₁ (A ∩ Cj) and A B = (A \ B) B, where these unions are disjoint, so A \ B and A B . It now follows by induction that if A₁…, An ε, then n₁ Aj ; indeed, by inductive hypothesis we may assume that A₁,…, An−1 are disjoint, and then , which is a disjoint union. To see that is closed under complements, suppose A₁,… An ε and with B¹m,…, BJmm disjoint members of ε. Then

which is in .

Exercises

1. A family of sets ⊂ (X) is called a ring if it is closed under finite unions and differences (i.e., if E₁,…, En , then n₁ Ej , and if E, F , then E \ F ). A ring that is closed under countable unions is called a σ-ring.

a. Rings (resp. σ-rings) are closed under finite (resp. countable) intersections.

b. If is a ring (resp. σ-ring), then is an algebra (resp. σ-algebra) iff X .

c. If is a σ-ring, then {E ⊂ X : E or Ec } is a σ-algebra.

d. If is a σ-ring, then {E ⊂ X : E ∩ F for all F } is a σ-algebra.

2. Complete the proof of Proposition 1.2.

3. Let be an infinite σ-algebra.

a. contains an infinite sequence of disjoint sets.

b. card() ≥ .

4. An algebra is a σ-algebra iff is closed under countable increasing unions (i.e., if and , then .

5. If is the σ-algebra generated by ε, then is the union of the σ-algebras generated by as ranges over all countable subsets of ε. (Hint: Show that the latter object is a σ-algebra.)

1.3 MEASURES

Let X be a set equipped with a σ-algebra . A measure on (or on (X, ), or simply on X if is understood) is a function μ: → [0, ∞] such that

i. μ(Ø) = 0,

ii. if is a sequence of disjoint sets in , then .

Property (ii) is called countable additivity. It implies finite additivity:

ii′. if E₁,… En are disjoint sets in , then ,

because one can take Ej = Ø for j > n. A function μ that satisfies (i) and (ii’) but not necessarily (ii) is called a finitely additive measure.

If X is a set and ⊂ (X) is a σ-algebra, (X, ) is called a measurable space and the sets in are called measurable sets. If μ is a measure on (X, ), then (X, , μ) is called a measure space.

Let (X, , μ) be a measure space. Here is some standard terminology concerning the “size” of μ. If μ(X) < ∞ (which implies that μ(E) < ∞ for all E since μ(X) = μ(E) + μ(Ec)), μ is called finite. If X = ∞₁ Ej where Ej and μ(Ej) < ∞ for all j, μ is called σ-finite. More generally, if E = ∞₁ Ej where Ej and μ(Ej) < ∞ for all j, the set E is said to be σ-finite for μ. (It would be correct but more cumbersome to say that E is of σ-finite measure.) If for each E with μ(E) = ∞ there exists F with F ⊂ E and 0 < μ(F) < ∞, μ is called semifinite.

Every σ-finite measure is semifinite (Exercise 13), but not conversely. Most measures that arise in practice are σ-finite, which is fortunate since non-σ-finite measures tend to exhibit pathological behavior. The properties of non-σ-finite measures will be explored from time to time in the exercises.

Let us examine a few examples of measures. These examples are of a rather trivial nature, although the first one is of practical importance. The construction of more interesting examples is a task to which we shall turn in the next two sections.

Let X be any nonempty set, = (X), and f any function from X to [0, ∞]. Then f determines a measure μ on by the formula μ(E) = ∑xE f(x). (For the definition of such possibly uncountable sums, see §0.5.) The reader may verify that μ is semifinite iff f(x) < ∞ for every x X, and μ is σ-finite iff μ is semifinite and {x : f(x) > 0} is countable. Two special cases are of particular significance: If f(x) = 1 for all x, μ is called counting measure; and if, for some x₀ X, f is defined by f(x₀) = 1 and f(x) = 0 for x ≠ x₀, μ is called the point mass or Dirac measure at x₀. (The same names are also applied to the restrictions of these measures to smaller σ-algebras on X.)

Let X be an uncountable set, and let be the σ-algebra of countable or co-countable sets. The function μ on defined by μ(E) = 0 if E is countable and μ(E) = 1 if E is co-countable is easily seen to be a measure.

Let X be an infinite set and = (X). Define μ(E) = 0 if E is finite, μ(E) = ∞ if E is infinite. Then μ is a finitely additive measure but not a measure.

The basic properties of measures are summarized in the following theorem.

1.8 Theorem. Let (X, , μ) be a measure space.

a. (Monotonicity) If E, F and E ⊂ F, then μ(E) ≤ μ(F).

b. (Subadditivity) If , then .

c. (Continuity from below) If and E₁ ⊂ E₂ ⊂ · · ·, then .

d. (Continuity from above) If , and μ(E₁) < ∞, then .

Proof. (a) If E ⊂ F, then μ(F) = μ(E) + μ(F \ E) ≥ μ(E).

(b) Let F₁ = E₁ and Fk = Ek for k > 1. Then the Fk’s are disjoint and for all n. Therefore, by (a),

(c) Setting E₀ = Ø, we have

(d) Let Fj = E₁ \ Ej; then F₁ ⊂ F₂ ⊂ · · ·, μ(E₁) = μ(Fj) + μ(Ej), and ∞₁ Fj = E₁ \ (∩∞₁ Ej). By (c), then,

Since μ(E₁) < ∞, we may subtract it from both sides to yield the desired result.

We remark that the condition μ(E₁) < ∞ in part (d) could be replaced by μ(En) < ∞ for some n > 1, as the first n − 1 Ej’s can be discarded from the sequence without affecting the intersection. However, some finiteness assumption is necessary, as it can happen that μ(Ej) = ∞ for all . (For example, let μ be counting measure on and let ; then

If (X, , μ) is a measure space, a set E such that μ(E) = 0 is called a null set. By subadditivity, any countable union of null sets is a null set, a fact which we shall use frequently. If a statement about points x X is true except for x in some null set, we say that it is true almost everywhere (abbreviated a.e.), or for almost every x. (If more precision is needed, we shall speak of a μ-null set, or μ-almost everywhere).

If μ(E) = 0 and F ⊂ E, then μ(F) = 0 by monotonicity provided that F , but in general it need not be true that F . A measure whose domain includes all subsets of null sets is called complete. Completeness can sometimes obviate annoying technical points, and it can always be achieved by enlarging the domain of μ, as follows.

1.9 Theorem. Suppose that (X, , μ) is a measure space. Let = {N : μ(N) = 0} and and F ⊂ N for some N }. Then is a σ-algebra, and there is a unique extension of μ to a complete measure on .

Proof. Since and are closed under countable unions, so is . If E F where E and F ⊂ N , we can assume that E ∩ N = Ø (otherwise, replace F and N by F \ E and N \ E). Then , so . But and N \ F ⊂ N, so that . Thus is a σ-algebra.

If E F as above, we set . This is well defined, since if E₁ F₁ = E₂ F₂ where Fj ⊂ Nj , then E₁ ⊂ E₂ N₂ and so μ(E₁) ≤ μ(E₂) + μ(N₂) = μ(E₂), and likewise μ(E₂) ≤ μ(E₁). It is easily verified that is a complete measure on , and that is the only measure on that extends μ; details are left to the reader (Exercise 6).

The measure μ in Theorem 1.9 is called the completion of μ, and is called the completion of with respect to μ.

Exercises

6. Complete the proof of Theorem 1.9.

7. If μ₁,…, μn are measures on (X, ) and a₁,…, an [0, ∞), then is a measure on (X, ).

8. If (X, , μ) is a measure space and , then μ(lim inf Ej) ≤ lim inf μ(Ej). Also, μ(lim sup Ej) ≥ lim sup μ(Ej) provided that .

9. If (X, , μ) is a measure space and E, F , then .

10. Given a measure space (X, , μ) and E , define for A . Then μE is a measure.

11. A finitely additive measure μ is a measure iff it is continuous from below as in Theorem 1.8c. If μ(X) < ∞, μ is a measure iff it is continuous from above as in Theorem 1.8d.

12. Let (X, , μ) be a finite measure space.

a. If E, F and μ(E Δ F) = 0, then μ(E) = μ(F).

b. Say that E ~ F if μ(E Δ F) = 0; then ~ is an equivalence relation on .

c. For E, F , define ρ(E, F) = μ(E Δ F). Then ρ(E, G) ≤ ρ(E, F) + ρ(F, G), and hence ρ defines a metric on the space / ~ of equivalence classes.

13. Every σ-finite measure is semifinite.

14. If μ is a semifinite measure and μ(E) = ∞, for any C > 0 there exists F ⊂ E with C < μ(F) < ∞.

15. Given a measure μ on (X, ), define μ₀ on by μ₀(E) = sup{μ(F) : F ⊂ E and μ(F) < ∞}.

a. μ₀ is a semifinite measure. It is called the semifinite part of μ.

b. If μ is semifinite, then μ = μ₀. (Use Exercise 14.)

c. There is a measure v on (in general, not unique) which assumes only the values 0 and ∞ such that μ = μ₀ + v.

16. Let (X, , μ) be a measure space. A set E ⊂ X is called locally measurable if E ∩ A for all A such that μ(A) < ∞. Let be the collection of all locally measurable sets. Clearly ⊂ ; if = , then μ is called saturated.

a. If μ is σ-finite, then μ is saturated.

b. is a σ-algebra.

c. Define on by (E) = μ(E) if E and (E) = ∞ otherwise. Then is a saturated measure on , called the saturation of μ.

d. If μ is complete, so is .

e. Suppose that μ is semifinite. For E , define μ(E) = sup{μ(A) : A and A ⊂ E}. Then μ is a saturated measure on that extends μ.

f. Let X₁, X₂ be disjoint uncountable sets, X = X₁ X₂, and the σ-algebra of countable or co-countable sets in X. Let μ₀ be counting measure on (X₁), and define μ on by μ(E) = μ₀(E ∩ X₁). Then μ is a measure on , = (X), and in the notation of parts (c) and (e), ≠ μ.

1.4 OUTER MEASURES

In this section we develop the tools we shall use to construct measures. To motivate the ideas, it may be useful to recall the procedure used in calculus to define the area of a bounded region E in the plane ². One draws a grid of rectangles in the plane and approximates the area of E from below by the sum of the areas of the rectangles in the grid that are subsets of E, and from above by the sum of the areas of the rectangles in the grid that intersect E. The limits of these approximations as the grid is taken finer and finer give the “inner area” and “outer area” of E, and if they are equal, their common value is the “area” of E. (We shall discuss these matters in more detail in §2.6.) The key idea here is that of outer area, since if R is a large rectangle containing E, the inner area of E is just the area of R minus the outer area of R \ E.

The abstract generalization of the notion of outer area is as follows. An outer measure on a nonempty set X is a function μ* : (X) → [0, ∞] that satisfies

μ* (Ø) = 0,

μ*(A) ≤ μ*(B) if A ⊂ B,

μ*(∞₁ Aj) ≤ ∑∞₁ μ* (Aj).

The most common way to obtain outer measures is to start with a family ε of “elementary sets” on which a notion of measure is defined (such as rectangles in the plane) and then to approximate arbitrary sets “from the outside” by countable unions of members of ε. The precise construction is as follows.

1.10 Proposition. Let ε ⊂ (X) and ρ : ε → [0, ∞] be such that Ø ε X ε, and ρ(Ø) = 0. For any A ⊂ X, define

Then μ* is an outer measure.

Proof. For any A ⊂ X there exists {Ej}∞₁ ⊂ ε such that A (take Ej = X for all j) so the definition of μ* makes sense. Obviously μ*(Ø) = 0 (take Ej = Ø for all j), and μ* (A) ≤ μ* (B) for A ⊂ B because the set over which the infimum is taken in the definition of μ* (A) includes the corresponding set in the definition of μ*(B). To prove the countable subadditivity, suppose and ε > 0. For each j there exists such that and . But then if A = ∞₁ Aj, we have and , whence μ* (A) ≤ ∑j μ*(Aj) + ε. Since ε is arbitrary, we are done.

The fundamental step that leads from outer measures to measures is as follows. If μ* is an outer measure on X, a set A ⊂ X is called μ*-measurable if

Of course, the inequality holds for any A and E, so to prove that A is μ*-measurable, it suffices to prove the reverse inequality. The latter is trivial if μ*(E) = ∞, so we see that A is μ*-measurable iff

Some motivation for the notion of μ*-measurability can be obtained by referring to the discussion at the beginning of this section. If E is a “well-behaved” set such that E ⊃ A, the equation says that the outer measure of A, μ*(A), is equal to the “inner measure” of A, μ* (E) − μ*(E ∩ Ac). The leap from “well-behaved” sets containing A to arbitrary subsets of X is a large one, but it is justified by the following theorem.

1.11 Carathéodory’s Theorem. If μ* is an outer measure on X, the collection of μ*-measurable sets is a σ-algebra, and the restriction of μ* to is a complete measure.

Proof. First, we observe that is closed under complements since the definition of μ*-measurability of A is symmetric in A and Ac. Next, if A, B and E ⊂ X,

and hence

It follows that A B , so is an algebra. Moreover, if A, B and A ∩ B = Ø,

so μ* is finitely additive on .

To show that is a σ-algebra, it will suffice to show that is closed under countable disjoint unions. If {Aj}∞₁ is a sequence of disjoint sets in , let and . Then for any E ⊂ X,

so a simple induction shows that . Therefore,

and letting n → ∞ we obtain

All the inequalities in this last calculation are thus equalities. It follows that , so μ* is countably additive on . Finally, if μ*(A) = 0, for any E ⊂ X we have

so that A . Therefore μ* | is a complete measure.

Our first applications of Carathéodory’s theorem will be in the context of extending measures from algebras to σ-algebras. More precisely, if ⊂ (X) is an algebra, a function μ₀ : → [0, ∞] will be called a premeasure if

μ₀(Ø) = 0,

if {Aj}∞₁ is a sequence of disjoint sets in such that ∞₁ Aj ε then .

In particular, a premeasure is finitely additive since one can take Aj = Ø for j large. The notions of finite and σ-finite premeasures are defined just as for measures. If μ₀ is a premeasure on ⊂ (X), it induces an outer measure on X in accordance with Proposition 1.10, namely,

(1.12) equation

1.13 Proposition. If μ₀ is a premeasure on and μ* is defined by (1.12), then

a. μ*| = μ₀;

b. every set in is μ*-measurable.

Proof. (a) Suppose E . If E ⊂ ∞₁ Aj with Aj , let Bn = E ∩ . Then the Bn’s are disjoint members of whose union is E, so . It follows that μ₀(E) ≤ μ*(E), and the reverse inequality is obvious since where A₁ = E and Aj = Ø for j > 1.

(b) If A , E ⊂ X, and ε > 0, there is a sequence with and . Since μ₀ is additive on ,

Since ε is arbitrary, A is μ*-measurable.

1.14 Theorem. Let ⊂ (X) be an algebra, μ₀ a premeasure on , and the σ-algebra generated by . There exists a measure μ on whose restriction to is μ₀ — namely, μ = μ*| where μ* is given by (1.12). If v is another measure on that extends μ₀, then v(E) ≤ μ(E) for all E , with equality when μ(E) < ∞. If μ₀ is σ-finite, then μ is the unique extension of μ₀ to a measure on .

Proof. The first assertion follows from Carathéodory’s theorem and Proposition 1.13 since the σ-algebra of μ*-measurable sets includes and hence . As for the second assertion, if E and where Aj , then v(E) ≤ . whence v(E) ≤ μ(E). Also, if we set A = , we have

If μ(E) < ∞, we can choose the Aj’s so that μ(A) < μ(E) + ε, hence μ(A\E) < ε, and

Since ε is arbitrary, μ(E) = v(E). Finally, suppose with μ₀(Aj) < ∞, where we can assume that the Aj’s are disjoint. Then for any E ,

so v = μ.

The proof of this theorem yields more than the statement. Indeed, μ₀ may be extended to a measure on the algebra * of all μ*-measurable sets. The relation between and * is explored in Exercise 22 (along with Exercise 20b, which ensures that the outer measures induced by μ₀ and μ are the same).

Exercises

17. If μ* is an outer measure on X and is a sequence of disjoint μ*-measurable sets, then for any E ⊂ X.

18. Let ⊂ (X) be an algebra, σ the collection of countable unions of sets in , and σδ the collection of countable intersections of sets in σ. Let μ₀ be a premeasure on and μ* the induced outer measure.

a. For any E ⊂ X and ε > 0 there exists A σ with E ⊂ A and μ*(A) ≤ μ*(E) + ε.

b. If μ*(E) < ∞, then E is μ*-measurable iff there exists B σδ with E ⊂ B and μ*(B \ E) = 0.

c. If μ₀ is σ-finite, the restriction μ*(E) < ∞ in (b) is superfluous.

19. Let μ* be an outer measure on X induced from a finite premeasure μ₀. If E ⊂ X, define the inner measure of E to be μ*(E) = μ₀(X) − μ*(Ec). Then E is μ*-measurable iff μ*(E) = μ*(E). (Use Exercise 18.)

20. Let μ* be an outer measure on X, * the σ-algebra of μ*-measurable sets, = μ*|*, and μ⁺ the outer measure induced by as in (1.12) (with and * replacing μ₀ and ).

a. If E ⊂ X, we have μ*(E) ≤ μ⁺(E), with equality iff there exists A * with A ⊃ E and μ*(A) = μ*(E).

b. If μ* is induced from a premeasure, then μ* = μ⁺. (Use Exercise 18a.)

c. If X = {0, 1}, there exists an outer measure μ* on X such that μ* ≠ μ⁺.

21. Let μ* be an outer measure induced from a premeasure and the restriction of μ* to the μ*-measurable sets. Then is saturated. (Use Exercise 18.)

22. Let (X, , μ) be a measure space, μ* the outer measure induced by μ according to (1.12), * the σ-algebra of μ*-measurable sets, and .

a. If μ is σ-finite, then is the completion of μ. (Use Exercise 18.)

b. In general, is the saturation of the completion of μ. (See Exercises 16 and 21.)

23. Let be the collection of finite unions of sets of the form where -∞ ≤ a < b ≤ ∞.

a. is an algebra on . (Use Proposition 1.7.)

b. The σ-algebra generated by is ().

c. Define μ₀ on by μ₀(Ø) = 0 and μ₀(A) = ∞ for A ≠ Ø. Then μ₀ is a premeasure on , and there is more than one measure on () whose restriction to is μ₀.

24. Let μ be a finite measure on (X, ), and let μ* be the outer measure induced by μ. Suppose that E ⊂ X satisfies μ*(E) = μ*(X) (but not that E ).

a. If A, B and A ∩ E = B ∩ E, then μ(A) = μ(B).

b. Let , and define the function v on E defined by (which makes sense by (a)). Then E is a σ-algebra on E and v is a measure on E.

1.5 BOREL MEASURES ON THE REAL LINE

We are now in a position to construct a definitive theory for measuring subsets of based on the idea that the measure of an interval is its length. We begin with a more general (but only slightly more complicated) construction that yields a large family of measures on whose domain is the Borel σ-algebra ; such measures are called Borel measures on .

To motivate the ideas, suppose that μ is a finite Borel measure on , and let F(x) = μ((−∞, x]). (F is sometimes called the distribution function of μ.) Then F is increasing by Theorem 1.8a and right continuous by Theorem 1.8d since whenever xn x. (Recall the discussion of increasing functions in §0.5.) Moreover, if . Our procedure will be to turn this process around and construct a measure μ starting from an increasing, right-continuous function F. The special case F(x) = x will yield the usual “length” measure.

The building blocks for our theory will be the left-open, right-closed intervals in — that is, sets of the form (a, b] or (a, ∞) or Ø, where -∞ ≤ a < b < ∞. In this section we shall refer to such sets as h-intervals (h for “half-open”). Clearly the intersection of two h-intervals is an h-interval, and the complement of an h-interval is an h-interval or the disjoint union of two h-intervals. By Proposition 1.7, the collection of finite disjoint unions of h-intervals is an algebra, and by Proposition 1.2, the σ-algebra generated by is .

1.15 Proposition. Let F : → be increasing and right continuous. If (aj, bj] (j = 1,…, n) are disjoint h-intervals, let

and let μ₀(Ø) = 0. Then μ₀ is a premeasure on the algebra .

Proof. First we must check that μ₀ is well defined, since elements of can be represented in more than one way as disjoint unions of h-intervals. If {(aj, bj]}n₁ are disjoint and , then, after perhaps relabeling the index j, we must have a = a₁ < b₁ = a₂ < b₂ = … < bn = b, so . More generally, if and are finite sequences of disjoint h-intervals such that , this reasoning shows that

Thus μ₀ is well defined, and it is finitely additive by construction.

It remains to show that if {Ij}∞₁ is a sequence of disjoint h-intervals with then . Since is a finite union of h-intervals, the sequence {Ij}∞₁ can be partitioned into finitely many subsequences such that the union of the intervals in each subsequence is a single h-interval. By considering each subsequence separately and using the finite additivity of μ₀, we may assume that is an h-interval I = (a, b]. In this case, we have

Letting n → ∞, we obtain . To prove the reverse inequality, let us suppose first that a and b are finite, and let us fix ε > 0. Since F is right continuous, there exists δ > 0 such that F(a + δ) − F(a) < ε, and if Ij = (aj, bj], for each j there exists δj > 0 such that F(bj + δj) − F(bj) < ε2−j. The open intervals (aj, bj + δj) cover the compact set [a + δ, b], so there is a finite subcover. By discarding any (aj, bj + δj) that is contained in a larger one and relabeling the index j, we may assume that

the intervals (a₁, b₁ + δ₁),…, (aN, bN + δN) cover [a + δ, b],

bj + δj (aj+1, bj+1 + δj+1) for j = 1,…, N − 1.

But then

Since ε is arbitrary, we are done when a and b are finite. If a = -∞, for any M < ∞ the intervals (aj, bj + δj) cover [−M, b], so the same reasoning gives , whereas if b = ∞, for any M < ∞ we likewise obtain . The desired result then follows by letting ε → 0 and M → ∞.

1.16 Theorem. If F : → is any increasing, right continuous function, there is a unique Borel measure μF on such that μF((a, b]) = F(b) — F(a) for all a, b. If G is another such function, we have μF = μG iff F − G is constant. Conversely, if μ, is a Borel measure on that is finite on all bounded Borel sets and we define

then F is increasing and right continuous, and μ = μF.

Proof. Each F induces a premeasure on by Proposition 1.15. It is clear that F and G induce the same premeasure iff F − G is constant, and that these premeasures are σ-finite (since . The first two assertions therefore follow from Theorem 1.14. As for the last one, the monotonicity of μ, implies the monotonicity of F, and the continuity of μ from above and below implies the right continuity of F for x ≥ 0 and x < 0. It is evident that μ = μF on , and hence μ = μF on by the uniqueness in Theorem 1.14.

Several remarks are in order. First, this theory could equally well be developed by using intervals of the form [a, b) and left continuous functions F. Second, if μ is a finite Borel measure on , then μ = μF where F(x) = μ((−∞, x]) is the cumulative distribution function of μ; this differs from the F specified in Theorem 1.16 by the constant μ((−∞, 0]). Third, the theory of §1.4 gives, for each increasing and right continuous F, not only the Borel measure μF but a complete measure F whose domain includes . In fact, F is just the completion of μF (Exercise 22a or Theorem 1.19 below), and one can show that its domain is always strictly larger than . We shall usually denote this complete measure also by μF; it is called the Lebesgue-Stieltjes measure associated to F.

Lebesgue-Stieltjes measures enjoy some useful regularity properties that we now investigate. In this discussion we fix a complete Lebesgue-Stieltjes measure μ on associated to the increasing, right continuous function F, and we denote by μ the domain of μ. Thus, for any E μ,

We first observe that in the second formula for μ(E) we can replace h-intervals by open h-intervals:

1.17 Lemma. For any E μ,

Proof. Let us call the quantity on the right v(E). Suppose . Each (aj, bj) is a countable disjoint union of h-intervals specifically, where {cj} is any sequence such that c¹j = aj and ckj increases to bj as k → ∞. Thus , so

and hence v(E) ≥ μ(E), On the other hand, given ε > 0 there exists with and and for each j there exists δj > 0 such that F(bj + δj) − F(bj) < ε2−j. Then and

so that v(E) ≥ μ(E).

1.18 Theorem. If E μ, then

Proof. By Lemma 1.17, for any ε > 0 there exist intervals (aj, bj) such that and then U is open, U ⊃ E, and μ(U) ≤ μ(E) + ε. On the other hand, μ(U) ≥ μ(E) whenever U ⊃ E, so the first equality is valid. For the second one, suppose first that E is bounded. If E is closed, then E is compact and the equality is obvious. Otherwise, given ε > 0 we can choose an open such that . Let K = Then K is compact, K ⊂ E, and

If E is unbounded, let Ej = E ∩ (j, j + 1]. By the preceding argument, for any ε > 0 there exist compact Kj ⊂ Ej with . Let Hn = . Then Hn is compact, Hn ⊂ E, and . Since , the result follows.

1.19 Theorem. If E ⊂ , the following are equivalent.

a. E μ.

b. E = V \ N₁ where V is a Gδ set and μ(N₁) = 0.

c. E = H N₂ where H is an Fσ set and μ(N₂) = 0.

Proof. Obviously (b) and (c) each imply (a) since μ is complete on μ. Suppose E μ and μ(E) < ∞. By Theorem 1.18, for j we can choose an open Uj ⊃ E and a compact Kj ⊂ E such that

Let and H = . Then H ⊂ E ⊂ V and μ(V) = μ(H) = μ(E) < ∞, so μ(V \ E) = μ(E \ H) = 0. The result is thus proved when μ(E) < ∞; the extension to the general case is left to the reader (Exercise 25).

The significance of Theorem 1.19 is that all Borel sets (or, more generally, all sets in μ) are of a reasonably simple form modulo sets of measure zero. This contrasts markedly with the machinations necessary to construct the Borel sets from the open sets when null sets are not excepted; see Proposition 1.23 below. Another version of the idea that general measurable sets can be approximated by “simple” sets is contained in the following proposition, whose proof is left to the reader (Exercise 26):

1.20 Proposition. If E μ and μ(E) < ∞, then for every ε > 0 there is a set A that is a finite union of open intervals such that μ(E Δ A) < ε.

We now examine the most important measure on , namely, Lebesgue measure: This is the complete measure μF associated to the function F(x) = x, for which the measure of an interval is simply its length. We shall denote it by m. The domain of m is called the class of Lebesgue measurable sets, and we shall denote it by . We shall also refer to the restriction of m to as Lebesgue measure.

Among the most significant properties of Lebesgue measure are its invariance under translations and simple behavior under dilations. If E ⊂ and s, r , we define

1.21 Theorem. If E , then E + s and r E for all s, r . Moreover, m(E + s)= m(E) and m(rE) = |r|m(E).

Proof. Since the collection of open intervals is invariant under translations and dilations, the same is true of . For E , let ms(E) = m(E + s) and mr(E) = m(rE). Then ms and mr clearly agree with m and |r|m on finite unions of intervals, hence on by Theorem 1.14. In particular, if E and m(E) = 0, then m(E + s) = m(rE) = 0, from which it follows that the class of sets of Lebesgue measure zero is preserved by translations and dilations. It follows that (the members of which are a union of a Borel set and a Lebesgue null set) is preserved by translation and dilations and that m(E + s) = m(E) and m(rE) = |r|m(E) for all E .

The relation between the measure-theoretic and topological properties of subsets of is delicate and contains some surprises. Consider the following facts. Every singleton set in has Lebesgue measure zero, and hence so does every countable In particular, m() = 0. Let {rj}∞₁ be an enumeration of the rational numbers in [0, 1], and given ε > 0, let Ij be the open interval centered at rj of length ε2−j. Then the set is open and dense in [0, 1], but its complement K = [0, 1] \ U is closed and nowhere dense, but m(K) ≥ 1 − ε. Thus a set that is open and dense, and hence topologically “large,” can be measure-theoretically small, and a set that is nowhere dense, and hence topologically “small,” can be measure-theoretically large. (A nonempty open set cannot have Lebesgue measure zero, however.)

The Lebesgue null sets include not only all countable sets but many sets having the cardinality of the continuum. We now present the standard example, the Cantor set, which is also of interest for other reasons.

Each x [0, 1] has a base-3 decimal expansion where aj = 0, 1, or 2. This expansion is unique unless x is of the form p3−k for some integers p, k, in which case x has two expansions: one with aj = 0 for j > k and one with aj = 2 for j > k. Assuming p is not divisible by 3, one of these expansions will have ak = 1 and the other will have ak = 0 or 2. If we agree always to use the latter expansion, we see that

and so forth. It will also be useful to observe that if and , then x < y iff there exists an n such that an < bn and aj = bj for j < n.

The Cantor set C is the set of all x [0, 1] that have a base-3 expansion with aj ≠ 1 for all j. Thus C is obtained from [0, 1] by removing the open middle third , then removing the open middle thirds and of the two remaining intervals, and so forth. The basic properties of C are summarized as follows:

1.22 Proposition. Let C be the Cantor set.

a. C is compact, nowhere dense, and totally disconnected (i.e., the only connected subsets of C are single points). Moreover, C has no isolated points.

b. m(C) = 0.

c. card(C) = .

Proof. We leave the proof of (a) to the reader (Exercise 27). As for (b), C is obtained from [0, 1] by removing one interval of length , two intervals of length , and so forth. Thus

Lastly, suppose x C, so that where aj = 0 or 2 for all j. Let where bj = aj/2. The series defining f(x) is the base-2 expansion of a number in [0, 1], and any number in [0, 1] can be obtained in this way. Hence f maps C onto [0, 1], and (c) follows.

Let us examine the map f in the preceding proof more closely. One readily sees that if x, y C and x < y, then f(x) < f(y) unless x and y are the two endpoints of one of the intervals removed from [0, 1] to obtain C. In this case f(x) = p2−k for some integers p, k, and f(x) and f(y) are the two base-2 expansions of this number. We can therefore extend f to a map from [0, 1] to itself by declaring it to be constant on each interval missing from C. This extended f is still increasing, and since its range is all of [0, 1] it cannot have any jump discontinuities; hence it is continuous. f is called the Cantor function or Cantor-Lebesgue function.

The construction of the Cantor set by starting with [0, 1] and successively removing open middle thirds of intervals has an obvious generalization. If I is a bounded interval and α (0, 1), let us call the open interval with the same midpoint as I and length equal to α times the length of I the “open middle αth” of I. If {αj}∞₁ is any sequence of numbers in (0, 1), then, we can define a decreasing sequence {Kj} of closed sets as follows: K₀ = [0, 1], and Kj is obtained by removing the open middle αjth from each of the intervals that make up Kj−1. The resulting limiting set K = is called a generalized Cantor set. Generalized Cantor sets all share with the ordinary Cantor set the properties (a) and (c) in Proposition 1.22. As for their Lebesgue measure, clearly m(Kj) = (1 − αj)m(Kj−1), so m(K) is the infinite product . If the αj are all equal to a fixed α (0, 1) (for example, α = for the ordinary Cantor set), we have m(K) = 0. However, if αj → 0 sufficiently rapidly as j → ∞, m(K) will be positive, and for any β (0, 1) one can choose αj so that m(K) will equal β; see Exercise 32. This gives another way of constructing nowhere dense sets of positive measure.

Not every Lebesgue measurable set is a Borel set. One can display examples of sets in by using the Cantor function; see Exercise 9 in Chapter 2. Alternatively, one can observe that since every-subset of the Cantor set is Lebesgue measurable, we have card() = card, whereas card() = . The latter fact follows from Proposition 1.23 below.

Exercises

25. Complete the proof of Theorem 1.19.

26. Prove Proposition 1.20. (Use Theorem 1.18.)

27. Prove Proposition 1.22a. (Show that if x, y C and x < y, there exists z C such that x < z < y.)

28. Let F be increasing and right continuous, and let μF be the associated measure. Then μF({a}) = F(a) − F(a−), μF([a, b)) = F(b−) − F(a−), μF([a, b) = F(b) − F(a−), and μF((a, b)) = F(b−) − F(a).

29. Let E be a Lebesgue measurable set.

a. If E ⊂ N where N is the nonmeasurable set described in §1.1, then m(E) =0.

b. If m(E) > 0, then E contains a nonmeasurable set. (It suffices to assume E ⊂ [0, 1]. In the notation of §1.1,

30. If E and m(E) > 0, for any α < 1 there is an open interval I such that m(E ∩ I) > αm(I).

31. If E and m(E) > 0, the set E − E = {x − y : x, y E} contains an interval centered at 0. (If I is as in Exercise 30 with α > , then E − E contains (−m(I), 1/2m(I)).)

32. Suppose .

a. 0 iff . (Compare

b. Given β (0, 1), exhibit a sequence {αj} such that .

33. There exists a Borel set A ⊂ [0, 1] such that 0 < m(A ∩ I) < m(I) for every subinterval I of [0, 1]. (Hint: Every subinterval of [0, 1] contains Cantor-type sets of positive measure.)

1.6 NOTES AND REFERENCES

The history of measure theory is intimately connected with the history of integration theory, comments on which will be made in §2.7.

§1.1: The Banach-Tarski paradox appeared first in [11], but the following variant goes back to Hausdorff [68]:

The unit sphere in ³, {x ³ : |x| = 1}, is the disjoint union of four sets E₁, …, E₄ such that (a) E₁ is countable and (b) the sets E₂, E₃, E₄, and E₃ E₄ are all images of each other under rotations.

An elementary exposition of the Banach-Tarski paradox and Hausdorff’s result can be found in Stromberg [146].

§1.2: Our characterization of the σ-algebra (ε) generated by a family ε ⊂ (X) is nonconstructive, and one might ask how to obtain (ε) explicitly from ε. The answer is rather complicated. One can begin as follows: Let ε₁ = ε {Ec : E ε}, and for j > 1 define εj to be the collection of all sets that are countable unions of sets in εj−1 or complements of such. Let εω = ∞₁ εj: is εω = (ε)? In general, no. εω is closed under complements, but if Ej εj \ εj−1 for each j, there is no reason for ∞₁ Ej to be in εω. So one must start all over again. More precisely, one must define εα for every countable ordinal α by transfinite induction: If α has an immediate predecesor β, εα is the collection of sets that are countable unions of sets in εβ or complements of such; otherwise, . Then:

1.23 Proposition. , where Ω is the set of countable ordinals.

Proof. Transfinite induction shows that for all , and hence . The reverse inclusion follows from the fact that any sequence in Ω has a supremum in Ω, (Proposition 0.19): If Ej εαj for j and α = sup{αj}, then Ej εα for all j and hence where β is the successor of α.

Combining this with Proposition 0.14, we see that if card() ≤ card(ε) ≤ , then card((ε)) = . (Cf. Exercise 3.)

§1.3: Some authors prefer to take the domains of measures to be σ-rings rather than σ-algebras (see Exercise 1). The reason is that in dealing with “very large” spaces one can avoid certain pathologies by not attempting to measure “very large” sets. However, this point of view also has technical disadvantages, and it is no longer much in favor.

§1.4: Carathéodory’s theorem appears in his treatise [22]. Theorem 1.14 has been attributed in the literature to Hahn, Carathéodory, and E. Hopf, but it is originally due to Fréchet [54]. The proof via Carathéodory’s theorem was discovered independently by Hahn [60] and Kolmogorov [85].

See König [86] for a deeper study of the problem of constructing measures from more primitive data.

§1.5: Lebesgue originally defined the outer measure m*(E) of a set E ⊂ in terms of countable coverings by intervals, as we have done. He then defined a bounded set E to be measurable if m*(E) + m*((a, b) \ E) = b − a, where (a, b) is an interval containing E, and an unbounded set to be measurable if its intersection with any bounded interval is measurable. Carathéodory’s characterization of measurability, which is technically eaiser to work with, came later. For the equivalence of the two definitions, see Exercise 19.

One should convince oneself that the remarkably fussy proof of Proposition 1.15 is necessary by contemplating the complicated ways in which an h-interval can be decomposed into a disjoint union of h-subintervals. In any such decomposition the collection of right endpoints of the subintervals, when ordered from right to left, is a well ordered set, but it can be order isomorphic to any initial segment of the set of countable ordinals.

Lebesgue measure can be extended to a translation-invariant measure on σ-algebras that properly include ; see Kakutani and Oxtoby [81]. Of course, such σ-algebras can never contain the nonmeasurable set discussed in §1. However, Lebesgue measure can be extended to a translation-invariant finitely additive measure on (), and its 2-dimensional analogue (see §2.6) can be extended to a finitely additive measure on (²) that is invariant under translations and rotations; see Banach [8]. The Banach-Tarski paradox prevents this result from being extended to higher dimensions.

In connection with the existence of nonmeasurable sets, Solovay [138] has proved a remarkable theorem which says in effect that it is impossible to prove the existence of Lebesgue nonmeasurable sets without using the axiom of choice. (The precise statement of the theorem involves some technical points of axiomatic set theory, which we shall not discuss here.) From the point of view of the working analyst, the effect of Solovay’s theorem is to reaffirm the adequacy of the Lebesgue theory for all practical purposes.

See Rudin [124] for a terse solution of Exercise 33.

Chapter 2

Integration

In the classical theory of integration on , ∫ba f(x) dx is defined as a limit of Riemann sums, which are integrals of functions that approximate f and are constant on subintervals of [a, b]. Similarly, on any measure space there is an obvious notion of integral for functions that are, in a suitable sense, locally constant, and it can be extended to an integral for more general functions. In this chapter, we develop the theory of integration on abstract measure spaces, paying particular attention to the Lebesgue integral on and its generalization to n.

2.1 MEASURABLE FUNCTIONS

We begin our study of integration theory with a discussion of measurable mappings, which are the morphisms in the category of measurable spaces.

We recall that any mapping f : X → Y between two sets induces a mapping f⁻¹ : (Y) → (X), defined by f⁻¹(E) = {x X : f(x) E}, which preserves unions, intersections, and complements. Thus, if is a σ-algebra on Y, {f⁻¹(E) : E } is a σ-algebra on X. If (X, ) and (Y, ) are measurable spaces, a mapping f : X → Y is called (, )-measurable, or just measurable when and are understood, if f⁻¹(E) for all E

It is obvious that the composition of measurable mappings is measurable; that is, if f : X → Y is (, )-measurable and g : Y → Z is (, )-measurable, then g → f is (, )-measurable.

2.1 Proposition. If is generated by ε, then f : X → Y is (, )-measurable iff f⁻¹(E) for all E ε.

Proof. The “only if” implication is trivial. For the converse, observe that {E ⊂ Y : f⁻¹(E) } is a σ-algebra that contains ε; it therefore contains .

2.2 Corollary. If X and Y are metric (or topological) spaces, every continuous f : X → Y is (X, Y)-measurable.

Proof, f is continuous iff f⁻¹ (U) is open in X for every open U ⊂ Y.

If (X, ) is a measurable space, a real- or complex-valued function f on X will be called -measurable, or just measurable, if it is (, ) or (, ) measurable. or is always understood as the σ-algebra on the range space unless otherwise specified. In particular, f : → is Lebesgue (resp. Borel) measurable if it is (resp. .

Warning: If f, g : → are Lebesgue measurable, it does not follow that f g is Lebesgue measurable, even if g is assumed continuous. (If E we have f⁻¹(E) , but unless f⁻¹(E) there is no guarantee that g⁻¹ (f⁻¹(E)) will be in . See Exercise 9.) However, if f is Borel measurable, then f g is Lebesgue or Borel measurable whenever g is.

2.3 Proposition. If (X, ) is a measurable space and f : X → , the following are equivalent:

a. f is -measurable.

b. f⁻¹ ((a, ∞)) for all a .

c. f⁻¹ ([a, ∞)) for all a .

d. f⁻¹ ((−∞, a)) for all a .

e. f⁻¹ ((−∞, a]) for all a .

Proof. This follows from Propositions 1.2 and 2.1.

Sometimes we wish to consider measurability on subsets of X. If (X, ) is a measurable space, f is a function on X, and E , we say that f is measurable on E if f⁻¹ (B) ∩ E for all Borel sets B. (Equivalently, f|E is E-measurable, where E = {F ∩ E : F }.)

Given a set X, if {(Yα, α)}αA is a family of measurable spaces, and fα : X → Yα is a map for each α A, there is a unique smallest σ-algebra on X with respect to which the fα’s are all measurable, namely, the σ-algebra generated by the sets f⁻¹α(Eα) with Eα α and α A. It is called the σ-algebra generated by {fα}αA. In particular, if X = ΠαA Yα, we see that the product σ-algebra on X, as defined in §1.2, is the σ-algebra generated by the coordinate maps πα : X → Yα.

2.4 Proposition. Let (X, ) and (Yα, α) (α A) be measurable spaces, Y = , and πα : Y → Yα the coordinate maps. Then f : X → Y is (, )-measurable iff fα = πα f is (, α)-measurable for all α.

Proof. If f is measurable, so is each fα since the composition of measurable maps is measurable. Conversely, if each fα is measurable, then for all Eα α, f⁻¹(π⁻¹α (Eα)) = f⁻¹α (Eα) , whence f is measurable by Proposition 2.1.

2.5 Corollary. A function f : X → is -measurable iff Re f and Im f are -measurable.

Proof. This follows since by Proposition 1.5.

It is sometimes convenient to consider functions with values in the extended real number system = [∞, ∞]. We define Borel sets in by : E ∩ }. (This coincides with the usual definition of the Borel σ-algebra if we make into a metric space with metric ρ(x, y) = |A(x) − A(y)|, where A(x) = arctan x.) It is easily verified as in Proposition 2.3 that is generated by the rays (a, ∞] or [−∞, a) (a ), and we define f : X → to be -measurable if it is ()-measurable. See Exercise 1.

We now establish that measurability is preserved under the familiar algebraic and limiting operations.

2.6 Proposition. If f, g : X → are -measurable, then so are f + g and fg.

Proof. Define F : , and : by F(x) = (f(x), g(x)), ϕ(z, w) = z + w, (z, w) = zw. Since by Proposition 1.5, F is -measurable by Proposition 2.4, whereas ϕ and are (x, )-measurable by Corollary 2.2. Thus f + g = ϕ F and f g = F are -measurable.

Proposition 2.6 remains valid for -valued functions provided one takes a little care with the indeterminate expressions ∞ − ∞ and 0 · ∞. (Recall, however, that by convention we always define 0 · ∞ to be 0.) See Exercise 2.

2.7 Proposition. If {fj} is a sequence of -valued measurable functions on (X, ), then the functions

are all measurable. If f(x) = limj→∞ fj(x) exists for every x X, then f is measurable.

Proof. We have

so g₁ and g₂ are measurable by Proposition 2.3. More generally, if hk(x) = supj>k fj(x) then hk is measurable for each k, so g₃ = infk hk is measurable, and likewise for g₄. Finally, if f exists then f = g₃ = g₄, so f is measurable.

2.8 Corollary. If f, g : X → are measurable, then so are max(f, g) and min(f, g).

2.9 Corollary. If {fj} is a sequence of complex-valued measurable functions and f(x) = limj→∞ fj(x) exists for all x, then f is measurable.

Proof. Apply Corollary 2.5.

For future reference we present two useful decompositions of functions. First, if f : X → , we define the positive and negative parts of f to be

Then f = f⁺ − f⁻. If f is measurable, so are f⁺ and f⁻, by Corollary 2.8. Second, if f : X → , we have its polar decomposition:

Again, if f is measurable, so are |f| and sgn f. Indeed, z → |z| is continuous on , and z → sgn z is continuous except at the origin. If U ⊂ is open, sgn⁻¹ (U) is either open or of the form V ∪ {0} where V is open, so sgn is Borel measurable. Therefore |f| = |·| f and sgn f = sgn f are measurable.

We now discuss the functions that are the building blocks for the theory of integration. Suppose that (X, ) is a measurable space. If E ⊂ X, the characteristic function χE of E (sometimes called the indicator function of E and denoted by 1E) is defined by

It is easily checked that χE is measurable iff E . A simple function on X is a finite linear combination, with complex coefficients, of characteristic functions of sets in . (We do not allow simple functions to assume the values ±∞.) Equivalently, f : X → is simple iff f is measurable and the range of f is a finite subset of . Indeed, we have

We call this the standard representation of f. It exhibits f as a linear combination, with distinct coefficients, of characteristic functions of disjoint sets whose union is X. Note: One of the coefficients zj may well be 0, but the term zjχEj is still to be envisioned as part of the standard representation, as the set Ej may have a role to play when f interacts with other functions.

It is clear that if f and g are simple functions, then so are f + g and fg. We now show that arbitrary measurable functions can be approximated in a nice way by simple functions.

2.10 Theorem. Let (X, ) be a measurable space.

a. If f : X → [0, ∞] is measurable, there is a sequence {ϕn} of simple functions such that 0 ≤ ϕ₁ ≤ ϕ₂ ≤ · · · ≤ f, ϕn → f pointwise, and ϕn → f uniformly on any set on which f is bounded.

b. If f : X → is measurable, there is a sequence {ϕn} of simple functions such that 0 ≤ |ϕ₁| ≤ |ϕ₂| ≤ · · · ≤ |f|, ϕn → f pointwise, and ϕn → f uniformly on any set on which f is bounded.

Proof. (a) For n = 0, 1, 2,… and 0 ≤ k ≤ 22n − 1, let

and define

(This formula is messy in print but easily understood graphically; see Figure 2.1.) It is easily checked that ϕn ≤ ϕn+1 for all n, and 0 ≤ f − ϕn ≤ 2−n on the set where f ≤ 2n. The result therefore follows.

Fig. 2.1 The functions ϕ₀ (left) and ϕ₁ (right) in the proof of Theorem 2.10a.

(b) If f = g + ih, we can apply part (a) to the positive and negative parts of g and h, obtaining sequences ⁺n, ⁻n, ζ⁺n, ζ⁻n of nonnegative simple functions that increase to g⁺, g⁻, h⁺, h⁻. Let ϕn = ⁺n − ⁻n + i(ζ⁺n − ζ⁻n); it is then a simple exercise to verify that ϕn has the desired properties.

If μ is a measure on (X, ), we may wish to except μ-null sets from consideration in studying measurable functions. In this respect, life is a bit simpler if μ is complete.

2.11 Proposition. The following implications are valid iff the measure μ is complete:

a. If f is measurable and f = g μ-a.e., then g is measurable.

b. If fn is measurable for n and fn → f μ-a.e., then f is measurable.

The proof is left to the reader (Exercise 10).

On the other hand, the following result shows that one is unlikely to commit any serious blunders by forgetting to worry about completeness of the measure.

2.12 Proposition. Let (X, , μ) be a measure space and let be its completion. If f is an -measurable function on X, there is an -measurable function g such that f = g -almost everywhere.

Proof. This is obvious from the definition of if f = χE where E , and hence if f is an -measurable simple function. For the general case, choose a sequence {ϕn} of -measurable simple functions that converge pointwise to f according to Theorem 2.10, and for each n let n be an -measurable simple function with n = ϕn except on a set En with (En) = 0. Choose N such that μ(N) = 0 and N ⊃ ∪∞₁ En, and set g = limχX/N n. Then g is -measurable by Corollary 2.9, and g = f on Nc.

Exercises

In Exercises 1–7, (X, ) is a measurable space.

1. Let f : X → and Y = f⁻¹(). Then f is measurable iff f⁻¹({-∞}) , f⁻¹({∞}) , and f is measurable on Y.

2. Suppose f, g : X → are measurable.

a. fg is measurable (where 0 · (±∞) = 0).

b. Fix a and define h(x) = a if f(x) = −g(x) = ±∞ and h(x) = f(x) + g(x) otherwise. Then h is measurable.

3. If {fn} is a sequence of measurable functions on X, then {x : lim fn(x) exists} is a measurable set.

4. If f : X → and f⁻¹ ((r, ∞]) for each r , then f is measurable.

5. If X = A ∪ B where A, B , a function f on X is measurable iff f is measurable on A and on B.

6. The supremum of an uncountable family of measurable -valued functions on X can fail to be measurable (unless the σ-algebra is very special).

7. Suppose that for each α are given a set Eα such that Eα ⊂ Eβ whenever , and . Then there is a measurable function f : X → such that f(x) ≤ α on Eα and f(x) ≥ α on Ecα for every α. (Use Exercise 4.)

8. If f : → is monotone, then f is Borel measurable.

9. Let f : [0, 1] → [0, 1] be the Cantor function (§1.5), and let g(x) = f(x) + x.

a. g is a bijection from [0, 1] to [0, 2], and h = g⁻¹ is continuous from [0, 2] to [0, 1].

b. If C is the Cantor set, m(g(C)) = 1.

c. By Exercise 29 of Chapter 1, g(C) contains a Lebesgue nonmeasurable set A. Let B = g⁻¹(A). Then B is Lebesgue measurable but not Borel.

d. There exist a Lebesgue measurable function F and a continuous function G on such that F G is not Lebesgue measurable.

10. Prove Proposition 2.11.

11. Suppose that f is a function on × k such that f(x, ·) is Borel measurable for each x and f(·, y) is continuous for each y k. For n , define fn as follows. For i let ai = i/n, and for ai ≤ x ≤ ai+1 let

Then fn is Borel measurable on × k and fn → f pointwise; hence f is Borel measurable on × k. Conclude by induction that every function on n that is continuous in each variable separately is Borel measurable.

2.2 INTEGRATION OF NONNEGATIVE FUNCTIONS

In this section we fix a measure space (X, , μ), and we define

L⁺ = the space of all measurable functions from X to [0, ∞].

If ϕ is a simple function in L⁺ with standard representation ϕ = ∑n₁ ajχEj, we define the integral of ϕ with respect to μ by

(with the convention, as always, that 0 · ∞ = 0). We note that ∫ ϕ dμ may equal ∞. When there is no danger of confusion, we shall also write ∫ ϕ for ∫ ϕ dμ. Also, it is sometimes convenient to display the argument of ϕ explicitly, especially when ϕ(x) is given by a formula in terms of x or when there are other variables involved; in this case we shall use the notation ∫ ϕ(x) dμ(x). (Some authors prefer to write ∫ ϕ(x) μ(dx) instead.) Finally, if A , then ϕχA is also simple (viz., ϕχA = ∑ajχA∩Ej), and we define ∫A ϕ dμ (or ∫A ϕ or ∫A ϕ(x) dμ(x)) to be ∫ ϕχA dμ. The same notational conventions will also apply to the inegrals of more general functions to be defined below. To summarize:

2.13 Proposition. Let ϕ and be simple functions in L⁺.

a. If c ≥ 0, ∫ cϕ = c ∫ ϕ.

b. ∫(ϕ + ) = ∫ ϕ + ∫ .

c. If ϕ ≤ , then ∫ ϕ ≤ ∫ .

d. The map A → ∫A ϕ dμ, is a measure on .

Proof, (a) is trivial. For (b), let ∑n₁ ajχEj and ∑m₁ bkχFk be the standard representations of ϕ and . Then Ej = ∪mk=1 (Ej ∩ Fk) and Fk = ∪nj=1 (Ej ∩ Fk) since ∪n₁ Ej = ∪m₁ Fk = X, and these unions are disjoint. Hence the finite additivity of μ implies that

and the same reasoning show that the sum on the right equals ∫ (ϕ + ). Moreover, if ϕ ≤ , then aj ≤ bk whenever Ej ∩ Fk ≠ Ø, so

which proves (c). Finally, if {Ak} is a disjoint sequence in and A = ∪∞₁ Ak,

which establishes (d).

We now extend the integral to all functions f L⁺ by defining

By Proposition 2.13c, the two definitions of ∫ f agree when f is simple, as the family of simple functions over which the supremum is taken includes f itself. Moreover, it is obvious from the definition that

The next step is to establish one of the fundamental convergence theorems.

2.14 The Monotone Convergence Theorem. If {fn} is a sequence in L⁺ such that fj ≤ fj+1 for all j, and f = limn→∞ fn (= supn fn), then ∫ f = limn→∞ ∫ fn.

Proof. {∫ fn} is an increasing sequence of numbers, so its limit exists (possibly equal to ∞). Moreover, ∫ fn ≤ ∫ f for all n, so lim ∫ fn ≤ ∫ f. To establish the reverse inequality, fix α (0, 1), let ϕ be a simple function with 0 ≤ ϕ ≤ f, and let En = {x : fn(x) ≥ αϕ(x)}. Then {En} is an increasing sequence of measurable sets whose union is X, and we have ∫ fn ≥ ∫En fn ≥ α ∫En ϕ. By Proposition 2.13d and Theorem 1.8c, lim ∫En ϕ = ∫ ϕ, and hence lim ∫ fn ≥ α ∫ ϕ. Since this is true for all α < 1, it remains true for α = 1, and taking the supremum over all simple ϕ ≤ f, we obtain lim ∫ fn ≥ ∫ f.

The monotone convergence theorem is an essential tool in many situations, but its immediate significance for us is as follows. The definition of ∫ f involves the supremum over a huge (usually uncountable) family of simple functions, so it may be difficult to evaluate ∫ f directly from the definition. The monotone convergence theorem, however, assures us that to compute ∫ f it is enough to compute lim ∫ ϕn where {ϕn} is any sequence of simple functions that increase to f, and Theorem 2.10 guarantees that such sequences exist. As a first application, we establish the additivity of the integral.

2.15 Theorem. If {fn} is a finite or infinite sequence in L⁺ and f = ∑n fn, then ∫ f = ∑n ∫ fn.

Proof. First consider two functions f₁ and f₂. By Theorem 2.10 we can find sequences {ϕj} and {j} of nonnegative simple functions that increase to f₁ and f₂. Then {ϕj + j} increases to f₁ + f₂, so by the monotone convergence theorem and Theorem 2.13b,

Hence, by induction, ∫ ∑N₁ fn = ∑N₁ ∫ fn for any finite N. Letting N → ∞ and applying the monotone convergence theorem again, we obtain ∫ ∑∞₁ fn = ∑∞₁ ∫ fn.

2.16 Proposition. If f L⁺, then ∫ f = 0 iff f = 0 a.e.

Proof. This is obvious if f is simple: if f = ∑n₁ ajχEj with aj ≥ 0, then ∫ f = 0 iff for each j either aj = 0 or μ(Ej) = 0. In general, if f = 0 a.e. and ϕ is simple with 0 ≤ ϕ ≤ f, then ϕ = 0 a.e., hence ∫ f = supϕ≤f ∫ ϕ = 0. On the other hand, {x : f(x) > 0} = ∪∞₁ En where En = {x : f(x) > n⁻¹}, so if it is false that f = 0 a.e., we must have μ(En) > 0 for some n. But then f > n⁻¹ χEn, so ∫ f ≥ n⁻¹ μ(En) > 0.

2.17 Corollary. If {fn} ⊂ L⁺, f L⁺, and fn(x) increases to f(x) for a.e. x, then ∫ f = lim ∫ fn.

Proof. If fn(x) increases to f(x) for x E where μ(Ec) = 0, then f-fχE = 0 a.e. and fn − fnχE = 0 a.e., so by the monotone convergence theorem, ∫ f = ∫ fχE = lim ∫ fnχE = lim ∫ fn.

The hypothesis that the sequence {fn} be increasing, at least a.e., is essential for the monotone convergence theorem. For example, if X is and μ is Lebesgue measure, we have χ(n, n+1) → 0 and nχ(0, 1/n) → 0 pointwise, but ∫ χ(n, n+1) = ∫ nχ(0, 1/n) = 1 for all n. As one sees by sketching the graphs, the trouble in these examples is that the area under the graph “escapes to infinity” as n → ∞, so the area in the limit is less than one would expect. This is typical of the cases when the integral of the limit is not the limit of the integrals, but in this situation there is still an inequality that remains valid. We deduce it from the following general result.

2.18 Fatou’s Lemma. If {fn} is any sequence in L⁺, then

Proof. For each k ≥ 1 we have infn≥k fn ≤ fj for j ≥ k, hence ∫ infn≥k fn ≤ ∫ fj for j ≥ k, hence ∫ infn≥k fn ≤ infj≥k ∫ fj. Now let k → ∞ and apply the monotone convergence theorem:

2.19 Corollary. If {fn} ⊂ L⁺, f L⁺, and fn → f a.e., then ∫ f ≤ lim inf ∫ fn.

Proof. If fn → f everywhere, the result is immediate from Fatou’s lemma, and this can be achieved by modifying fn and f on a null set without affecting the integrals, by Proposition 2.16.

2.20 Proposition. If f L⁺ and ∫ f < ∞, then {x : f(x) = ∞} is a null set and {x : f(x) > 0} is σ-finite.

The proof is left to the reader (Exercise 12).

Exercises

12. Prove Proposition 2.20. (See Proposition 0.20, where a special case is proved.)

13. Suppose {fn} ⊂ L⁺, fn → f pointwise, and ∫ f = lim ∫ fn < ∞. Then ∫E f = lim ∫E fn for all E However, this need not be true if ∫ f = lim ∫ fn = ∞.

14. If f L⁺, let λ(E) = ∫E f dμ for E . Then λ is a measure on , and for any g L⁺, ∫ g dλ = ∫ f g dμ. (First suppose that g is simple.)

15. If {fn} ⊂ L⁺, fn decreases pointwise to f, and ∫ f₁ < ∞, then ∫ f = lim ∫ fn.

16. If f L⁺ and ∫ f < ∞, for every ε > 0 there exists E such that μ(E) < ∞ and E f > (∫ f) − ε.

17. Assume Fatou’s lemma and deduce the monotone convergence theorem from it.

2.3 INTEGRATION OF COMPLEX FUNCTIONS

We continue to work on a fixed measure space (X, , μ). The integral defined in the previous section can be extended to real-valued measurable functions f in an obvious way; namely, if f⁺ and f⁻ are the positive and negative parts of f and at least one of ∫ f⁺ and ∫ f⁻ is finite, we define

We shall be mainly concerned with the case where ∫ f⁺ and ∫ f⁻ are both finite; we then say that f is integrable. Since |f| = f⁺ + f⁻, it is clear that f is integrable iff ∫ |f| < ∞.

2.21 Proposition. The set of integrable real-valued functions on X is a real vector space, and the integral is a linear functional on it.

Proof. The first assertion follows from the fact that |af + bg| ≤ |a||f| + |b||g|, and it is easy to check that ∫ af = a ∫ f for any a . To show additivity, suppose that f and g are integrable and let h = f + g. Then h⁺ − h⁻ = f⁺ − f⁻ + g⁺ − g⁻, so h⁺ + f⁻ + g⁻ = h⁻ + f⁺ + g⁺. By Theorem 2.15,

and regrouping then yields the desired result:

Next, if f is a complex-valued measurable function, we say that f is integrable if ∫ |f| < ∞. More generally, if E , f is integrable on E if ∫E |f| < ∞. Since |f| ≤ |Re f| + |Im f| ≤ 2|f|, f is integrable iff Re f and Im f are both integrable, and in this case we define

It follows easily that the space of complex-valued integrable functions is a complex vector space and that the integral is a complex-linear functional on it. We denote this space — provisionally — by L¹(μ) (or L¹(X, μ), or L¹(X), or simply L¹, depending on the context). The superscript 1 is standard notation, but it will not assume any significance for us until Chapter 6.

2.22 Proposition. If f L¹, then |∫ f| ≤ ∫ |f|.

Proof. This is trivial if ∫ f = 0 and almost trivial if f is real, since

If f is complex-valued and ∫ f ≠ 0, let α = . Then |∫ f| = α ∫ f = ∫ αf. In particular, ∫ α f is real, so

2.23 Proposition.

a. If f L¹, then {x : f(x) ≠ 0} is σ-finite.

b. If f, g L¹, then ∫E f = ∫E g far all E iff ∫ |f-g| = 0 iff f = g a.e.

Proof. (a) and the second equivalence in (b) follow from Propositions 2.20 and 2.16. If ∫|f − g| = 0, then by Proposition 2.22, for any E ,

so that ∫E f = ∫E g. On the other hand, if u = Re(f − g), v = Im(f − g), and it is false that f = g a.e., then at least one of u⁺, u⁻, v⁺, and v⁻ must be nonzero on a set of positive measure. If, say, E = {x : u⁺(x) > 0} has positive measure, then Re(∫E f − ∫E g) = ∫E u⁺ > 0 since u⁻ = 0 on E; likewise in the other cases.

This proposition shows that for the purposes of integration it makes no difference if we alter functions on null sets. Indeed, one can integrate functions f that are only defined on a measurable set E whose complement is null simply by defining f to be zero (or anything else) on Ec. In this fashion we can treat -valued functions that are finite a.e. as real-valued functions for the purposes of integration.

With this in mind, we shall find it more convenient to redefine L¹(μ) to be the set of equivalence classes of a.e.-defined integrable functions on X, where f and g are considered equivalent iff f = g a.e. This new L¹(μ) is still a complex vector space (under pointwise a.e. addition and scalar multiplication). Although we shall henceforth view L¹(μ) as a space of equivalence classes, we shall still employ the notation “f L¹(μ)” to mean that f is an a.e.-defined integrable function. This minor abuse of notation is commonly accepted and rarely causes any confusion.

The new definition of L¹(μ) has two further advantages. First, if is the completion of μ, Proposition 2.12 yields a natural one-to-one correspondence between and L¹(μ), so we can (and shall) identify these spaces. Second, L¹ is a metric space with distance function ρ(f, g) = ∫ |f − g|. (The triangle inequality is easily verified, and obviously ρ(f, g) = ρ(g, f); but to obtain the condition that ρ(f, g) = 0 only when f = g, one must identify functions that are equal a.e., according to Proposition 2.23b.) We shall refer to convergence with respect to this metric as convergence in L¹; thus fn → f in L¹ iff ∫ |fn − f| → 0.

We now present the last of the three basic convergence theorems (the other two being the monotone convergence theorem and Fatou’s lemma) and derive some useful consequences from it. In the context of integration on with Lebesgue measure as in the discussion preceding Fatou’s lemma, the idea behind this theorem is that if fn → f a.e. and the graph of |fn| is confined to a region of the plane with finite area so that the area beneath it cannot escape to infinity, then ∫ fn → ∫ f.

2.24 The Dominated Convergence Theorem. Let {fn} be a sequence in L¹ such that (a) fn → f a.e., and (b) there exists a nonnegative g L¹ such that |fn| ≤ g a.e. for all n. Then f L¹ and ∫ f = limn→∞ ∫ fn.

Proof. f is measurable (perhaps after redefinition on a null set) by Propositions 2.11 and 2.12, and since |f| ≥ g a.e., we have f L¹. By taking real and imaginary parts it suffices to assume that fn and f are real-valued, in which case we have g + fn ≥ 0 a.e. and g − fn ≥ 0 a.e. Thus by Fatou’s lemma,

Therefore, lim inf ∫ fn ≥ ∫ f ≥ lim sup ∫ fn, and the result follows.

2.25 Theorem. Suppose that {fj} is a sequence in L¹ such that ∑∞₁ ∫ |fj| < ∞. Then ∑∞₁ fj converges a.e. to a function in L¹, and ∫ ∑∞₁ fj = ∑∞₁ ∫ fj.

Proof. By Theorem 2.15, ∫ ∑∞₁ |fj| = ∑∞₁ ∫ |fj| < ∞, so the function g = ∑∞₁ |fj| is in L¹. In particular, by Proposition 2.20, ∑∞₁ |fj(x)| is finite for a.e. x, and for each such x the series ∑∞₁ fj(x) converges. Moreover, |∑n₁ fj| ≤ g for all n, so we can apply the dominated convergence theorem to the sequence of partial sums to obtain ∫ ∑∞₁ = fj = ∑∞₁ ∫ fj.

2.26 Theorem. If f L¹(μ) and ε > 0, there is an integrable simple function ϕ = ∑ ajχEj such that ∫ |f − ϕ|dμ < ε. (That is, the integrable simple functions are dense in L¹ in the L¹ metric.) If μ is a Lebesgue-Stieltjes measure on , the sets Ej in the definition of ϕ can be taken to be finite unions of open intervals; moreover, there is a continuous function g that vanishes outside a bounded interval such that ∫ |f − g| dμ < ε.

Proof. Let {ϕn} be as in Theorem 2.10b; then ∫ |ϕn − f| < ε for n sufficiently large by the dominated convergence theorem, since |ϕn − f| ≤ 2|f|. If ϕn = ∑ ajχEj where the Ej are disjoint and the aj are nonzero, we observe that μ(Ej) = |aj|⁻¹ ∫Ej |ϕn| ≤ |aj|⁻¹ ∫ |f| < ∞. Moreover, if E and F are measurable sets, we have μ(E Δ F) = ∫ |χE − χF|. Thus if μ is a Lebesgue-Stieltjes measure on , by Proposition 1.20 we can approximate χEj arbitrarily closely in the L¹ metric by finite sums of functions χIk where the Ik’s are open intervals. Finally, if Ik = (a, b) we can approximate χIk in the L¹ metric by continuous functions that vanish outside (a, b). (For example, given ε > 0, take g to be the continuous function that equals 0 on (−∞, a] and [b, ∞), equals 1 on [a + ε, b − ε], and is linear on [a, a + ε] and [b − ε, b].) Putting these facts together, we obtain the desired assertions.

The next theorem gives a criterion, less restrictive than those found in most advanced calculus books, for the validity of interchanging a limit or a derivative with an integral.

2.27 Theorem. Suppose that f : X × [a, b] → (−∞ < a < b < ∞) and that f(·, t) : X → is integrable for each t [a, b]. Let F(t) = ∫X f(x, t) dμ(x).

a. Suppose that there exists g L¹(μ) such that |f(x, t)| ≤ g(x) for all x, t. If limt→t₀ f(x, t) = f (x, t₀) for every x, then limt→t₀ F(t) = F(t₀); in particular, if f(x, ·) is continuous for each x, then F is continuous.

b. Suppose that ∂ f / ∂t exists and there is a g L¹ (μ) such that |(∂f/∂t)(x, t)| ≤ g(x) for all x, t. Then F is differentiable and F’(t) = ∫(∂ f/∂t)(x, t) dμ(x).

Proof. For (a), apply the dominated convergence theorem to fn(x) = f(x, tn) where {tn} is any sequence in [a, b] converging to t₀. For (b), observe that

{tn} again being any sequence converging to t₀. It follows that ∂f/∂t is measurable, and by the mean value theorem,

so the dominated convergence theorem can be invoked again to give

The device of using sequences converging to t₀ in the preceding proof is technically necessary because the dominated convergence theorem deals only with sequences of functions. However, in such situations we shall usually just say “let t → t₀” with the understanding that sequential convergence is underlying the argument.

It is important to note that in Theorem 2.27 the interval [a, b] on which the estimates on f or ∂f/∂t hold might be a proper subinterval of an open interval I (perhaps itself) on which f(x, ·) is defined. If the hypotheses of (a) or (b) hold for all [a, b] ⊂ I, perhaps with the dominating function g depending on a and b, one obtains the continuity or differentiability of the integrated function F on all of I, as these properties are local in nature.

In the special case where the measure μ is Lebesgue measure on , the integral we have developed is called the Lebesgue integral. At this point it is appropriate to study the relation between the Lebesgue and Riemann integrals on . We shall use Darboux’s characterization of the Riemann integral in terms of upper and lower sums, which we now recall.

Let [a, b] be a compact interval. By a partition of [a, b] we shall mean a finite sequence P = {tj}n₀ such that a = t₀ < t₁ < · · · < tn = b. Let f be an arbitrary bounded real-valued function on [a, b]. For each partition P we define

where Mj and mj are the supremum and infimum of f on [tj−1, tj]. Then we define

where the infimum and supremum are taken over all partitions P. If , their common value is the Riemann integral ∫ba f(x) dx, and f is called Riemann integrable.

2.28 Theorem. Let f be a bounded real-valued function on [a, b].

a. If f is Riemann integrable, then f is Lebesgue measurable (and hence integrable on [a, b] since it is bounded), and ∫ba f(x) dx = ∫[a, b] f dm.

b. f is Riemann integrable iff{x [a, b] : f is discontinuous at x} has Lebesgue measure zero.

Proof. Suppose that f is Riemann integrable. For each partition P let

(with the same notation as above), so that SP f = ∫ GP dm and sP f = ∫ gP dm. There is a sequence {Pk} of partitions whose mesh (i.e., maxj(tj − tj−1)) tends to zero, each of which includes the preceding one (so that gPk increases with k while GPk decreases), such that SPk f and sPk f converge to ∫ba f(x) dx. Let G = lim GPk and g = lim gPk. Then g ≤ f ≤ G, and by the dominated convergence theorem, ∫ G dm = ∫ g dm = ∫ba f(x) dx. Hence ∫(G − g)dm = 0, so G = g a.e. by Proposition 2.16, and thus G = f a.e. Since G is measurable (being the limit of a sequence of simple functions) and m is complete, f is measurable and ∫[a,b] f dm = ∫ G dm = ∫ba f(x) dx. This proves (a), and the proof of (b) is outlined in Exercise 23.

The (proper) Riemann integral is thus subsumed in the Lebesgue integral. Some improper Riemann integrals (the absolutely convergent ones) can be interpreted directly as Lebesgue integrals, but others still require a limiting procedure. For example, if f is Riemann integrable on [0, b] for all b > 0 and Lebesgue integrable on [0, ∞), then ∫[0, ∞) f dm = limb→∞ ∫b₀ f(x) dx (by the dominated convergence theorem), but the limit on the right can exist even when f is not integrable. (Example: f = ∑∞₁ n⁻¹(−1)n χ(n, n+1].) Henceforth we shall generally use the notation ∫ba f(x) dx for Lebesgue integrals.

A few remarks comparing the construction of the Lebesgue and Riemann integrals may be helpful. Let f be a bounded measurable function on [a, b], and for simplicity let us assume that f ≥ 0. To compute the Riemann integral of f, one partitions the interval [a, b] into subintervals and approximates f from above and below by functions that are constant on each subinterval. To compute the Lebesgue integral of f, one picks a sequence of simple functions that increase to f. In particular, if one picks the sequence constructed in the proof of Theorem 2.10a (see Figure 2.1), one is in effect partitioning the range of f into subintervals Ij and approximating f by a constant on each of the sets f⁻¹(Ij). This procedure requires a more sophisticated theory of measure to begin with since the sets f⁻¹(Ij) can be complicated, even when f is continuous; but it is better adapted to the particular f under consideration and therefore more flexible — and more susceptible to generalization. (In the Lebesgue theory, the assumption that f is measurable removes the necessity of considering both upper and lower approximations; however, the latter point of view can also be made to work in the abstract setting. See Exercise 24.)

The Lebesgue theory offers two real advantages over the Riemann theory. First, much more powerful convergence theorems, such as the monotone and dominated convergence theorems, are available. These not only yield results previously unobtainable but also reduce the labor in proving classical theorems. Second, a wider class of functions can be integrated. For example, if R is the set of rational numbers in [0, 1], χR is not Riemann integrable, being everywhere discontinuous on [0, 1], but it is Lebesgue integrable, and ∫ χR dm = 0. (Actually, this is in some sense a trivial example since χR agrees a.e. with the constant function 0. For a more interesting example, see Exercise 25.) Of course, virtually all functions that one meets in classical analysis are (locally) Riemann integrable, so this added generality is rarely used in computing specific integrals. However, it has the crucial effect that various metric spaces of functions whose metrics are defined in terms of integrals are complete when Lebesgue integrable functions are used but not when one considers only Riemann integrable functions. We shall investigate this situation more thoroughly later, especially in Chapter 6. (We have already proved the completeness of L¹(μ), disguised as Theorem 2.25. To remove the disguise, see Theorem 5.1.)

We conclude this section by introducing the most ubiquitous of the higher transcendental functions, the gamma function Γ, which will play a role in a number of places later on. If z and Re z > 0, define fz : (0, ∞) → by fz(t) = tz−1e−t. (Here tz−1 = exp[(z − 1) logt].) Since |tz−1| = tRe z−1, we have |fz(t)| ≤ tRe z−1, and also |fz(t)| ≤ Cze−t/2 for t ≥ 1. (The precise value of Cz can easily be found by maximizing tRe z−1e−t/2, but it is of no importance here.) Since ∫¹₀ ta dt < ∞ for a > −1 and ∫∞₁ e−t/2 dt < ∞, we see that fz L¹((0, ∞)) for Re z > 0, and we define

Since

by integration by parts, by letting ε → 0 and N → ∞ we see that for Re z > 0, Γ satisfies the functional equation

This equation can then be used to extend Γ to (almost) the entire complex plane. Namely, for −1 < Re z ≤ 0 we can define Γ(z) to be Γ(z + 1)/z, and by induction, having defined Γ(z) for Re z > −n, we define Γ(z) for Re z > −n − 1 to be Γ(z + 1)/z. The result is a function defined on all of except for singularities at the nonpositive integers where the algorithm just described involves division by zero.

We have Γ(1) = ∫∞₀ e−t dt = −e−t|∞₀ = 1, so an n-fold application of the functional equation shows that Γ(n +1) = n!. (Another proof of this fact is outlined in Exercise 29.) Many of the applications of the gamma function involve the fact that it provides an extension of the factorial function to nonintegers.

Exercises

18. Fatou’s lemma remains valid if the hypothesis that fn L⁺ is replaced by the hypothesis that fn is measurable and fn ≥ −g where g L⁺ ∩ L¹. What is the analogue of Fatou’s lemma for nonpositive functions?

19. Suppose {fn} ⊂ L¹(μ) and fn → f uniformly.

a. If μ(X) < ∞, then f L¹(μ) and ∫ fn → ∫ f.

b. If μ(X) = ∞, the conclusions of (a) can fail. (Find examples on with Lebesgue measure.)

20. (A generalized Dominated Convergence Theorem) If fn, gn, f, g L¹, fn → f and gn → g a.e., |fn| ≤ gn, and ∫ gn → ∫ g, then ∫ fn → ∫ f. (Rework the proof of the dominated convergence theorem.)

21. Suppose fn, f L¹ and fn → f a.e. Then ∫|fn − f| → 0 iff ∫|fn| → ∫|f|. (Use Exercise 20.)

22. Let μ be counting measure on . Interpret Fatou’s lemma and the monotone and dominated convergence theorems as statements about infinite series.

23. Given a bounded function f : [a, b] → , let

Prove Theorem 2.28b by establishing the following lemmas:

a. H(x) = h(x) iff f is continuous at x.

b. In the notation of the proof of Theorem 2.28a, H = G a.e. and h = g a.e. Hence H and h are Lebesgue measurable, and ∫[a, b] H dm = and .

24. Let (X, , μ) be a measure space with μ(X) < ∞, and let be its completion. Suppose f : X → is bounded. Then f is -measurable (and hence in L¹()) iff there exist sequences {ϕn} and {n} of -measurable simple functions such that ϕn ≤ f ≤ n and ∫(n − ϕn) dμ < n⁻¹. In this case, lim ∫ ϕn dμ = lim ∫ n .

25. Let f(x) = x−1/2 if 0 < x < 1, f(x) = 0 otherwise. Let {rn}∞₁ be an enumeration of the rationals, and set g(x) = ∑∞₁ 2−n f(x − rn).

a. g L¹(m), and in particular g < ∞ a.e.

b. g is discontinuous at every point and unbounded on every interval, and it remains so after any modification on a Lebesgue null set.

c. g² < ∞ a.e., but g² is not integrable on any interval.

26. If f L¹(m) and F(x) = ∫x-∞ f(t) dt, then F is continuous on .

27. Let fn(x) = ae−nax − be−nbx where 0 < a < b.

a. ∑∞₁ ∫∞₀ |fn(x)| dx = ∞.

b. ∑∞₁ ∫∞₀ fn(x) dx = 0.

c. ∑∞₁ fn L¹([0, ∞), m), and ∫∞₀ ∑∞₁ fn(x) dx = log(b/a).

28. Compute the following limits and justify the calculations:

a. limn→∞ ∫∞₀(1 + (x/n))−n sin(x/n) dx.

b. limn→∞ ∫¹₀ (1 + nx²)(1 + x²)−n dx.

c. limn→∞ ∫∞₀ n sin(x/n)[x(1 + x²)]⁻¹ dx.

d. limn→∞ ∫∞₀ n(1 + n²x²)⁻¹ dx. (The answer depends on whether a > 0, a = 0, or a < 0. How does this accord with the various convergence theorems?)

29. Show that ∫∞₀ xne−x dx = n! by differentiating the equation ∫∞₀ e−tx dx = 1/t. Similarly, show that ∫∞-∞ x2n e−x² dx = (2n)!! by differentiating the equation ∫∞-∞ e−tx² dx = (see Proposition 2.53).

30. Show that limk→∞ ∫k₀ xn(1 − k⁻¹x)k dx = n!.

31. Derive the following formulas by expanding part of the integrand into an infinite series and justifying the term-by-term integration. Exercise 29 may be useful. (Note: In (d) and (e), term-by-term integration works, and the resulting series converges, only for a > 1, but the formulas as stated are actually valid for all a > 0.)

a. For a > 0, ∫∞-∞ e−x² cos ax dx = .

b. For a > −1, ∫¹₀ xa(1 − x)⁻¹ log x dx = − ∑∞₁(a + k)⁻².

c. For a > 1, ∫∞₀ xa−1 (ex − 1)⁻¹ dx = Γ(a)ζ(a), where ζ(a) = ∑∞₁ n−a.

d. For a > 1, ∫∞₀ e−axx⁻¹ sin x dx = arctan(a⁻¹).

e. For a > 1, ∫∞₀ e−ax J₀(x) dx = (s² + 1)−1/2, where J₀(x) = ∑∞₀(−1)nx2n/4n(n!)² is the Bessel function of order zero.

2.4 MODES OF CONVERGENCE

If {fn} is a sequence of complex-valued functions on a set X, the statement “fn → f as n → ∞” can be taken in many different senses, for example, pointwise or uniform convergence. If X is a measure space, one can also speak of a.e. convergence or convergence in L¹. Of course, uniform convergence implies pointwise convergence, which in turn implies a.e. convergence (and not conversely, in general), but these modes of convergence do not imply L¹ convergence or vice versa. It will be useful to keep in mind the following examples on (with Lebesgue measure):

i. fn = n⁻¹ χ(0, n).

ii. fn = χ(n, n+1).

iii. fn = nχ[0, 1/n].

iv. f₁ = χ[0, 1], f₂ = χ[0, 1/2], f₃ = χ[1/2, 1], f₄ = χ[0, 1/4], f₅ = χ[1/4, 1/2], f₆ = χ[1/2, 3/4], f₇ = χ[3/4, 1], and in general, fn = χ[j/2k, (j+1)/2k] where n = 2k + j with 0 ≤ j < 2k.

In (i), (ii), and (iii), fn → 0 uniformly, pointwise, and a.e., repectively, but (in fact ∫ |fn| = ∫ fn = 1 for all n). In (iv), fn → 0 in L¹ since ∫ |fn| = 2−k for 2k ≤ n < 2k+1 but fn(x) does not converge for any x [0, 1] since there are infinitely many n for which fn(x) = 0 and infinitely many for which fn(x) = 1.

On the other hand, if fn → f a.e. and |fn| ≤ g L¹ for all n, then fn → f in L¹. (This is clear from the dominated convergence theorem since |fn − f| ≤ 2g.) Also, we shall see below that if fn → f in L¹ then some subsequence converges to f a.e.

Another mode of convergence that is frequently useful is convergence in measure. We say that a sequence {fn} of measurable complex-valued functions on (X, , μ) is Cauchy in measure if for every ε > 0,

and that {fn} converges in measure to f if for every ε > 0,

For example, the sequences (i), (iii), and (iv) above converge to zero in measure, but (ii) is not Cauchy in measure.

2.29 Proposition. If fn → f in L¹, then fn → f in measure.

Proof. Let En, ε = {x : |fn(x)-f(x)| ≥ ε}. Then ∫ |fn-f| ≥ En, ε |fn-f| ≤ εμ(En, ε), so μ(En, ε) ≤ ⁻¹ ∫ |fn − f| → 0.

The converse of Proposition 2.29 is false, as examples (i) and (iii) show.

2.30 Theorem. Suppose that {fn} is Cauchy in measure. Then there is a measurable function f such that fn → f in measure, and there is a subsequence {fnj} that converges to f a.e. Moreover, if also fn → g in measure, then g = f a.e.

Proof. We can choose a subsequence {gj} = {fnj} of {fn} such that if Ej = {x : |gj(x) − gj+1(x)| ≥ 2−j}, then μ(Ej) ≤ 2−j If Fk = ∪∞j=k Ej, then μ(Fk) ≤ ∑∞k 2−j = 21-k and if x Fk, for i ≥ j ≥ k we have

(2.31) equation

Thus {gj} is pointwise Cauchy on Fck. Let F = ∩∞₁ Fk = lim sup Ej. Then μ(F) = 0, and if we set f(x) = lim gj(x) for x F and f(x) = 0 for x F, then f is measurable (see Exercises 3 and 5) and gj → f a.e. Also, (2.31) shows that |gj(x) − f(x)| ≤ 21-j for x Fk and j ≥ k. Since μ(Fk) → 0 as k → ∞, it follows that gj → f in measure. But then fn → f in measure, because

and the sets on the right both have small measure when n and j are large. Likewise, if fn → g in measure,

for all n, hence μ({x : |f(x) − g(x)| ≥ }) = 0 for all . Letting tend to zero through some sequence of values, we conclude that f = g a.e.

2.32 Corollary. If fn → f in L¹, there is a subsequence {fnj} such that fnj → f a.e.

Proof. Combine Proposition 2.29 and Theorem 2.30.

If fn → f a.e., it does not follow that fn → f in measure, as example (ii) shows. However, this conclusion does hold on a finite measure space, where something considerably stronger is true.

2.33 Egoroff’s Theorem. Suppose that μ(X) < ∞, and f₁, f₂,… and f are measurable complex-valued functions on X such that fn → f a.e. Then for every > 0 there exists E ⊂ X such that μ(E) < and fn → f uniformly on Ec.

Proof. Without loss of generality we may assume that fn → f everywhere on X. For k, n let

Then, for fixed k, En(k) decreases as n increases, and ∩∞n=1 En(k) = Ø, so since μ(X) < ∞ we conclude that μ(En(k)) → 0 as n → ∞. Given > 0 and k , choose nk so large that μ(Enk (k)) < 2−k and let E = ∪∞k=1 Enk (k). Then μ(E) < , and we have |fn(x) − f(x)| < k⁻¹ for n > nk and x E. Thus fn → f uniformly on Ec.

The type of convergence involved in the conclusion of Egoroff’s theorem is sometimes called almost uniform convergence. It is not hard to see that almost uniform convergence implies a.e. convergence and convergence in measure (Exercise 39).

Exercises

32. Suppose μ(X) < ∞. If f and g are complex-valued measurable functions on X, define

Then ρ is a metric on the space of measurable functions if we identify functions that are equal a.e., and fn → f with respect to this metric iff fn → f in measure.

33. If fn ≥ 0 and fn → f in measure, then ∫ f ≤ lim inf ∫ fn.

34. Suppose |fn| ≤ g L¹ and fn → f in measure.

a. ∫ f = lim ∫ fn.

b. fn → f in L¹.

35. fn → f in measure iff for every > 0 there exists N such that μ({x : |fn(x) − f(x)| ≥ }) < for n ≥ N.

36. If μ(En) < ∞ for n and χEn → f in L¹, then f is (a.e. equal to) the characteristic function of a measurable set.

37. Suppose that fn and f are measurable complex-valued functions and ϕ : → .

a. If ϕ is continuous and fn → f a.e., then ϕ fn → ϕ f a.e.

b. If ϕ is uniformly continuous and fn → f uniformly, almost uniformly, or in measure, then ϕ fn → ϕ f uniformly, almost uniformly, or in measure, respectively.

c. There are counterexamples when the continuity assumptions on ϕ are not satisfied.

38. Suppose fn → f in measure and gn → g in measure.

a. fn + gn → f + g in measure.

b. fngn → fg in measure if μ(X) < ∞, but not necessarily if μ(X) = ∞.

39. If fn → f almost uniformly, then fn → f a.e. and in measure.

40. In Egoroff’s theorem, the hypothesis “μ(X) < ∞” can be replaced by “|fn| ≤ g for all n, where g L¹ (μ).”

41. If μ is σ-finite and fn → f a.e., there exist measurable E₁, E₂,… ⊂ X such that μ((∪∞₁ Ej)c) = 0 and fn → f uniformly on each Ej.

42. Let μ be counting measure on . Then fn → f in measure iff fn → f uniformly.

43. Suppose that μ(X) < ∞ and f : X × [0, 1] → is a function such that f(·, y) is measurable for each y [0, 1] and f(x, ·) is continuous for each x X.

a. If 0 < , δ < 1 then E, δ = {x : |f(x, y) − f(x, 0)| ≤ for all y < δ} is measurable.

b. For any > 0 there is a set E ⊂ X such that μ(E) < and f(·, y) → f(·, 0) uniformly on Ec as y → 0.

44. (Lusin’s Theorem) If f : [a, b] → is Lebesgue measurable and > 0, there is a compact set E ⊂ [a, b] such that μ(Ec) < and f|E is continuous. (Use Egoroff’s theorem and Theorem 2.26.)

2.5 PRODUCT MEASURES

Let (X, , μ) and (Y, , v) be measure spaces. We have already discussed the product σ-algebra on X × Y; we now construct a measure on that is, in an obvious sense, the product of μ and v.

To begin with, we define a (measurable) rectangle to be a set of the form A × B where A and B . Clearly

Therefore, by Proposition 1.7, the collection of finite disjoint unions of rectangles is an algebra, and of course the σ-algebra it generates is .

Suppose A × B is a rectangle that is a (finite or countable) disjoint union of rectangles Aj × Bj. Then for x X and y Y,

If we integrate with respect to x and use Theorem 2.15, we obtain

In the same way, integration in y then yields

It follows that if E is the disjoint union of rectangles A₁ × B₁,…, An × Bn, and we set

(with the usual convention that 0 · ∞ = 0), then π is well defined on (since any two representations of E as a finite disjoint union of rectangles have a common refinement), and π is a premeasure on . According to Theorem 1.14, therefore, π generates an outer measure on X × Y whose restriction to is a measure that extends π. We call this measure the product of μ and v and denote it by μ × v. Moreover, if μ and v are σ-finite—say, X = ∪∞₁ Aj and Y = ∪∞₁ Bk with μ(Aj) < ∞ and v(Bk) < ∞ — then X × Y = ∪j, k Aj × Bk, and μ × v(Aj × Bk) < ∞, so μ × v is also σ-finite. In this case, by Theorem 1.14, μ × v is the unique measure on such that μ × v(A × B) = μ(A)v(B) for all rectangles A × B.

The same construction works for any finite number of factors. That is, suppose (Xj, j, μj) are measure spaces for j = 1,…, n. If we define a rectangle to be a set of the form A₁ × · · · × An with Aj j, then the collection of finite disjoint unions of rectangles is an algebra, and the same procedure as above produces a measure μ₁ × · · · × μn on such that

Moreover, if the μj’s are σ-finite so that the extension from to is uniquely determined, the obvious associativity properties hold. For example, if we identify X₁ × X₂ × X₃ with (X₁ × X₂) × X₃, we have (the former being generated by sets of the form A₁ × A₂ × A₃ with Aj j, and the latter by sets of the form B × A₃ with and A₃ ₃), and μ₁ × μ₂ × μ₃ = (μ₁ × μ₂) × μ₃ (since they agree on sets of the form A₁ × A₂ × A₃, and hence in general by uniqueness). Details are left to the reader (Exercise 45). All of our results below have obvious extensions to products with n factors, but we shall stick to the case n = 2 for simplicity.

We return to the case of two measure spaces (X, , μ) and (Y, , v). If E ⊂ X x Y, for x X and y Y we define the x-section Ex and the y-section Ey of E by

Also, if f is a function on X × Y we define the x-section fx and the y-section fy of f by

Thus, for example, (χE)x = χEx and (χE)y = χEy.

2.34 Proposition.

a. If E , then Ex for all × X and Ey for all y Y.

b. If f is -measurable, then fx is -measurable for all × X and fy is -measurable for all y Y.

Proof. Let be the collection of all subsets E of X × Y such that Ex for all x and Ey for all y. Then obviously contains all rectangles (e.g., (A × B)x = B if x A, = Ø otherwise). Since (∪∞₁ Ej)x = ∪∞₁(Ej)x and (Ec)x = (Ex)c, and likewise for y-sections, is a σ-algebra. Therefore , which proves (a). (b) follows from (a) because (fx)⁻¹ (B) = (f⁻¹(B))x and (fy)⁻¹(B) = (f⁻¹(B))y.

Before proceeding further we need a technical lemma. We define a monotone class on a space X to be a subset of (X) that is closed under countable increasing unions and countable decreasing intersections (that is, if Ej and E₁ ⊂ E₂ ⊂ ···, then ∪ Ej , and likewise for intersections). Clearly every σ-algebra is a monotone class. Also, the intersection of any family of monotone classes is a monotone class, so for any ε ⊂ (X) there is a unique smallest monotone class containing ε, called the monotone class generated by ε.

2.35 The Monotone Class Lemma. If is an algebra of subsets of X, then the monotone class generated by coincides with the σ-algebra generated by .

Proof. Since is a monotone class, we have ⊂ ; and if we can show that is a σ-algebra, we will have ⊂ . To this end, for E let us define

Clearly Ø and E are in (E), and E iff F . Also, it is easy to check that is a monotone class. If E , then for all F because is an algebra; that is, , and hence , Therefore, if , then F for all E . But this means that for all E , so that and hence . Conclusion: If E, F , then and E ∩ F are in . Since X is therefore an algebra. But then if {Ej}∞₁ ⊂ , we have ∪n₁ Ej for all n, and since is closed under countable increasing unions it follows that ∪∞₁ Ej . In short, is a σ-algebra, and we are done.

We now come to the main results of this section, which relate integrals on X × Y to integrals on X and Y.

2.36 Theorem. Suppose (X, , μ) and (Y, , v) are σ-finite measure spaces. If E , then the functions × → v(Ex) and y → μ(Ey) are measurable on X and Y, respectively, and

Proof. First suppose that μ and v are finite, and let be the set of all E for which the conclusions of the theorem are true. If E = A × B, then v(Ex) = χA(x)v(B) and μ(Ey) = μ(A)χB(y), so clearly E . By additivity it follows that finite disjoint unions of rectangles are in , so by Lemma 2.35 it will suffice to show that is a monotone class. If {En} is an increasing sequence in and E = ∪∞₁ En, then the functions fn(y) = μ((En)y) are measurable and increase pointwise to f(y) = μ(Ey). Hence f is measurable, and by the monotone convergence theorem,

Likewise μ × v(E) = ∫ v(Ex) dμ(x), so E . Similarly, if {En} is a decreasing sequence in and ∩∞₁ En, the function y → μ((E₁)y) is in L¹(v) because μ((E₁)y) ≤ μ(X) < ∞ and v(Y) < ∞, so the dominated convergence theorem can be applied to show that E . Thus is a monotone class, and the proof is complete for the case of finite measure spaces.

Finally, if μ and v are σ-finite, we can write X × Y as the union of an increasing sequence {Xj × Yj} of rectangles of finite measure. If E , the preceding argument applies to E ∩ (Xj × Yj) for each j to give

and a final application of the monotone convergence theorem then yields the desired result.

2.37 The Fubini-Tonelli Theorem. Suppose that (X, , μ) and (Y, , v) are σ-finite measure spaces.

a. (Tonelli) If f L⁺(X × Y), then the functions g(x) = ∫ fx dv and h(y) = ∫ fy dμ are in L⁺(X) and L⁺(Y), respectively, and

(2.38) equation

b. (Fubini) If f L¹(μ × v), then fx L¹(v) for a.e. × X, fy L¹(μ) for a.e. y Y, the a.e.-defined functions g(x) = ∫ fx dv and h(y) = ∫ fy dv are in L¹(μ) and L¹(v), respectively, and (2.38) holds.

Proof. Tonelli’s theorem reduces to Theorem 2.36 in case f is a characteristic function, and it therefore holds for nonnegative simple functions by linearity. If f L⁺(X × Y), let {fn} be a sequence of simple functions that increase pointwise to f as in Theorem 2.10. The monotone convergence theorem implies, first, that the corresponding gn and hn increase to g and h (so that g and h are measurable), and, second that

which is (2.38). This establishes Tonelli’s theorem and also shows that if f L⁺(X × Y) and ∫ f d(μ × v) < ∞, then g < ∞ a.e. and h < ∞ a.e., that is, fx L¹(v) for a.e. x and fy L¹(μ) for a.e. y. If f L¹(μ × v), then, the conclusion of Fubini’s theorem follows by applying these results to the positive and negative parts of the real and imaginary parts of f.

A few remarks are in order:

We shall usually omit the brackets in the iterated integrals in (2.38), thus:

The hypothesis of σ-finiteness is necessary; see Exercise 46.

The hypothesis f L⁺(X × Y) or f L¹(μ × v) is necessary, in two respects. First, it is possible for fx and fy to be measurable for all x, y and for the iterated integrals ∫∫ f dμ dv and ∫∫ f dv dμ to exist even if f is not -measurable. However, the iterated integrals need not then be equal; see Exercise 47. Second, if f is not nonnegative, it is possible for fx and fy to be integrable for all x, y and for the iterated integrals ∫∫ f dμ dv and ∫∫ f dv dμ to exist even if ∫ |f| d(μ × v) = ∞. But again, the iterated integrals need not be equal; see Exercise 48.

The Fubini and Tonelli theorems are frequently used in tandem. Typically one wishes to reverse the order of integration in a double integral ∫∫ f dμ dv. First one verifies that ∫|f| d(μ × v) < ∞ by using Tonelli’s theorem to evaluate this integral as an iterated integral; then one applies Fubini’s theorem to conclude that ∫∫ f dμ dv = ∫∫ f dv dμ. For examples, see the exercises in §2.6.

Even if μ and v are complete, μ × v is almost never complete. Indeed, suppose that there is a nonempty A with μ(A) = 0 and that ≠ (Y). (This is the case with μ = v = Lebesgue measure on , for example.) If E (Y) \ , then by Proposition 2.34, but A × E ⊂ A × Y, and μ × v(A × Y) = 0.

If one wishes to work with complete measures, of course, one can consider the completion of μ × v. In this setting the relationship between the measurability of a function on X × Y and the measurability of its x-sections and y-sections is not so simple. However, the Fubini-Tonelli theorem is still valid when suitably reformulated:

2.39 The Fubini-Tonelli Theorem for Complete Measures. Let (X, , μ) and (Y, , v) be complete, σ-finite measure spaces, and let (X × Y, , λ) be the completion of (X × Y, , μ × v). If f is -measurable and either (a) f ≥ 0 or (b) f L¹(λ), then fx is -measurable for a.e. × and fy is -measurable for a.e. y, and in case (b) fx and fy are also integrable for a.e. x and y. Moreover, x → ∫ fx dv and y → ∫ fy dμ, are measurable, and in case (b) also integrable, and

This theorem is a fairly easy corollary of Theorem 2.37; the proof is outlined in Exercise 49.

Exercises

45. If (Xj, j) is a measurable space for j = 1, 2, 3, then . Moreover, if μj is a σ-finite measure on (Xj, j), then μ₁ × μ₂ × μ₃ = (μ₁ × μ₂) × μ₃.

46. Let X = Y = [0, 1], = = [0, 1], μ = Lebesgue measure, and v = counting measure. If D = {(x, x) : x [0, 1]} is the diagonal in X × Y, then ∫∫ χD dμ dv, ∫∫ χD dv dμ, and ∫ χD d(μ × v) are all unequal. (To compute ∫ χD d(μ × v) = μ × v(D), go back to the definition of μ × v.)

47. Let X = Y be an uncountable linearly ordered set such that for each x X, {y X : y < x} is countable. (Example: the set of countable ordinals.) Let = be the σ-algebra of countable or co-countable sets, and let μ = v be defined on by μ(A) = 0 if A is countable and μ(A) = 1 if A is co-countable. Let E = {(x, y) X × X : y < x}. Then Ex and Ey are measurable for all x, y, and ∫∫ χE dμ dv and ∫∫ χE dv dμ exist but are not equal. (If one believes in the continuum hypothesis, one can take X = [0, 1] [with a nonstandard ordering] and thus obtain a set E ⊂ [0, 1]² such that Ex is countable and Ey is co-countable [in particular, Borel] for all x, y, but E is not Lebesgue measurable.)

48. Let X = Y = , = , = (), μ = v = counting measure. Define f(m, n) = 1 if m = n, f(m, n) = −1 if m = n + 1, and f(m, n) = 0 otherwise. Then ∫|f| d(μ × v) = ∞, and ∫∫ f dμ dv and ∫∫ f dv dμ exist and are unequal.

49. Prove Theorem 2.39 by using Theorem 2.37 and Proposition 2.12 together with the following lemmas.

a. If E × and μ × v(E) = 0, then v(Ex) = μ(Ey) = 0 for a.e. x and y.

b. If f is -measurable and f = 0 λ-a.e., then fx and fy are integrable for a.e. x and y, and ∫ fx dv = ∫ fy dμ = 0 for a.e. x and y. (Here the completeness of μ and v is needed.)

50. Suppose (X, , μ) is a σ-finite measure space and f L⁺(X). Let

Then Gf is × -measurable and μ × m(Gf) = ∫ f dμ; the same is also true if the inequality y ≤ f(x) in the definition of Gf is replaced by y < f(x). (To show measurability of Gf, note that the map (x, y) f(x) − y is the composition of (x, y) (f(x), y) and (z, y) → z − y.) This is the definitive statement of the familiar theorem from calculus, “the integral of a function is the area under its graph.”

51. Let (X, , μ) and (Y, , v) be arbitrary measure spaces (not necessarily σ-finite).

a. If f : X → is -measurable, g : Y → is -measurable, and h(x, y) = f(x)g(y), then h is -measurable.

b. If f L¹(μ) and g L¹(v), then h L¹(μ × v) and ∫ h d(μ, × v) = [∫ f dμ][∫ g dv].

52. The Fubini-Tonelli theorem is valid when (X, , μ) is an arbitrary measure space and Y is a countable set, = (Y), and v is counting measure on Y. (Cf. Theorems 2.15 and 2.25.)

2.6 THE n-DIMENSIONAL LEBESGUE INTEGRAL

Lebesgue measure mn on n is the completion of the n-fold product of Lebesgue measure on with itself, that is, the completion of m × · · · × m on , or equivalently the completion of m × · · · × m on . The domain is the class of Lebesgue measurable sets in n; sometimes we shall also consider mn as a measure on the smaller domain n. When there is no danger of confusion, we shall usually omit the superscript n and write m for mn, and as in the case n = 1, we shall usually write ∫ f(x) dx for ∫ f dm.

We begin by establishing the extensions of some of the results in §1.5 to the n-dimensional case. In what follows, if E = Πn₁ Ej is a rectangle in n, we shall refer to the sets Ej ⊂ as the sides of E.

2.40 Theorem. Suppose E .

a. m(E) = inf{m(U) : U ⊃ E, U open} = sup{m(K) : K ⊂ E, K compact}.

b. E = A₁ ∪ N₁ = A₂ \ N₂ where A₁ is an Fσ set, A₂ is a Gδ set, and m(N₁) = m(N₂) = 0.

c. If m(E) < ∞ for any > 0 there is a finite collection {Rj}N₁ of disjoint rectangles whose sides are intervals such that m(E Δ ∪N₁ Rj) < .

Proof. By the definition of product measures, if and > 0 there is a countable family {Tj} of rectangles such that E ⊂ ∪∞₁ Tj and ∑∞₁ m(Tj) ≤ m(E) + . For each j, by applying Theorem 1.18 to the sides of Tj we can find a rectangle Uj ⊃ Tj whose sides are open sets such that m(Uj) < m(Tj) + 2−j. If U = ∪∞₁ Uj, then U is open and m(U) ≤ ∑∞₁ m(Uj) ≤ m(E) + 2. This proves the first equation in part (a); the second one, and part (b), then follow as in the proofs of Theorems 1.18 and 1.19. Next, if m(E) < ∞ then m(Uj) < ∞ for all j. Since the sides of Uj are countable unions of open intervals, by taking suitable finite subunions we obtain rectangles Vj ⊂ Uj whose sides are finite unions of intervals such that m(Vj) ≥ m(Uj) − 2−j. If N is sufficiently large, then, we have

and

so that m(E Δ ∪N₁ Vj) < 3. Since ∪N₁ Vj can be expressed as a finite disjoint union of rectangles whose sides are intervals, we have proved (c).

2.41 Theorem. If f L¹(m) and > 0, there is a simple function ϕ = ∑N₁ ajχRj, where each Rj is a product of intervals, such that ∫|f − ϕ| < , and there is a continuous function g that vanishes outside a bounded set such that ∫|f − g| < .

Proof. As in the proof of Theorem 2.26, approximate f by simple functions, then use Theorem 2.40c to approximate the latter by functions ϕ of the desired form. Finally, approximate such ϕ’s by continuous functions by applying an obvious generalization of the argument in the proof of Theorem 2.26.

2.42 Theorem. Lebesgue measure is translation-invariant. More precisely, for a n define τa : n → n by τa(x) = x + a.

a. If E n, then τa(E) n and m(τa(E)) = m(E).

b. If f : n → is Lebesgue measurable, then so is f τa. Moreover, if either f ≥ 0 or f L¹(m), then ∫(f τa) dm = ∫ f dm.

Proof. Since τa and its inverse τ-a are continuous, they preserve the class of Borel sets. The formula m(τa(E)) = m(E) follows easily from the one-dimensional result (Theorem 1.21) if E is a rectangle, and it then follows for general Borel sets since m is determined by its action on rectangles (the uniqueness in Theorem 1.14). In particular, the collection of Borel sets E such that m(E) = 0 is invariant under τa. Assertion (a) now follows immediately.

If f is Lebesgue measurable and B is a Borel set in , we have f⁻¹(B) = E ∪ N where E is Borel and m(N) = 0. But τ⁻¹a (E) is Borel and m(τ⁻¹a (N)) = 0, so (f and is Lebesgue measurable. The equality ∫(f τa) dμ = ∫ f dμ reduces to the equality m(τ-a(E)) = m(E) when f = χE. It is then true for simple functions by linearity, and hence for nonnegative measurable functions by the definition of the integral. Taking positive and negative parts of real and imaginary parts then yields the result for f L¹(m).

Let us now compare Lebesgue measure on n to the more naive theory of n-dimensional measure usually found in advanced calculus books. In this discussion, a cube in n is a Cartesian product of n closed intervals whose side lengths are all equal.

For k , let be the collection of cubes whose side length is 2−k and whose vertices are in the lattice (2−k)n. (That is, Πn₁[aj, bj] k iff 2kaj and 2kbj are integers and bj − aj = 2−k for all j.) Note that any two cubes in k have disjoint interiors, and that the cubes in k+1 are obtained from the cubes in k by bisecting the sides.

If E ⊂ n, we define the inner and outer approximations to E by the grid of cubes k to be

(See Figure 2.2.) The measure of A(E, k) (in either the naive geometric sense or the Lebesgue sense) is just 2−nk times the number of cubes in k that lie in A(E, k), and we denote it by m(A(E, k)); likewise for . Also, the sets A(E, k) increase with k while the sets (E, k) decrease, because each cube in k is a union of cubes in k+1. Hence the limits

exist. They are called the inner and outer content of E, and if they are equal, their common value κ(E) is the Jordan content of E.

Fig. 2.2 Approximations to the inner and outer content of a set.

Two comments: First, Jordan content is usually defined using general rectangles whose sides are intervals rather than our dyadic cubes, but the result is the same. Second, although all the definitons above make sense for arbitrary E ⊂ n, the theory of Jordan content is meaningful only if E is bounded, for otherwise (E) always equals ∞.

Let

Then and are Borel sets, and κ(E) = m(A(E)) and . Thus the Jordan content of E exists iff m((E)\A(E)) = 0, which implies that E is Lebesgue measurable and m(E) = κ(E).

To clarify further the relationship between Lebesgue measure and the approximation process leading to Jordan content, we establish the following lemma. (The second part of the lemma will be used later.)

2.43 Lemma. If U ⊂ n is open, then U = A(U). Moreover, U is a countable union of cubes with disjoint interiors.

Proof. If x U, let δ = inf{|y − x| : y U}, which is positive since U is open. If Q is a cube in k that contains x, then every y Q is at a distance at most 2−k from x (the worst case being when |xj − yj| = 2−k for all j), so we will have Q ⊂ U provided k is large enough so that . But then x A(U, k) ⊂ A(U).

This shows that A(U) = U, and the second assertion follows by writing A(U) = A(U, 0) ∪ ∪∞₁[A(U, k) \ A(U, k − 1)]. A(U, 0) is a (countable) union of cubes in , and for k ≥ 1, the closure of A(U, k) \ A(U, k − 1) is a (countable) union of cubes in . These cubes all have disjoint interiors, and the result follows.

Lemma 2.43 immediately implies that the Lebesgue measure of any open set is equal to its inner content. On the other hand, suppose that F ⊂ n is compact. We can find a large cube, say Q₀ = {x : max|xj| ≤ 2M}, whose interior int(Q₀) contains F. If Q and Q ⊂ Q₀ then either Q ∩ F ≠ Ø or Q ⊂ (Q₀ \ F), so m((F, k)) + m(A(Q₀ \ F, k)) = m(Q₀). Letting k → ∞, we see that κ(F) + κ(Q₀ \ F) = m(Q₀). But Q₀ \ F is the union of the open set int(Q₀) \ F and the boundary of Q₀, which has content zero, so that κ(Q₀ \ F) = κ(int(Q₀) \ F) = m(Q₀ \ F). It follows that the Lebesgue measure of any compact set is equal to its outer content.

Combining these results with Theorem 2.40a, we can see exactly how Lebesgue measure compares to Jordan content. The Jordan content of E is defined by approximating E from the inside and the outside by finite unions of cubes. The Lebesgue measure of E, on the other hand, is given by a two-step approximation process: First one approximates E from the outside by open sets and from the inside by compact sets, and then approximates the open sets from the inside and the compact sets from the outside by finite unions of cubes. The Lebesgue measurable sets are precisely those for which these outer-inner and inner-outer approximations give the same answer in the limit. (Cf. Exercise 19 in §1.4.)

We now investigate the behavior of the Lebesgue integral under linear transformations. We identify a linear map T : n → n with the matrix (Tij) = (ei · Tej) where {ej} is the standard basis for n. We denote the determinant of this matrix by det T and recall that det(T S) = (det T)(det S). Furthermore, we employ the standard notation GL(n, ) (the “general linear” group) for the group of invertible linear transformations of n. We shall need the fact from elementary linear algebra that every T GL(n, ) can be written as the product of finitely many transformations of three “elementary” types. The first type multiplies one coordinate by a nonzero constant c and leaves the others fixed; the second type adds a multiple of one coordinate to some other coordinate and leaves all but the latter fixed; the third type interchanges two coordinates and leaves the others fixed. In symbols:

That every invertible transformation is a product of transformations of these three types is simply the fact that every nonsingular matrix can be row-reduced to the identity matrix.

2.44 Theorem. Suppose T GL(n, ).

a. If f is a Lebesgue measurable Junction on n, so is f T. If f ≥ 0 or f L¹(m), then

(2.45) equation

b. If E , then T(E) and m(T(E)) = |det T|m(E).

Proof. First suppose that f is Borel measurable. Then f T is Borel measurable since T is continuous. If (2.45) is true for the transformations T and S, it is also true for T S, since

Hence it suffices to prove (2.45) when T is of the types T₁, T₂, T₃ described above. But this is a simple consequence of the Fubini-Tonelli theorem. For T₃ we interchange the order of integration in the variables xj and xk, and for T₁ and T₂ we integrate first with respect to xj and use the one-dimensional formulas

which follow from Theorem 1.21. Since it is easily verified that det T₁ = c, det T₂ = 1, and det T3 = −1, (2.45) is proved. Moreover, if E is a Borel set, so is T(E) (since T⁻¹ is continuous), and by taking f = χT(E), we obtain m(T(E)) = |det T|m(E). In particular, the class of Borel null sets is invariant under T and T⁻¹, and hence so is n. The result for Lebesgue measurable functions and sets now follows as in the proof of Theorem 2.42.

2.46 Corollary. Lebesgue measure is invariant under rotations.

Proof. Rotations are linear maps satisfying TT* = I where T* is the transpose of T. Since det T = det T*, this condition implies that | det T| = 1.

Next we shall generalize Theorem 2.44 to differentiable maps. This result will not be used elsewhere in this book and may be omitted on a first reading. We shall prove a generalization of it, by somewhat different methods, in §11.2.

Let G = (g₁,…, gn) be a map from an open set Ω ⊂ n into n whose components gj are of class C¹, i.e., have continuous first-order partial derivatives. We denote by DxG the linear map defined by the matrix ((∂gi/∂xj)(x)) of partial derivatives at x. (Observe that if G is linear, then DxG = G for all x.) G is called a C¹ diffeomorphism if G is injective and DxG is invertible for all x Ω. In this case, the inverse function theorem guarantees that G⁻¹ : G(Ω) → Ω is also a C¹ diffeomorphism and that Dx(G⁻¹) = [DG-1(x)G]⁻¹ for all x G(Ω).

2.47 Theorem. Suppose that Ω is an open set in n and G : Ω → n is a C¹ diffeomorphism.

a. If f is a Lebesgue measurable function on G(Ω), then f G is Lebesgue measurable on Ω. If f ≥ 0 or f L¹(G(Ω), m), then

b. If E ⊂ Ω and E n, then G(E) n and m(G(E)) = ∫E | det DxG| dx.

Proof. It suffices to consider Borel measurable functions and sets. Since G and G⁻¹ are both continuous, there are no measurability problems in this case, and the general case follows as in the proof of Theorem 2.42.

A bit of notation: For x n and T = (Tij) GL(n, ), we set

We then have ||Tx|| ≤ ||T|| ||x||, and {x : ||x − a|| ≤ h} is the cube of side length 2h centered at a.

Let Q be a cube in Ω, say Q = {x : ||x − a|| ≤ h}. By the mean value theorem, gj(x) − gj(a) = ∑j(xj − aj)(∂g/∂xj) (y) for some y on the line segment joning x and a, so that for x Q, ||G(x) − G(a)|| ≤ h(supyQ ||DyG||). In other words, G(Q) is contained in a cube of side length supyQ ||DyG|| times that of Q, so that by Theorem 2.44, m(G(Q)) ≤ (supyQ ||DyG||)nm(Q). If T GL(n, ), we can apply this formula with G replaced by T⁻¹ G together with Theorem 2.44 to obtain

(2.48) equation

Since DyG is continuous in y, for any > 0 we can choose δ > 0 so that ||(DzG)⁻¹DyG||n ≤ 1 + if y, z Q and ||y − z|| ≤ δ. Let us now subdivide Q into subcubes Q₁,… QN whose interiors are disjoint, whose side lengths are at most δ, and whose centers are x₁,… xN. Applying (2.48) with Q replaced by Qj and with T = Dxj G, we obtain

This last sum is the integral of ∑N₁ | det DxjG|χQj (x), which tends uniformly on Q to | det DxG| as δ → 0 since DxG is continuous. Thus, letting δ → 0 and → 0, we find that

We claim that this estimate holds with Q replaced by any Borel set in Ω. Indeed, if U ⊂ Ω is open, by Lemma 2.43 we can write U = ∪∞₁ Qj where the Qj’s are cubes with disjoint interiors. Since the boundaries of the cubes have Lebesgue measure zero, we have

Next, let WK = Ω ∩ {x : |x| < K and | det DxG| < K}. If E is a Borel subset of WK, by Theorem 2.40 there is a decreasing sequence of open sets Uj ⊂ WK+1 such that E ⊂ ∪∞₁ Uj and m(∩∞₁ Uj \ E) = 0. By the preceding estimate and the dominated convergence theorem,

Finally, if E is any Borel subset of Ω, we apply this argument to E ∩ WK, let K → ∞, and conclude via the monotone convergence theorem that m(G(E)) ≤ ∫E|det DxG| dx.

If f = ∑ aj χAj is a nonnegative simple function on G(Ω), we therefore have

Theorem 2.10 and the monotone convergence theorem then imply that

for any nonnegative measurable f. But the same reasoning applies with G replaced by G⁻¹ and f replaced by f G, so that

This establishes (a) for f ≥ 0, and the case f L¹ follows immediately. Since (b) is just the special case of (a) where f = χG(E), the proof is complete.

Exercises

53. Fill in the details of the proof of Theorem 2.41.

54. How much of Theorem 2.44 remains valid if T is not invertible?

55. Let E = [0, 1] × [0, 1]. Investigate the existence and equality of ∫E f dm², ∫¹₀ ∫¹₀ f(x, y) dx dy, and ∫¹₀ ∫¹₀ f(x, y) dy dx for the following f.

a. f(x, y) = (x² − y²)(x² + y²)⁻².

b. f(x, y) = (1 − xy)−a (a > 0).

c. f(x, y) = (x − 1/2)⁻³ if 0 < y < |x − 1/2|, f(x, y) = 0 otherwise.

56. If f is Lebesgue integrable on (0, a) and g(x) = ∫ax t⁻¹ f(t)dt, then g is integrable on (0, a) and ∫a₀ g(x) dx = ∫a₀ f(x) dx.

57. Show that ∫∞₀ e−sxx⁻¹ sin x dx = arctan(s⁻¹) for s > 0 by integrating e-sxy sin x with aspect to x and y. (It may be useful to recall that tan(π/2 − θ) = (tan θ)⁻¹. Cf. Exercise 31d.)

58. Show that ∫ e−sxx⁻¹ sin² x dx = 1/4 log(1 + 4s⁻²) for s > 0 by integrating e−sx sin 2xy with respect to x and y.

59. Let f(x) = x⁻¹ sin x.

a. Show that ∫∞₀ |f(x)|dx = ∞.

b. Show that limb→∞ ∫b₀ f(x) dx = π by integrating e−xy sin x with respect to x and y. (In view of part (a), some care is needed in passing to the limit as b → ∞.)

60. Γ(x)Γ(y)/Γ(x + y) = ∫¹₀ tx−1 (1 − t)y − 1 dt for x, y > 0. (Recall that Γ was defined in §2.3. Write Γ(x)Γ(y) as a double integral and use the argument of the exponential as a new variable of integration.)

61. If f is continuous on [0, ∞), for α > 0 and x ≥ 0 let

Iα f is called the αth fractional integral of f.

a. Iα+β f = Iα(Iβ f) for all α, β > 0. (Use Exercise 60.)

b. If n , In f is an nth-order antiderivative of f.

2.7 INTEGRATION IN POLAR COORDINATES

The most important nonlinear coordinate systems in ² and ³ are polar coordinates (x = r cos θ, y = r sin θ) and spherical coordinates (x = r sin ϕ cos θ, y = r sin ϕ sin θ, z = r cos ϕ). Theorem 2.47, applied to these coordinates, yields the familiar formulas (loosely stated) dx dy = r dr dθ and dx dy dz = r² sin ϕ dr dθ dϕ. Similar coordinate systems exist in higher dimensions, but they become increasingly complicated as the dimension increases. (See Exercise 65.) For most purposes, however, it is sufficient to know that Lebesgue measure is effectively the product of the measure rn−1 dr on (0, ∞) and a certain “surface measure” on the unit sphere (dθ for n = 2, sin ϕ dθ dϕ for n = 3).

Our construction of this surface measure is motivated by a familiar fact from plane geometry. Namely, if Sθ is a sector of a disc of radius r with central angle θ (i.e., the region in the disc contained between the two sides of the angle), the area m(Sθ) is proportional to θ; in fact, m(Sθ) = r²θ. This equation can be solved for θ and hence used to define the angular measure θ in terms of the area m(Sθ). The same idea works in higher dimensions: We shall define the surface measure of a subset of the unit sphere in terms of the Lebesgue measure of the corresponding sector of the unit ball.

We shall denote the unit sphere {x n : |x| = 1} by Sn−1. If x n \ {0}, the polar coordinates of x are

The map Φ(x) = (r, x’) is a continuous bijection from n \ {0} to (0, ∞) × Sn−1 whose (continuous) inverse is Φ⁻¹ (r, x’) = rx’. We denote by m* the Borel measure on (0, ∞) × Sn−1 induced by Φ from Lebesgue measure on n, that is, m*(E) = m(Φ⁻¹(E)). Moreover, we define the measure ρ = ρn on (0, ∞) by ρ(E) = ∫E rn−1 dr.

2.49 Theorem. There is a unique Borel measure σ = σn−1 on Sn−1 such that m* = ρ × σ. If f is Borel measurable on n and f ≥ 0 or f L¹ (m), then

(2.50) equation

Proof. Equation (2.50), when f is a characteristic function of a set, is merely a restatement of the equation m* = ρ × σ, and it follows for general f by the usual linearity and approximation arguments. Hence we need only to construct σ.

If E is a Borel set in Sn−1, for a > 0 let

If (2.50) is to hold when f = χE₁, we must have

We therefore define σ(E) to be n · m(E₁). Since the map E → E₁ takes Borel sets to Borel sets and commutes with unions, intersections, and complements, it is clear that σ is a Borel measure on Sn−1. Also, since Ea is the image of E₁ under the map x → ax, it follows from Theorem 2.44 that m(Ea) = anm(E₁), and hence, if 0 < a < b,

Fix E Sn−1 and let E be the collection of finite disjoint unions of sets of the form (a, b] × E. By Proposition 1.7, E is an algebra on (0, ∞) × E that generates the σ-algebra E = {A × E : A (0, ∞)}. By the preceding calculation we have m* = ρ × σ on E, and hence by the uniqueness assertion of Theorem 1.14, m* = ρ × σ on E. But ∪{E : E Sn−1} is precisely the set of Borel rectangles in (0, ∞) × Sn−1, so another application of the uniqueness theorem shows that m* = ρ × σ on all Borel sets.

Of course, (2.50) can be extended to Lebesgue measurable functions by considering the completion of the measure σ. Details are left to the reader.

2.51 Corollary. If f is a measurable function on n, nonnegative or integrable, such that f(x) = g(|x|) for some function g on (0, ∞), then

2.52 Corollary. Let c and C denote positive constants, and let B = {x n : |x| < c}. Suppose that f is a measurable Junction on n.

a. If |f(x)| ≤ C|x|−a on B for some a < n, then f L¹(B). However, if |f(x)| ≥ C|x|−n on B, then f L¹(B).

b. If|f(x)| ≤ C|x|−a on Bc for some a > n, then f L¹(Bc). However, if |f(x)| ≥ C|x|−n on Bc, then f L¹(Bc).

Proof. Apply Corollary 2.51 to |x|−a χB and |x|−a χBc.

We shall compute σ(Sn−1) shortly. Of course, we know that σ(S¹) = 2π; this is just the definition of 2π as the ratio of the circumference of a circle to its radius. Armed with this fact, we can compute a very important integral.

2.53 Proposition. If a > 0,

Proof. Denote the integral on the left by In. For n = 2, by Corollary 2.51 we have

Since exp(−a|x|²) = Πn₁ exp(−ax²j), Tonelli’s theorem implies that In = (I₁)n. In particular, I₁ = (I₂)1/2, so In = (I₂)n/2 = (π/a)n/2.

Once we know this result, the device used in its proof can be turned around to compute σ(Sn−1) for all n in terms of the gamma function introduced in §2.3.

2.54 Proposition. .

Proof. By Corollary 2.51, Proposition 2.53, and the substitution s = r²,

2.55 Corollary. If Bn = {x n : |x| < 1}, then m(Bn) = .

Proof. m(Bn) = n⁻¹ σ(Sn−1) by definition of σ, and by the functional equation for the gamma function.

We observed in §2.3 that Γ(n) = (n − 1)!. Now we can also evaluate the gamma function at the half-integers:

2.56 Proposition. .

Proof. We have by the functional equation, and by Proposition 2.53 and the substitution s = r²,

An amusing consequence of Proposition 2.56 and the formula Γ(n) = (n − 1)! is that the surface measure of the unit sphere and the Lebesgue measure of the unit ball in n are always rational multiples of integer powers of π, and the power of π increases by 1 when n increases by 2.

Exercises

62. The measure σ on Sn−1 is invariant under rotations.

63. The technique used to prove Proposition 2.54 can also be used to integrate any polynomial over Sn−1. In fact, suppose is a monomial. Then ∫ f dσ = 0 if any αj is odd, and if all αj’s are even,

64. For which real values of a and b is |x|a|log|x| |b integrable over {x n : |x| < 1/2}? Over {x n : |x| > 2}?

65. Define G : n → n by G(r, ϕ₁,…, ϕn−2, θ) = (x₁,…, xn) where

a. G maps n onto n, and |G(r, ϕ₁,…, ϕn−2, θ)| = |r|.

b. det D(r, ϕ₁,…, ϕn − 2, θ) G = rn−1 sinn−2 ϕ₁ sinn−3 ϕ₂ · · · sin ϕn−2.

c. Let Ω = (0, ∞) × (0, π)n−2 × (0, 2π). Then G|Ω is a diffeomorphism and m(n \ G(Ω)) = 0.

d. Let F(ϕ₁,…, ϕn−2, θ) = G(1, ϕ₁,…, ϕn−2, θ) and Ω’ = (0, π)n−2 × (0, 2π). Then (F|Ω’)⁻¹ defines a coordinate system on Sn−1 except on a σ-null set, and the measure σ is given in these coordinates by

2.8 NOTES AND REFERENCES

The history of modern measure and integration theory can fairly be said to have begun with the publication of Lebesgue’s thesis [91] in 1902, although of course Lebesgue was building on earlier works of other mathematicians, and some of his results were obtained independently by Vitali and W. H. Young. The theory of the Lebesgue integral was extensively developed by a number of mathematicians in the ensuing decade, during which time most of the results in this chapter were first derived. In particular, Lebesgue himself proved the dominated convergence theorem and deduced the monotone convergence theorem from it in the case when the limit function f is integrable; when ∫ f = ∞ the latter theorem is due to B. Levi.

Lebesgue [92] studied more general measures on n (which he called “additive set functions”) in connection with the problem of generalizing the notion of indefinite integrals to functions of several variables. Radon [111] then developed the theory of integration with respect to what we now call regular Borel measures on n, which in particular yields the Lebesgue-Stieltjes integrals when n = 1. Finally, in 1915 Fréchet [53] pointed out that many of Radon’s ideas would work in the general setting of sets equipped with σ-algebras. Thus was abstract measure and integration theory born. It continued to develop until, by about 1950, it had assumed more or less the form in which we know it today. The first systematic modern treatise on the subject is Halmos [61].

For accounts of the prehistory and early history of the Lebesgue integral, see Hawkins [70]. References concerning the later development of the subject can be found in Saks [128].

We have adopted the point of view of beginning with measures and deriving integrals from them. However, it is also possible to go the other way, a procedure first developed by Daniell [29]. Roughly speaking, one starts with an “elementary integral”: a linear functional I defined on a suitable space of functions that satisfies some mild continuity conditions and is positive in the sense that I(f) ≥ 0 whenever f ≥ 0 (for example, the Riemann integral on the space of continuous functions on [a, b]). The Daniell theory provides an extension of I to a functional defined on a larger class of functions. Under appropriate hypotheses, the collection of sets E such that χE is in the domain of is then a σ-algebra, the function μ(E) = (χE) is a measure on , and is integration with respect to μ. See Royden [121] for a concise account of the Daniell theory and Pfeffer [108] for a comprehensive treatment, as well as König [86] for a somewhat different approach.

The Lebesgue theory is not the last word regarding integration on . Motivated partly by the problem of establishing the fundamental theorem of calculus in the greatest possible generality (about which we shall say more in §3.6), a number of theories of integration have been developed that include not only the Lebesgue integral but also certain “conditionally convergent” integrals. That is, they assign a meaning to ∫ f(x) dx for certain measurable functions f : → such that ∫ f⁺ = ∫ f⁻ = ∞, but for which the cancellation of positive and negative values in some way yields a reasonable definition of ∫ f(x) dx. (A standard example is f(x) = x⁻¹ sin x; see Exercise 59.) The first procedures for defining such integrals, due to Denjoy and Perron, were quite complicated. However, in the late 1950s, Henstock and Kurzweil independently discovered a modification of the classical Riemann integral that yields the same results.

The Henstock-Kurzweil integral on a bounded interval [a, b] is defined as follows. A tagged partition of [a, b] is a finite sequence {xj}N₀ such that a = x₀ < · · · < xN = b (i.e., a partition in the sense of §2.3) together with another finite sequence {tj}N₁ such that tj [xj − 1, xj]. A gauge on [a, b] is an (arbitrary!) function δ : [a, b] → (0, ∞). If P is a tagged partition and δ is a gauge, P is called δ-fine if xj − xj−1 < δ(tj) for all j. The compactness of [a, b] easily implies that for any gauge δ there is a δ-fine tagged partition of [a, b].

Now suppose f is a real-valued function on [a, b]. If P is a tagged partition of [a, b], the corresponding Riemann sum for f is ∑P f = ∑N₁ f(tj)(xj − xj−1). The function f is called Henstock-Kurzweil integrable on [a, b] if there exists c with the following property: For any > 0 there is a gauge δ such that if P is any δ-fine tagged partition of [a, b], then |∑P f − c| < . In this case the number c is unique, and it is called the Henstock-Kurzweil integral of f. The ordinary Riemann integral of f, in contrast, can be defined in exactly the same way except that one allows only constant gauges.

It turns out that the Henstock-Kurzweil integral coincides with the integrals of Denjoy and Perron. In particular, it coincides with the Lebesgue integral for nonnegative functions, but its domain includes many functions that have both positive and negative values and are not in L¹([a, b]). The definition of the Henstock-Kurzweil integral is easily extended to unbounded intervals. It also admits an n-dimensional version: One simply defines an n-interval to be a product of n one-dimensional intervals and a tagged partition of an n-interval I to be a finite collection {Ij} of n-intervals with disjoint interiors whose union is I together with a choice of tj Ij for each j; the definition of the integral then proceeds as above.

A good case can be made that the Henstock-Kurzweil integral ought to be the theory of integration on n that is generally taught to students, not just because of its added generality but (more cogently) because its definition is relatively simple and requires no measure theory to get started. On the other hand, it does not generalize as readily to spaces other than n, and although it can be developed in a rather abstract setting, it loses much of its appealing simplicity there. Moreover, although conditionally convergent integrals that cannot be obtained by a simple limiting procedure from absolutely convergent ones do turn up now and then in certain problems, their utility is not sufficently broad to make a compelling case for their study by nonspecialists.

In any case, in this book we shall content ourselves with the Lebesgue integral and the general theory of measure and integration of which it is a part. Readers who wish to learn more about the Henstock-Kurzweil integral can find a brief introduction in Bartle [13] and detailed treatments in McLeod [99] and Pfeffer [109]. See also Gordon [57] for a comprehensive account of the Denjoy, Perron, and Henstock-Kurzweil integrals on [a, b], and Henstock [72] for a development of the theory in a more abstract setting.

§2.1: A Borel isomorphism between two measurable spaces (X, ) and (Y, ) is a bijection f : X → Y such that f⁻¹ is a bijection from to . Unlike the related notion of homeomorphism for topological spaces (see Chapter 4) and notions of isomorphism in various other categories, the notion of Borel isomorphism is of limited utility, because it is too easy for two spaces to be Borel isomorphic. That this is so is clearly indicated by the single major theorem in the subject, due to Kuratowski:

Suppose that (X, ) is Borel isomorphic to a Borel subset E of a complete separable metric space Y (equipped with the σ-algebra {F Y : F ⊂ E}). Then either X is countable and = (X), or (X, ) is Borel isomorphic to (, ).

A proof of this theorem, as well as much additional information about Borel sets, can be found in Srivastava [139].

There is a hierarchy of Borel measurable functions on a metric space that corresponds roughly to the hierarchy of Borel sets (open and closed, Fσ and Gδ, etc.). Namely, let B₀ be the space of all continuous functions, and for each countable ordinal α define Bα recursively as follows. If α has an immediate predecessor β, Bα is the set of all limits of pointwise convergent sequences in Bβ; otherwise, Bα = ∪β<α Bβ. Functions in Bα are said to be of Baire class α. For example, if f is everywhere differentiable on , f’ is of Baire class 1.

Exercise 11 is a result from Lebesgue’s first published paper. See Rudin [123] for a discussion of it.

§2.3: The blurring of the distinction between individual measurable functions and equivalence classes of functions defined by almost-every where equality is often convenient and rarely disastrous. The most common situations where some care is needed involve the interplay of measurable and continuous functions (on n, say), for a function that is equal a.e. to a continuous function will not be continuous in general. See Zaanen [165] for a careful discussion of this point.

§2.4: An interesting discussion of Egoroff’s theorem, including some necessary and sufficient conditions for almost uniform convergence, can be found in Bartle [12]. For a simple proof of Lusin’s theorem (Exercise 44) that does not depend on Egoroff’s theorem, see Feldman [43]. We shall prove a more general form of this theorem in §7.2.

§2.5: The original theorems of Fubini and Tonelli pertained to Lebesgue measure in the plane. The theory of abstract product measures was developed independently by several people in the 1930s; the construction of μ × v presented here is that of Hahn [60]. It is also possible to define a product measure on the product of an infinite family {(Xα, α, μα)}αA of measure spaces provided that μα(Xα) = 1 for all but finitely many α; see Saeki [127], Halmos [61, §38], or Hewitt and Stromberg [76, §22]. We shall present a version of this result in §7.4 (Theorem 7.28).

Using the axiom of choice but not the continuum hypothesis, Sierpiński [134] has proved the existence of a Lebesgue nonmeasurable subset of ² whose intersection with any straight line contains at most two points. This should be compared with Exercise 47 (which is also due to Sierpiński).

The following generalization of the notion of product measures is useful in a number of situations: One is given a measurable space (X, ), a σ-finite measure space (Y, , v), and a family {μy : y Y} of finite measures on X such that the function y → μy (E) is measurable on Y for each E . One can then define a measure λ on X × Y such that ∫ f dλ = ∫∫ f(x, y) dμy(x) dv(y) for f L⁺(X × Y). See Johnson [79].

§2.6: Our proof of Theorem 2.47 follows J. Schwartz [131]. This theorem can also be proved under slightly weaker hypotheses on the transformation G; see Rudin [125, Theorem 7.26].

Chapter 3

Signed Measures and Differentiation

The principal theme of this chapter is the concept of differentiating a measure v with respect to another measure μ on the same σ-algebra. We do this first on the abstract level, then obtain a more refined result when μ is Lebesgue measure on n. When the latter is specialized to the case n = 1, it joins with classical real-variable theory to produce a version of the fundamental theorem of calculus for Lebesgue integrals.

In developing this program it is useful to generalize the notion of measure so as to allow measures to assume negative or even complex values. There are three reasons for this. First, in applications such “signed measures” can represent things such as electric charge that can be either positive or negative. Second, the differentiation theory proceeds more naturally in the more general setting. Finally, complex measures have a functional-analytic significance that will be explained in Chapter 7.

3.1 SIGNED MEASURES

Let (X, ) be a measurable space. A signed measure on (X, ) is a function v : → [−∞, ∞] such that

v(Ø) = 0;

v assumes at most one of the values ±∞

if {Ej} is a sequence of disjoint sets in , then v(∪∞₁ Ej) = ∑∞₁v(Ej), where the latter sum converges absolutely if v(∪∞₁) is finite.

Thus every measure is a signed measure; for emphasis we shall sometimes refer to measures as positive measures.

Two examples of signed measures come readily to mind. First, if μ₁, μ₂, are measures on and at least one of them is finite, then v = μ₁ − μ₂ is a signed measure. Second, if μ is a measure on and f : X → [−∞, ∞] is a measurable function such that at least one of ∫ f⁺ dμ, and ∫ f⁻ dμ is finite (in which case we shall call f an extended μ-integrable function), then the set function v defined by v(E) = ∫E f dμ is a signed measure. In fact, we shall see shortly that these are really the only examples: Every signed measure can be represented in either of these two forms.

3.1 Proposition. Let v be a signed measure on (X, ). If{Ej} is an increasing sequence in , then v(∪∞₁ Ej) = limj→∞ v(Ej). If{Ej} is a decreasing sequence in and v(E₁) is finite, then v(∩∞₁ Ej) = limj→∞ v(Ej).

The proof is essentially the same as for positive measures (Theorem 1.8) and is left to the reader (Exercise 1).

If v is a signed measure on (X, ), a set E is called positive (resp. negative, null) for v if v(F) ≥ 0 (resp. v(F) ≤ 0, v(F) = 0) for all F such that F ⊂ E. (Thus, in the example v(E) = ∫E f dμ described above, E is positive, negative, or null precisely when f ≥ 0, f ≤ 0, or f = 0 μ-a.e. on E.)

3.2 Lemma. Any measurable subset of a positive set is positive, and the union of any countable family of positive sets is positive.

Proof. The first assertion is obvious from the definition of positivity. If P₁, P₂,… are positive sets, let Qn = Pn \ ∪n−1₁ Pj. Then Qn ⊂ Pn, so Qn is positive. Hence if E ⊂ ∪∞₁ Pj, then v(E) = ∑∞₁ v(E ∩ Qj) ≥ 0, as desired.

3.3 The Hahn Decomposition Theorem. If v is a signed measure on (X, ), there exist a positive set P and a negative set N for v such that P ∪ N = X and P ∩ N = Ø. If P’, N’ is another such pair, then P Δ P’ (= N Δ N′) is null for v.

Proof. Without loss of generality, we assume that v does not assume the value +∞. (Otherwise, consider −v.) Let m be the supremum of v(E) as E ranges over all positive sets; thus there is a sequence {Pj} of positive sets such that v(Pj) → m. Let P = ∪∞₁ Pj. By Lemma 3.2 and Proposition 3.1, P is positive and v(P) = m; in particular, m < ∞. We claim that N = X \ P is negative. To this end, we assume that N is not negative and derive a contradiction.

First, notice that N cannot contain any nonnull positive sets. Indeed, if E ⊂ N is positive and v(E) > 0, then E ∪ P is positive and v(E ∪ P) = v(E) + v(P) > m, which is impossible.

Second, if A ⊂ N and v(A) > 0, there exists B ⊂ A with v(B) > v(A). Indeed, since A cannot be positive, there exists C ⊂ A with v(C) < 0; thus if B = A \ C we have v(B) = v(A) − v(C) > v(A).

If N is not negative, then, we can specify a sequence of subsets {Aj} of N and a sequence {nj} of positive integers as follows: n₁ is the smallest integer for which there exists a set B ⊂ N with v(B) > n⁻¹₁, and A₁ is such a set. Proceeding inductively, nj is the smallest integer for which there exists a set B ⊂ Aj−1 with v(B) > v(Aj−1) + n⁻¹j, and Aj is such a set.

Let A = ∩∞₁ Aj. Then ∞ > v(A) = limv(Aj) > ∑∞₁ n⁻¹j, so nj → ∞ as j → ∞. But once again, there exists B ⊂ A with v(B) > v(A) + n⁻¹ for some integer n. For j sufficiently large we have n < nj, and B ⊂ Aj−1, which contradicts the construction of nj and Aj. Thus the assumption that N is not negative is untenable.

Finally, if P’, N′ is another pair of sets as in the statement of the theorem, we have P \ P’ ⊂ P and P \ P’ ⊂ N’, so that P \ P’ is both positive and negative, hence null; likewise for P’ \ P.

The decomposition X = P ∪ N of X as the disjoint union of a positive set and a negative set is called a Hahn decomposition for v. It is usually not unique (v-null sets can be transferred from P to N or from N to P), but it leads to a canonical representation of v as the difference of two positive measures.

To state this result we need a new concept: We say that two signed measures μ and v on (X, ) are mutually singular, or that v is singular with respect to μ, or vice versa, if there exist E, F such that E ∩ F = Ø, E ∪ F = X, E is null for μ, and F is null for v. Informally speaking, mutual singularity means that μ and v “live on disjoint sets.” We express this relationship symbolically with the perpendicularity sign:

3.4 The Jordan Decomposition Theorem. If v is a signed measure, there exist unique positive measures v⁺ and v⁻ such that v = v⁺ − v⁻ and v⁺ ⊥ v⁻.

Proof. Let X = P ∪ N be a Hahn decomposition for v, and define v⁺ (E) = v(E ∩ P) and v⁻(E) = −v(E ∩ N). Then clearly v = v⁺ − v⁻ and v⁺ ⊥ v⁻. If also v = μ⁺ − μ⁻ and μ⁺ ⊥ μ⁻, let E, F be such that E∩F = Ø, E∪F = X, and μ⁺(F) = μ⁻(E) = 0. Then X = E ∪ F is another Hahn decomposition for v, so PΔE is v-null. Therefore, for any A , μ⁺(A) = μ⁺(A ∩ E) = v(A ∩ E) = v(A ∩ P) = v⁺(A), and likewise v⁻ = μ⁻.

The measures v⁺ and v⁻ are called the positive and negative variations of v, and v = v⁺ − v⁻ is called the Jordan decomposition of v, by analogy with the representation of a function of bounded variation on as the difference of two increasing functions (see §3.5). Furthermore, we define the total variation of v to be the measure |v| defined by

It is easily verified that E is v-null iff |v|(E) = 0, and v ⊥ μ iff |v| ⊥ μ, iff v⁺ ⊥ μ and v⁻ ⊥ μ (Exercise 2.)

We observe that if v omits the value ∞ then v⁺ (X) = v(P) < ∞, so that v⁺ is a finite measure and v is bounded above by v⁺ (X); similarly if v omits the value −∞. In particular, if the range of v is contained in , then v is bounded. We observe also that v is of the form v(E) = ∫E f dμ, where μ = |v| and f = χP − χN, X = P ∪ N being a Hahn decomposition for v.

Integration with respect to a signed measure v is defined in the obvious way: We set

One more piece of terminology: a signed measure v is called finite (resp. σ-finite) if |v| is finite (resp. σ-finite).

Exercises

1. Prove Proposition 3.1.

2. If v is a signed measure, E is v-null iff |v|(E) = 0. Also, if v and μ are signed measures, v ⊥ μ iff |v| ⊥ μ iff v⁺ ⊥ μ and v⁻ ⊥ μ.

3. Let v be a signed measure on (X, ).

a. L¹ (v) = L¹ (|v|).

b. If f L¹ (v), |∫ f dv| ≤ ∫ |f|d|v|.

c. If E , |v|(E) = sup{|∫E f dv| : |f| ≤ 1}.

4. If v is a signed measure and λ, μ are positive measures such that v = λ − μ, then λ ≥ v⁺ and μ ≥ v⁻.

5. If v₁, v₂ are signed measures that both omit the value +∞ or −∞, then |v₁+v₂| ≤ |v₁| + |v₂|. (Use Exercise 4.)

6. Suppose v(E) = ∫ f dμ, where μ is a positive measure and f is an extended μ-integrable function. Describe the Hahn decompositions of v and the positive, negative, and total variations of v in terms of f and μ.

7. Suppose that v is a signed measure on (X, ) and E .

a. v⁺(E) = sup{v(F) : F , F ⊂ E} and v⁻(E) = −inf{v(F) : F , F ⊂ E}.

b. |v|(E) = sup{∑n₁ |v(Ej)| : n , E₁,…, En are disjoint, and ∪n₁ Ej = E}.

3.2 THE LEBESGUE-RADON-NIKODYM THEOREM

Suppose that v is a signed measure and μ, is a positive measure on (X, ). We say that v is absolutely continuous with respect to μ and write

if v(E) = 0 for every E for which μ(E) = 0. It is easily verified that v μ iff |v| μ iff v⁺ μ and v⁻ μ (Exercise 8). Absolute continuity is in a sense the antithesis of mutual singularity. More precisely, if v ⊥ μ and v μ, then v = 0, for if E and F are disjoint sets such that E ∪ F = X and μ(E) = |v|(F) = 0, then the fact that v μ implies that |v|(E) = 0, whence |v| = 0 and v = 0. One can extend the notion of absolute continuity to the case where μ is a signed measure (namely, v μ iff v |μ|), but we shall have no need of this more general definition.

The term “absolute continuity” is derived from real-variable theory; see §3.5. For finite signed measures it is equivalent to another condition that is obviously a form of continuity.

3.5 Theorem. Let v be a finite signed measure and μ a positive measure on (X, ). Then v μ iff for every > 0 there exists δ > 0 such that |v(E)| < whenever μ(E) < δ.

Proof. Since v μ iff |v| μ and |v(E)| ≤ |v|(E), it suffices to assume that v = |v| is positive. Clearly the -δ condition implies that v μ. On the other hand, if the -δ condition is not satisfied, there exists > 0 such that for all n we can find En with μ(En) < 2−n and v(En) > . Let Fk = ∪∞k En and F = ∩∞₁ Fk. Then μ(Fk) < ∑∞k 2−n = 21−k, so μ(F) = 0; but v(Fk) ≥ for all k and hence, since v is finite, v(F) = lim v(Fk) ≥ . Thus it is false that v μ.

If μ is a measure and f is an extended μ-integrable function, the signed measure v defined by v(E) = ∫Ef dμ is clearly absolutely continuous with respect to μ it is finite iff f L¹(μ). For any complex-valued f L¹(μ), the preceding theorem can be applied to Re f and Im f, and we obtain the following useful result:

3.6 Corollary. If f L¹(μ), for every > 0 there exists δ > 0 such that |∫E f dμ|, < whenever μ(E) < δ.

We shall use the following notation to express the relationship v(E) = ∫E f dμ:

Sometimes, by a slight abuse of language, we shall refer to “the signed measure f dμ.”

We now come to the main theorem of this section, which gives a complete picture of the structure of a signed measure relative to a given positive measure. First, a technical lemma.

3.7 Lemma. Suppose that v and μ are finite measures on (X, ). Either v ⊥ μ, or there exist > 0 and E such that μ(E) > 0 and v ≥ μ on E (that is, E is a positive set for v − μ).

Proof. Let X = Pn ∪ Nn be a Hahn decomposition for v − n⁻¹ μ, and let P = ∪∞₁ Pn and N = ∩∞₁ Nn = Pc. Then N is a negative set for v − n⁻¹ μ for all n, i.e., 0 ≤ v(N) ≤ n⁻¹ μ(N) for all n, so v(N) = 0. If μ(P) = 0, then v ⊥ μ. If μ(P) > 0, then μ(Pn) > 0 for some n, and Pn is a positive set for v − n⁻¹ μ.

3.8 The Lebesgue-Radon-Nikodym Theorem. Let v be a σ-finite signed measure and μ a σ-finite positive measure on (X, ). There exist unique σ-finite signed measures λ, ρ on (X, ) such that

Moreover, there is an extended μ-integrable function f : X → such that dρ = f dμ, and any two such functions are equal μ-a.e.

Proof. Case I: Suppose that v and μ are finite positive measures. Let

is nonempty since 0 . Also, if f, g , then h = max(f, g) , for if A = {x : f(x) > g(x)}, for any E we have

Let a = sup{∫ f dμ : f }, noting that a ≤ v(X) < ∞, and choose a sequence {fn} ⊂ such that ∫ fndμ → a. Let gn = max(f₁,…, fn) and f = supn fn. Then gn , gn increases pointwise to f, and ∫ gn dμ ≥ ∫ fn dμ. It follows that lim ∫ gn dμ = a and hence, by the monotone convergence theorem, that f and ∫ f dμ = a. (In particular, f < ∞ a.e., so we may take f to be real-valued everywhere.)

We claim that the measure dλ = dv − f dμ (which is positive since f ) is singular with respect to μ. If not, by Lemma 3.7 there exist E and > 0 such that μ(E) > 0 and λ ≥ μ on E. But then χE dμ ≤ dλ − dv − f dμ, that is, (f + χE) dμ ≤ dv, so f + χE and ∫(f + χE) dμ = a + μ(E) > a, contradicting the definition of a.

Thus the existence of λ, f, and dρ = f dμ is proved. As for uniqueness, if also dv = dλ’ + f’dμ, we have dλ − dλ’ = (f’ − f) dμ. But λ − λ’ ⊥ μ (see Exercise 9), while (f’ − f) dμ dμ hence dλ − dλ’ = (f’ − f) dμ = 0, so that λ = λ’ and (by Proposition 2.23) f = f’ μ-a.e. Thus we are done in the case when μ and v are finite measures.

Case II: Suppose that μ and v are σ-finite measures. Then X is a countable disjoint union of μ-finite sets and a countable disjoint union of v-finite sets; by taking intersections of these we obtain a disjoint sequence {Aj} ⊂ such that μ(Aj) and v(Aj) are finite for all j and X = ∪∞₁ Aj. Define μj (E) = μ(E ∩ Aj) and vj(E) = v(E ∩ Aj). By the reasoning above, for each j we have dvj = dλj + fj dμj where λj ⊥ μj. Since μj (Acj) = vj(Acj) = 0, we have λj(Acj) = vj(Acj) − ∫Acj f dμj = 0, and we may assume that fj = 0 on Acj. Let λ = ∑∞₁ λj and f = ∑∞₁ fj. Then dv = dλ + f dμ, λ ⊥ μ (see Exercise 9), and dλ and f dμ are σ-finite, as desired. Uniqueness follows as before.

The General Case: If v is a signed measure, we apply the preceding argument to v⁺ and v⁻ and subtract the results.

The decomposition v = λ + ρ where λ ⊥ μ and ρ μ is called the Lebesgue decomposition of v with respect to μ. In the case where v μ, Theorem 3.8 says that dv = f dμ, for some f. This result is usually known as the Radon-Nikodym theorem, and f is called the Radon-Nikodym derivative of v with respect to μ. We denote it by dv/dμ:

(Strictly speaking, dv/dμ should be construed as the class of functions equal to f μ-a.e.) The formulas suggested by the differential notation dv/dμ are generally correct. For example, it is obvious that d(v₁ + v₂)/dμ = (dv₁/dμ) + (dv₂/dμ), and we have the chain rule:

3.9 Proposition. Suppose that v is a σ-finite signed measure and μ, λ are σ-finite measures on (X, ) such that v μ and μ λ.

a. If g L¹(v), then g(dv/dμ) L¹(μ) and

b. We have v λ, and

Proof. By considering v⁺ and v⁻ separately, we may assume that v ≥ 0. The equation ∫ g dv = ∫ g (dv/dμ) dμ is true when g = χE by definition of dv/dμ. It is therefore true for simple functions by linearity, then for nonnegative measurable functions by the monotone convergence theorem, and finally for functions in L¹(v) by linearity again. Replacing v, μ by μ, λ and setting g = χE(dv/dμ), we obtain

for all E , whence (dv/dλ) = (dv/dμ)(dμ/dλ) λ-a.e. by Proposition 2.23.

3.10 Corollary. If μ λ and λ μ, then (dλ/dμ)(dμ/dλ) = 1 a.e. (with respect to either λ or μ).

Nonexample: Let μ be Lebesgue measure and v the point mass at 0 on (, ). Clearly v ⊥ μ. The nonexistent Radon-Nikodym derivative dv/dμ is popularly known as the Dirac δ-function.

We conclude this section with a simple but important observation:

3.11 Proposition. If μ₁,…, μn are measures on (X, ), there is a measure μ such that μj μ for all j − namely, μ = ∑n₁ μj.

The proof is trivial.

Exercises

8. v μ iff |v| μ iff v⁺ μ and v⁻ μ.

9. Suppose {vj} is a sequence of positive measures. If vj ⊥ μ for all j, then ∑∞₁ vj ⊥ μ and if vj μ for all j, then ∑∞₁ vj μ.

10. Theorem 3.5 may fail when v is not finite. (Consider dv(x) = dx/x and dμ(x) = dx on (0, 1), or v = counting measure and μ(E) = ∑nE 2−n on .)

11. Let μ be a positive measure. A collection of functions {fα}αA ⊂ L¹(μ) is called uniformly integrable if for every > 0 there exists δ > 0 such that |∫E fα dμ| < for all α A whenever μ(E) < δ.

a. Any finite subset of L¹ (μ) is uniformly integrable.

b. If {fn} is a sequence in L¹(μ) that converges in the L¹ metric to f L¹(μ), then {fn} is uniformly integrable.

12. For j = 1, 2, let μj, vj be σ-finite measures on (Xj, j) such that vj μj. Then v₁ x v₂ μ₁ x μ₂ and

13. Let X = [0, 1], = [0, 1], m = Lebesgue measure, and μ = counting measure on .

a. m μ but dm ≠ f dμ for any f.

b. μ has no Lebesgue decomposition with respect to m.

14. If v is an arbitrary signed measure and μ is a σ-finite measure on (X, ) such that v μ, there exists an extended μ-integrable function f : X → [−∞, ∞] such that dv = f dμ. Hints:

a. It suffices to assume that μ is finite and v is positive.

b. With these assumptions, there exists E that is σ-finite for v such that μ(E) ≥ μ(F) for all sets F that are σ-finite for v.

c. The Radon-Nikodym theorem applies on E. If F ∩ E = Ø, then either v(F) = μ(F) = 0 or μ(F) > 0 and |v(F)| = ∞.

15. A measure μ on (X, ) is called decomposable if there is a family ⊂ with the following properties: (i) μ(F) < ∞ for all F ; (ii) the members of are disjoint and their union is X; (iii) if μ(E) < ∞ then μ(E) = ∑F μ(E ∩ F);(iv) if E ⊂ X and E ∩ F for all F then E .

a. Every σ-finite measure is decomposable.

b. If μ is decomposable and v is any signed measure on (X, ) such that v μ, there exists a measurable f : X → [−∞, ∞] such that v(E) = ∫E f dμ for any E that is σ-finite for μ, and |f| < ∞ on any F that is σ-finite for v. (Use Exercise 14 if v is not σ-finite.)

16. Suppose that μ, v are σ-finite measures on (X, ) with v μ, and let λ = μ + v. If f = dv/dλ, then 0 ≤ f < 1 μ-a.e. and dv/dμ = f/(1 − f).

17. Let (X, , μ) be a finite measure space, a sub-σ-algebra of , and v = μ|. If f L¹(μ), there exists g L¹(v) (thus g is -measurable) such that ∫E f dμ = ∫E g dv for all E ; if g’ is another such function then g = g’ v-a.e. (In probability theory, g is called the conditional expectation of f on .)

3.3 COMPLEX MEASURES

A complex measure on a measurable space (X, ) is a map v : → such that

v(Ø) = 0;

if {Ej} is a sequence of disjoint sets in , then v(∪∞₁ Ej) = ∑∞₁ v(Ej), where the series converges absolutely.

In particular, infinite values are not allowed, so a positive measure is a complex measure only if it is finite. Example: If μ, is a positive measure and f L¹ (μ), then f dμ is a complex measure.

If v is a complex measure, we shall write vr and vi for the real and imaginary parts of v. Thus vr and vi are signed measures that do not assume the values ±∞ hence they are finite, and so the range of v is a bounded subset of .

The notions we have developed for signed measures generalize easily to complex measures. For example, we define L¹(v) to be L¹(vr) ∩ L¹(vi), and for f L¹(v), we set ∫ f dv = ∫ f dvr + i ∫ f dvi. If v and μ are complex measures, we say that v ⊥ μ if va ⊥ μb for a, b = r, i, and if λ is a positive measure, we say that v λ if vr λ and vi λ. The theorems of §3.2 also generalize; one has merely to apply them to the real and imaginary parts separately. In particular:

3.12 The Lebesgue-Radon-Nikodym Theorem. If v is a complex measure and μ is a σ-finite positive measure on (X, ), there exist a complex measure λ and an f L¹ (μ) such that λ ⊥ μ and dv = dΔ + f dμ. If also λ’ ⊥ μ and dv = dΔ’+ f dμ, then λ = λ’ and f = f’ μ-a.e.

As before, if v μ, we denote the f in Theorem 3.12 by dv/dμ.

The total variation of a complex measure v is the positive measure |v| determined by the property that if dv = f dμ where μ is a positive measure, then d|v| = |f|dμ. To see that this is well defined, we observe first that every v is of the form f dμ for some finite measure μ and some f L¹(μ); indeed, we can take μ = |vr| + |vi| and use Theorem 3.12 to obtain f. Second, if dv = f₁ dμ₁ = f₂ dμ₂, let ρ = μ₁ + μ₂. Then by Proposition 3.9,

so that f₁ (dμ₁/dρ) = f2 (dμ₂/dρ) ρ-a.e. Since dμj/dρ is nonnegative, we therefore have

and thus

Hence the definition of |v| is independent of the choice of μ and f. This definition agrees with the previous definition of |v| when v is a signed measure, for in that case dv = (χP − χN)d|v| where X = P ∪ N is a Hahn decomposition, and |χP−χN| = 1.

3.13 Proposition. Let v be a complex measure on (X, ).

a. |v(E)| ≤ |v|(E) for all E .

b. v |v|, and dv/d|v| has absolute value 1 |v|-a.e.

c. L¹(v) = L¹(|v|), and if f L¹(v), then |∫ f dv| ≤ ∫|f|d|v|.

Proof. Suppose dv = f dμ as in the definition of |v|. Then

This proves (a) and shows that v |v|. If g = dv/d|v|, then, we have f dμ = dv = g d|v| = g|f| dμ, so g|f| = f μ-a.e. and hence |v|-a.e. But clearly |f| > 0 |v|-a.e., whence |g| = 1 |v|-a.e. Part (c) is left to the reader (Exercise 18).

3.14 Proposition. If v₁, v₂ are complex measures on (X, ), then |v₁ + v₂| ≤ |v₁| + |v₂|.

Proof. By Proposition 3.11 we can write dvj = fj dμ, with the same μ, for j = 1, 2. But then d|v₁ + v₂| = |f₁ + f₂|dμ ≤ |f₁|dμ + |f₂| dμ = d|v₁| + d| + d|v₂|.

Exercises

18. Prove Proposition 3.13c.

19. If v, μ are complex measures and λ is a positive measure, then v ⊥ μ iff |v| ⊥ |μ|, and v λ iff |v| λ.

20. If v is a complex measure on (X, ) and v(X) = |v|(X), then v = |v|.

21. Let v be a complex measure on (X, ). If E , define

Then μ₁ = μ₂ = μ₃ = |v|. (First show that μ₁ ≤ μ₂ ≤ μ₃. To see that μ₃ = |v|, let and apply Proposition 3.13. To see that μ₃ ≤ μ₁, approximate f by simple functions.)

3.4 DIFFERENTIATION ON EUCLIDEAN SPACE

The Radon-Nikodym theorem provides an abstract notion of the “derivative” of a signed or complex measure v with respect to a measure μ. In this section we analyze more deeply the special case where (X, ) = (n, n) and μ = m is Lebesgue measure. Here one can define a pointwise derivative of v with respect to m in the following way. Let B(r, x) be the open ball of radius r about x in n; then one can consider the limit

when it exists. (One can also replace the balls B(r, x) by other sets which, in a suitable sense, shrink to x in a regular way; we shall examine this point later.) If v m, so that dv = f dm, then v(B(r, x))/m(B(r, x)) is simply the average value of f on B(r, x), so one would hope that F = f m-a.e. This turns out to be the case provided that v(B(r, x)) is finite for all r, x. From the point of view of the function f, this may be regarded as a generalization of the fundamental theorem of calculus: The derivative of the indefinite integral of f (namely, v) is f.

For the remainder of this section, terms such as “integrable” and “almost everywhere” refer to Lebesgue measure unless otherwise specified. We begin our analysis with a technical lemma that is of interest in its own right.

3.15 Lemma. Let be a collection of open balls in . If c < m(U), there exist disjoint B₁,…, such that ∑k₁ m(Bj) > 3−nc.

Proof. If c < m(U), by Theorem 2.40 there is a compact K ⊂ U with m(K) > c, and finitely many of the balls in — say, A₁,…, Am — cover K. Let B₁ be the largest of the Aj’s (that is, choose B₁ to have maximal radius), let B₂ be the largest of the Aj’s that are disjoint from B₁, B₃ the largest of the Aj’s that are disjoint from B₁ and B₂, and so on until the list of Aj’s is exhausted. According to this construction, if Ai is not one of the Bj’s, there is a j such that Ai ∩ Bj ≠ Ø, and if j is the smallest integer with this property, the radius of Ai is at most that of Bj. Hence Ai ⊂ B*j, where B*j is the ball concentric with Bj whose radius is three times that of Bj. But then K ⊂ ∪k₁ B*j, so

A measurable function f : n → is called locally integrable (with respect to Lebesgue measure) if ∫K |f(x)| dx < ∞ for every bounded measurable set K ⊂ n. We denote the space of locally integrable functions by L¹loc. If f L¹loc, x n, and r > 0, we define Arf(x) to be the average value of f on B(r, x):

3.16 Lemma. If f L¹loc, Arf(x) is jointly continuous in r and x(r > 0, x n).

Proof. From the results in §2.7 we know that m(B(r, x)) = crn where c = m(B(1, 0)), and m(S(r, x)) = 0 where S(r, x) = {y : |y − x| = r}. Moreover, as r → r₀ and x → x₀, χB(r, x) → χB(r₀, x₀) pointwise on n \ S(r₀, x₀). Hence χB(r, x) → χB(r₀, x₀) a.e., and if and . By the dominated convergence theorem, it follows that ∫B(r, x) f(y) dy is continuous in r and x, and hence so is Arf(x) = c⁻¹r−n ∫B(r, x) f(y) dy.

Next, if f L¹loc, we define its Hardy-Littlewood maximal function H f by

H f is measurable, for (H f)⁻¹((a, ∞)) = ∪r>0(Ar|f|)⁻¹((α, ∞)) is open for any a , by Lemma 3.16.

3.17 The Maximal Theorem. There is a constant C > 0 such that for all f L¹ and all α > 0,

Proof. Let Eα = {x : H f(x) > α}. For each x Eα we can choose rx > 0 such that Arx|f|(x) > α. The balls B(rx, x) cover Eα, so by Lemma 3.15, if c < m(Eα) there exist x₁,…, xk Eα such that the balls Bj = B(rxj, xj) are disjoint and ∑k₁ m(Bj) > 3−nc. But then

Letting c → m(Eα), we obtain the desired result.

With this tool in hand, we now present three successively sharper versions of the fundamental differentiation theorem. In the proofs we shall use the notion of limit superior for real-valued functions of a real variable,

and the easily verified fact that

3.18 Theorem. If f L¹loc, then limr→0 Arf(x) = f(x) for a.e. x n.

Proof. It suffices to show that for N , Arf(x) → f(x) for a.e. x with |x| ≤ N. But for |x| ≤ N and r ≤ 1 the values Arf(x) depend only on the values f(y) for |y| ≤ N + 1, so by replacing f with fχB(N+1,0) we may assume that f L¹.

Given > 0, by Theorem 2.41 we can find a continuous integrable function g such that ∫ ∫ |g(y) − f(y)|dy < . Continuity of g implies that for every x n and δ > 0 there exists r > 0 such that |g(y) − g(x)| < δ whenever |y − x| < r, and hence

Therefore Arg(x) → g(x) as r → 0 for every x, so

Hence, if

we have

But (α/2)m(Fα/2) ≤ ∫Fα/2 |f(x) − g(x)| dx < , so by the maximal theorem,

Since is arbitrary, m(Eα) = 0 for all α > 0. But limr→0 Arf(x) = f(x) for all x ∪∞₁ E1/n so we are done.

This result can be rephrased as follows: If f L¹loc,

(3.19) equation

Actually, something stronger is true: (3.19) remains valid if one replaces the integrand by its absolute value. That is, let us define the Lebesgue set Lf of f to be

3.20 Theorem. If f L¹loc then m((Lf)c) = 0.

Proof. For each c we can apply Theorem 3.18 to gc(x) = |f(x) − c| to conclude that, except on a Lebesgue null set Ec, we have

Let D be a countable dense subset of , and let E = ∪cD Ec. Then m(E) = 0, and if x E, for any > 0 we can choose c D with |f(x) − c| < , so that |f(y) − f(x)| < |f(y) − c| + , and it follows that

Since is arbitrary, the desired result follows.

Finally, we consider families of sets more general than balls. A family {Er>r0 of Borel subsets of n is said to shrink nicely to x n if

Er ⊂ B(r, x) for each r;

there is a constant α > 0, independent of r, such that m(Er) > αm(B(r, x)).

The sets Er need not contain x itself. For example, if U is any Borel subset of B(1, 0) such that m(U) > 0, and Er = {x + ry : y U}, then {Er} shrinks nicely to x. Here, then, is the final version of the differentiation theorem.

3.21 The Lebesgue Differentiation Theorem. Suppose f L¹loc. For every x in the Lebesgue set of f − in particular, for almost every x — we have

for every family {Er}r0 that shrinks nicely to x.

Proof. For some α > 0 we have

The first equality therefore follows from Theorem 3.20, and one sees immediately that it implies the second one by writing the latter in the form (3.19).

We now return to the study of measures. A Borel measure v on n will be called regular if

i. v(K) < ∞ for every compact K;

ii. v(E) = inf{v(U) : U open, E ⊂ U} for every E n.

(Condition (ii) is actually implied by condition (i). For n = 1 this follows from Theorems 1.16 and 1.18, and we shall prove it for arbitrary n in §7.2. For the time being, we assume (ii) explicitly.) We observe that by (i), every regular measure is σ-finite. A signed or complex Borel measure v will be called regular if |v| is regular.

For example, if f L⁺(n), the measure f dm is regular iff f L¹loc. Indeed, the condition f L¹loc is clearly equivalent to (i). If this holds, (ii) may be verified directly as follows. Suppose that E is a bounded Borel set. Given δ > 0, by Theorem 2.40 there is a bounded open U ⊃ E such that m(U) < m(E) + δ and hence m(U \ E) < δ. But then, given > 0, by Corollary 3.6 there is an open U ⊃ E such that ∫U\E f dm < and hence ∫U f dm < ∫E f dm + . The case of unbounded E follows easily by writing E = ∪∞₁ Ej where Ej is bounded and finding an open Uj ⊃ Ej such that ∫Uj\Ej f dm < 2−j.

3.22 Theorem. Let v be a regular signed or complex Borel measure on n, and let dv = dλ + f dm be its Lebesgue-Radon-Nikodym representation. Then for m-almost every x n,

for every family {Er}r0 that shrinks nicely to x.

Proof. It is easily verified that d|v| = d|λ| + |f| dm, so the regularity of v implies the regularity of both λ and f dm (Exercise 26). In particular, f L¹loc, so in view of Theorem 3.21, it suffices to show that if λ is regular and λ ⊥ m, then for m-almost every x, λ(Er)/m(Er) → 0 as r → 0 when Er shrinks nicely to x. It also suffices to take Er = B(r, x) and to assume that λ is positive, since for some α > 0 we have

Assuming λ ≥ 0, then, let A be a Borel set such that λ(A) = m(Ac) = 0, and let

We shall show that m(Fk) = 0 for all k, and this will complete the proof.

The argument is similar to the proof of the maximal theorem. By regularity of λ, given > 0 there is an open U ⊃ A such that λ(U) < . Each x Fk is the center of a ball Bx ⊂ U such that λ(Bx) > k⁻¹m(Bx). By Lemma 3.15, if V = ∪xFk Bx and c < m(V) there exist x₁,…, xJ such that Bx₁,…, BxJ are disjoint and

We conclude that m(V) ≤ 3nk, and since Fk ⊂ V and is arbitrary, m(Fk) = 0.

Exercises

22. If f L¹(n), f ≠ 0, there exist C, R > 0 such that H f(x) ≥ C|x|−n for |x| > R. Hence m({x : H f(x) > α}) ≥ C’/α when α is small, so the estimate in the maximal theorem is essentially sharp.

23. A useful variant of the Hardy-Littlewood maximal function is

Show that H f ≤ H* f ≤ 2nH f.

24. If f L¹loc and f is continuous at x, then x is in the Lebesgue set of f.

25. If E is a Borel set in n, the density DE(x) of E at x is defined as

whenever the limit exists.

a. Show that DE(x) = 1 for a.e. x E and DE(x) = 0 for a.e. x Ec.

b. Find examples of E and x such that DE(x) is a given number α (0,1), or such that DE(x) does not exist.

26. If λ and μ are positive, mutually singular Borel measures on n and λ + μ is regular, then so are λ and μ.

3.5 FUNCTIONS OF BOUNDED VARIATION

The theorems of the preceding section apply in particular on the real line, where, because of the correspondence between regular Borel measures and increasing functions that we established in §1.5, they yield results about differentiation and integration of functions. As in §1.5, we adopt the notation that if F is an increasing, right continuous function on , μF is the Borel measure determined by the relation μF((a, b]) = F(b) – F(a). Also, throughout this section the term “almost everywhere” will always refer to Lebesgue measure.

Our first result uses the Lebesgue differentiation theorem to prove the a.e. differentiability of increasing functions.

3.23 Theorem. Let F : → be increasing, and let G(x) = F(x+).

a. The set of points at which F is discontinuous is countable.

b. F and G are differentiable a.e., and F’ = G’ a.e.

Proof. Since F is increasing, the intervals (F(x−), F(x+)) (x ) are disjoint, and for |x| < N they lie in the interval (F(−N), F(N)). Hence

which implies that {x (−N, N) : F(x+) ≠ F(x−)} is countable. As this is true for all N, (a) is proved.

Next, we observe that G is increasing and right continuous, and G = F except perhaps where F is discontinuous. Moreover,

and the families {(x − r, x]} and {(x, x + r]} shrink nicely to x as r = |h| → 0. Thus, an application of Theorem 3.22 to the measure μG (which is regular by Theorem 1.18) shows that G’(x) exists for a.e. x. To complete the proof, it remains to show that if H = G − F, then H′ exists and equals zero a.e.

Let {xj} be an enumeration of the points at which H ≠ 0. Then H(xj) > 0, and as above we have ∑{j:|xj|N} H(xj) < ∞ for any N. Let δj be the point mass at xj and μ = ∑j H(xj)δj. Then μ is finite on compact sets by the preceding sentence, and hence μ is regular by Theorems 1.16 and 1.18; also, μ ⊥ m since m(E) = μ(Ec) = 0 where E = {xj}∞₁. But then

which tends to zero as h → 0 for a.e. x, by Theorem 3.22. Thus H′ = 0 a.e., and we are done.

As positive measures on are related to increasing functions, complex measures on are related to so-called functions of bounded variation. The definition of the latter concept is a bit technical, so some motivation may be appropriate. Intuitively, if F(t) represents the position of a particle moving along the real line at time t, the “total variation” of F over the interval [a, b] is the total distance traveled from time a to time b, as shown on an odometer. If F has a continuous derivative, this is just the integral of the “speed,” ∫ba |F’(t)| dt. To define the total variation without any smoothness hypotheses on F requires a different approach; namely, one partitions [a, b] into subintervals [tj−1, tj] and approximates F on each subinterval by the linear function whose graph joins (tj−1, F(tj−1)) to (tj, F(tj)), and then passes to a limit.

In making this precise, we begin with a slightly different point of view, taking a = −∞ and considering the total variation as a function of b. To wit, if F : → and x , we define

TF is called the total variation function of F. We observe that the sums in the definition of TF are made bigger if the additional subdivision points xj are added. Hence, if a < b, the definition of TF(b) is unaffected if we assume that a is always one of the subdivision points. It follows that

(3.24) equation

Thus TF is an increasing function with values in [0, ∞]. If TF(∞) = limx→∞ TF(x) is finite, we say that F is of bounded variation on , and we denote the space of all such F by BV.

More generally, the supremum on the right of (3.24) is called the total variation of F on [a, b]. It depends only on the values of F on [a, b], so we may define BV([a, b]) to be the set of all functions on [a, b] whose total variation on [a, b] is finite. If F BV, the restriction of F to [a, b] is in BV([a, b]) for all a, b; indeed, its total variation on [a, b] is nothing but TF(b) − TF(a). Conversely, if F BV([a, b]) and we set F(x) = F(a) for x < a and F(x) = F(b) for x > b, then F BV. By this device the results that we shall prove for BV can also be applied to BV([a, b]).

3.25 Examples.

a. If F : → is bounded and increasing, then F BV (in fact, TF(x) = F(x) − F(−∞)).

b. If F, G BV and a, b , then aF + bG BV.

c. If F is differentiable on and F’ is bounded, then F BV([a, b]) for −∞ < a < b < ∞ (by the mean value theorem).

d. If F(x) = sin x, then F BV([a, b]) for −∞ < a < b < ∞, but F BV.

e. If F(x) = x sin(x⁻¹) for x ≠ 0 and F(0) = 0, then F BV([a, b]) for a ≤ 0 < b or a < 0 ≤ b.

The verification of these examples is left to the reader (Exercise 27).

3.26 Lemma. If F BV is real-valued, then TF + F and TF − F are increasing.

Proof. If x < y and > 0, choose x₀ < ··· < xn = x such that

Then ∑n₁|F(xj) − F(xj−1)| + |F(y) − F(x)| is an approximating sum for TF(y), and F(y) = [F(y) − F(x)] + F(x), so

Since is arbitrary, TF(y) ± F(y) > TF(x) ± F(x), as desired.

3.27 Theorem.

a. F BV iff Re F BV and Im F BV.

b. If F : → , then F BV iff F is the difference of two bounded increasing functions; for F BV these functions may be taken to be (TF + F) and (TF − F).

c. If F BV, then F(x+) = and F(x−) = exist for all x as do F(±∞) = limy→±∞ F(y).

d. If F BV, the set of points at which F is discontinuous is countable.

e. If F BV and G(x) = F(x+), then F’ and G’ exist and are equal a.e.

Proof, (a) is obvious. For (b), the “if” implication is easy (see Examples 3.25a,b). To prove “only if,” observe that by Lemma 3.26, the equation F = (TF + F) − (TF − F) expresses F as the difference of two increasing functions. Also, the inequalities

imply that

so that F, and hence TF ± F, is bounded. Finally, (c), (d), and (e) follow from (a), (b), and Theorem 3.23.

The representation F = (TF + F) − (TF − F) of a real-valued F BV is called the Jordan decomposition of F, and (TF + F) and (TF − F) are called the positive and negative variations of F. Since x⁺ = max(x, 0) = (|x| + x) and x⁻ = max(−x, 0) = (|x| − x) for x , we have

Theorem 3.27(a,b) leads to the connection between BV and the space of complex Borel measures on . To make this precise, we introduce the space N BV (N for “normalized”) defined by

We observe that if F BV, then the function G defined by G(x) = F(x+)–F(−∞) is in N BV and G’ = F’ a.e. (That G BV follows easily from Theorem 3.27(a,b): if F is real and F = F₁ − F₂ where F₁, F₂ are increasing, then G(x) = F₁(x+) − [F₂(x+) + F(−∞)], which is again the difference of two increasing functions.)

3.28 Lemma. If F BV, then TF(−∞) = 0. If F is also right continuous, then so is TF.

Proof. If > 0 and x , choose x₀ < ··· < xn = x so that

From (3.24) we see that TF(x) − TF(x₀) ≥ TF(x) − , and hence TF(y) ≤ for y ≤ x₀. Thus TF(−∞) = 0.

Now suppose that F is right continuous. Given x and > 0, let α = TF(x+) − TF(x), and choose δ > 0 so that |F(x + h) − F(x)| < and TF(x + h) − TF(x+) < whenever 0 < h < δ. For any such h, by (3.24) there exist x₀ < ··· < xn = x + h such that

and hence

Likewise, there exist x = t₀ < ··· < tm = x₁ such that ∑n₁ |F(tj) − F(tj₋₁)| ≥ α, and hence

Thus α < 4, and since e is arbitrary, α = 0.

3.29 Theorem. If μ is a complex Borel measure on and F(x) = μ((−∞, x]), then F N BV. Conversely, if F N BV, there is a unique complex Borel measure μF such that F(x) = μF((−∞, x]); moreover, |μF| = μTF.

Proof. If μ is a complex measure, we have μ = μ⁺₁ − μ⁻₁ + i(μ⁺₂ − μ⁻₂) where the μ±j are finite measures. If F±j(x) = μ±j ((−∞, x]), then F±j is increasing and right continuous, F±j(−∞) = 0, and F±j(∞) = μ±j() < ∞. By Theorem 3.27(a,b), the function F = F⁺₁ − F⁻₁ + i(F⁺₂ − F⁻₂) is in N BV. Conversely, by Theorem 3.27 and Lemma 3.28, any F N BV can be written in this form with the F±j increasing and in N BV. Each F±j gives rise to a measure μ±j according to Theorem 1.16, so F(x) = μF ((−∞, x]) where μF = μ⁺₁ − μ⁻₁ + i(μ⁺₂ − μ⁻₂). The proof that |μF| = μTF is outlined in Exercise 28.

The next obvious question is: Which functions in N BV correspond to measures μ such that μ ⊥ m or μ m? One answer is the following:

3.30 Proposition. If F N BV, then F’ L¹(m). Moreover, μF ⊥ m iff F’ = 0 a.e., and μF m iff F(x) = ∫x−∞ F′(t) dt.

Proof. We have merely to observe that F’(x) = limr→0 μF(Er)/m(Er) where Er = (x, x + r] or (x − r, x] and apply Theorem 3.22. (The measure μF is automatically regular by Theorem 1.18.)

The condition μF m can also be expressed directly in terms of F, as follows. A function F : → is called absolutely continuous if for every > 0 there exists δ > 0 such that for any finite set of disjoint intervals (a₁, b₁),…, (aN, bN),

(3.31) equation

More generally, F is said to be absolutely continuous on [a, b] if this condition is satisfied whenever the intervals (aj, bj) all lie in [a, b]. Clearly, if F is absolutely continuous, then F is uniformly continuous (take N = 1 in (3.31)). On the other hand, if F is everywhere differentiable and F′ is bounded, then F is absolutely continuous, for |F(bj) − F(aj)| ≤ (max |F’|)(bj − aj) by the mean value theorem.

3.32 Proposition. If F N BV, then F is absolutely continuous iff μF m.

Proof. If μF m, the absolute continuity of F follows by applying Theorem 3.5 to the sets E = ∪N₁ (aj, bj]. To prove the converse, suppose that E is a Borel set such that m(E) = 0. If and δ are as in the definition of absolute continuity of F, by Theorem 1.18 we can find open sets U₁ ⊃ U₂ ⊃ ··· ⊃ E such that m(U₁) < δ (and thus μ(Uj) < δ for all j) and μF(Uj) → μF(E). Each Uj is a disjoint union of open intervals (akj, bkj), and

for all N. Letting N → ∞, we obtain |μF(Uj)| < and hence |μF(E)| ≤ . Since is arbitrary, μF(E) = 0, which shows that μF m.

3.33 Corollary. If f L¹(m), then the Junction F(x) = ∫x−∞ f(t) dt is in N BV and is absolutely continuous, and f = F’ a.e. Conversely, if F N BV is absolutely continuous, then F’ L¹(m) and F(x) = ∫x−∞ F’(t) dt.

Proof This follows immediately from Propositions 330 and 3.32.

If we consider functions on bounded intervals, this result can be refined a bit.

3.34 Lemma. If F is absolutely continuous on [a, b], then F BV([a, b]).

Proof. Let δ be as in the definition of absolute continuity, corresponding to = 1, and let N be the greatest integer less than δ⁻¹(b − a) + 1. If a = x₀ < ··· < xn = b, by inserting more subdivision points if necessary, we can collect the intervals (xj−1, xj) into at most N groups of consecutive intervals such that the sum of the lengths in each group is less than δ. The sum ∑|F(xj) − F(xj−1)| over each group is at most 1, and hence the total variation of F on [a, b] is at most N.

3.35 The Fundamental Theorem of Calculus for Lebesgue Integrals. If −∞ < a < b < ∞ and F : [a, b] → , the following are equivalent:

a. F is absolutely continuous on [a, b],

b. F(x) − F(a) = ∫xa f(t) dt for some f L¹([a, b], m).

c. F is differentiate a.e. on [a, b], F′ L¹([a, b],m), and F(x) − F(a) = ∫xa F’(t)dt.

Proof. To prove that (a) implies (c), we may assume by subtracting a constant from F that F(a) = 0. If we set F(x) = 0 for x < a and F(x) = F(b) for x > b, then F N BV by Lemma 3.34, so (c) follows from Corollary 3.33. That (c) implies (b) is trivial. Finally, (b) implies (a) by setting f(t) = 0 for t [a, b] and applying Corollary 3.33.

The following decomposition of Borel measures on n is sometimes important. A complex Borel measure μ on n is called discrete if there is a countable set {xj} ⊂ n and complex numbers cj such that ∑|cj| < ∞ and μ = ∑ cjδxj, where δx is the point mass at x. On the other hand, μ is called continuous if μ({x}) = 0 for all x n. Any complex measure μ can be written uniquely as μ = μd + μc where μd is discrete and μc is continuous. Indeed, let E = {x : μ({x}) ≠ 0}. For any countable subset F of E the series ∑xF μ({x}) converges absolutely (to μ(F)), so {x E : |μ({x})| > k⁻¹} is finite for all k, and it follows that E itself is countable. Hence μd(A) = μ(A ∩ E) is discrete and μc(A) = μ(A \ E) is continuous.

Obviously, if μ is discrete, then μ ⊥ m; and if μ m, then μ is continuous. Thus, by Theorem 3.22, any (regular) complex Borel measure on n can be written uniquely as

where μd is discrete, μac is absolutely continuous with respect to m, and μsc is a “singular continuous” measure, that is, μsc is continuous but μsc ⊥ m.

The existence of nonzero singular continuous measures in n is evident enough when n > 1; the surface measure on the unit sphere discussed in §2.7 is one example. Their existence when n = 1 is not quite so obvious; they correspond via Theorem 3.29 to nonconstant functions F N BV such that F is continuous but F′ = 0 a.e. One such function is the Cantor function constructed in §1.5 (extended to by setting F(x) = 0 for x < 0 and F(x) = 1 for x > 1). More surprisingly, there exist strictly increasing continuous functions F such that F′ = 0 a.e.; see Exercise 40.

If F N BV, it is customary to denote the integral of a function g with respect to the measure μF by ∫ g dF or ∫ g(x) dF(x); such integrals are called Lebesgue-Stieltjes integrals. We conclude by presenting an integration-by-parts formula for Lebesgue-Stieltjes integrals; other variants of this result can be found in Exercises 34 and 35.

3.36 Theorem. If F and G are in N BV and at least one of them is continuous, then for −∞ < a < b < ∞,

Proof. F and G are linear combinations of increasing functions in N BV by Theorem 3.27(a,b), so a simple calculation shows that it suffices to assume F and G increasing. Suppose for the sake of definiteness that G is continuous, and let Ω = {(x, y) : a < x ≤ y ≤ b}. We use Fubini’s theorem to compute μF x μG(Ω) in two ways:

and since G(x) = G(x−),

Subtracting these two equations, we obtain the desired result.

Exercises

27. Verify the assertions in Examples 3.25.

28. If F N BV, let G(x) = | μF|((−∞, x]). Prove that |μF| = μTF by showing that G = TF via the following steps.

a. From the definition of TF, TF ≤ G.

b. |μF(E)| ≤ μTF(E) when E is an interval, and hence when E is a Borel set.

c. |μF| ≤ μTF, and hence G ≤ TF. (Use Exercise 21.)

29. If F N BV is real-valued, then μ⁺F = μP and μ⁻F = μN where P and N are the positive and negative variations of F. (Use Exercise 28.)

30. Construct an increasing function on whose set of discontinuities is .

31. Let F(x) = x² sin(x⁻¹) and G(x) = x²sin(x⁻²) for x ≠ 0, and F(0) = G(0) = 0.

a. F and G are differentiable everywhere (including x = 0).

b. F BV([−1, 1]), but G BV([−1,1]).

32. If F₁, F₂,…, F N BV and Fj → F pointwise, then TF < lim inf TFj.

33. If F is increasing on , then F(b) − F(a) ≥ ∫ba F’(t) dt.

34. Suppose F, G N BV and −∞ < a < b < ∞.

a. By adapting the proof of Theorem 3.36, show that

b. If there are no points in [a, b] where F and G are both discontinuous, then

35. If F and G are absolutely continuous on [a, b], then so is FG, and

36. Let G be a continuous increasing function on [a, b] and let G(a) = c, G(b) = d.

a. If E ⊂ [c, d] is a Borel set, then m(E) = μG(G⁻¹ (E)). (First consider the case where E is an interval.)

b. If f is a Borel measurable and integrable function on [c, d], then ∫dc f(y) dy = ∫ba f(G(x)) dG(x). In particular, ∫dc f(y) dy = ∫ba f(G(x))G’(x) dx if G is absolutely continuous.

c. The validity of (b) may fail if G is merely right continuous rather than continuous.

37. Suppose F : → . There is a constant M such that |F(x) − F(y)| ≤ M|x − y| for all x, y (that is, F is Lipschitz continuous) iff F is absolutely continuous and |F’| ≤ M a.e.

38. If f : [a, b] → , consider the graph of f as a subset of , namely, {t + if(t) : t [a, b]}. The length L of this graph is by definition the supremum of the lengths of all inscribed polygons. (An “inscribed polygon” is the union of the line segments joining tj−1 + if(tj−1) to tj + if(tj), 1 ≤ j ≤ n, where a = t₀ < ··· < tn = b.)

a. Let F(t) = t + if(t); then L is the total variation of F on [a, b].

b. If f is absolutely continuous, L = ∫ba[1 + f’(t)²]1/2 dt.

39. If {Fj} is a sequence of nonnegative increasing functions on [a, b] such that F(x) = ∑∞₁ Fj(x) < ∞ for all x [a, b]. then F’(x) = ∑∞₁ F’j(x) for a.e. x [a, b]. (It suffices to assume Fj N BV. Consider the measures μFj.)

40. Let F denote the Cantor function on [0,1] (see §1.5), and set F(x) = 0 for x < 0 and F(x) = 1 for x > 1. Let {[an, bn]} be an enumeration of the closed subintervals of [0,1] with rational endpoints, and let Fn(x) = F((x − an)/(bn − an)). Then G = ∑∞₁ 2−n Fn is continuous and strictly increasing on [0, 1], and G′ = 0 a.e. (Use Exercise 39.)

41. Let A ⊂ [0, 1] be a Borel set such that 0 < m(A ∩ I) < m(I) for every subinterval I of [0, 1] (Exercise 33, Chapter 1).

a. Let F(x) = m([0, x] ∩ A). Then F is absolutely continuous and strictly increasing on [0, 1], but F′ = 0 on a set of positive measure.

b. Let G(x) = m([0, x] ∩ A) − m([0, x] \ A). Then G is absolutely continuous on [0, 1], but G is not monotone on any subinterval of [0, 1].

42. A function F : (a, b) → (−∞ ≤ a < b ≤ ∞) is called convex if

for all s, t (a, b) and λ (0, 1). (Geometrically, this says that the graph of F over the interval from s to t lies underneath the line segment joining (s, F(s)) to (t, F(t)).)

a. F is convex iff for all s, t, s’, t′ (a, b) such that s ≤ s’ < t′ and s < t ≤ t′,

b. F is convex iff F is absolutely continuous on every compact subinterval of (a, b) and F′ is increasing (on the set where it is defined).

c. If F is convex and t₀ (a, b), there exists β such that F(t) − F(t₀) ≥ β(t − t₀) for all t (a, b).

d. (Jensen’s Inequality) If (X, , μ) is a measure space with μ(X) = 1, g : X → (a, b) is in L¹(μ), and F is convex on (a, b), then

(Let t₀ = ∫ gdμ and t = g(x) in (c), and integrate.)

3.6 NOTES AND REFERENCES

§3.2: The Lebesgue-Radon-Nikodym theorem was proved by Lebesgue [92] in the case where μ is Lebesgue measure on n. Under the hypothesis v μ, it was generalized by Radon [111] to arbitrary regular Borel measures on n and by Nikodym [107] to measures on abstract spaces. The Lebesgue decomposition in the abstract setting appears in Saks [128]. The proof of the Lebesgue-Radon-Nikodym theorem in the text is similar to, but more efficient than, the one in Halmos [61]; I learned it from L. Loomis.

§3.3: The characterization

of the total variation of a complex measure v (see Exercise 21) is usually taken as the definition of |v|. Our definition seems more generally useful, and it is certainly easier to compute with.

§3.4: Theorems 3.21 and 3.22 are due to Lebesgue [92], but the line of argument we have presented is essentially that of Wiener [161], and the maximal function H f, in dimension one, was first studied in Hardy and Littlewood [64]. Our proof of Theorem 3.18 is illustrative of a general technique that has been much exploited in recent years, namely, controlling the limiting behavior of a family of operators by means of estimates on an appropriate maximal function.

Lemma 3.15, a simplified version of Wiener’s covering lemma, is taken from Rudin [125]. There is also an older and more delicate covering theorem, due to Vitali, which is used for similar purposes:

If E ⊂ n and is a family of cubes such that each x E is contained in members of of arbitrarily small diameter, then there is a (finite or infinite) disjoint sequence {Qj} ⊂ such that m(E \ ∪jQj) = 0.

Proofs can be found in many books, for example, Cohn [27, §6.2], Falconer [39, §1.3], and Hewitt and Stromberg [76, §17].

§3.5: The main results of this section are due to Lebesgue and Vitali; see Hawkins [70] for detailed references. Exercise 36 gives one form of the change-of-variable formula for Lebesgue integrals; others can be found in Serrin and Varberg [133]. Exercise 39 is a theorem of Fubini, and the example in Exercise 40 is due to Brown [21].

The Stieltjes integral ∫ba gdF was originally defined, under the hypothesis that F is an increasing function on [a, b], as a limit of Riemann sums ∑g(tj)[F(tj)-F(tj−1)]. The theory of such “Riemann-Stieltjes” integrals is much like that of the ordinary Riemann integral, but some care is needed to handle cases where g and F are both allowed to be discontinuous. See ter Horst [148], which contains the analogue of Theorem 2.28 for Stieltjes integrals.

The example of the Cantor function shows that a continuous, a.e.-differentiable function need not be the integral of its derivative. It is a highly nontrivial theorem that if F is continuous on [a, b], F’(x) exists for every x [a, b] \ A where A is countable, and F′ L¹, then F is absolutely continuous and hence can be recovered from F′ by integration. A proof can be found in Cohn [27, §6.3]; see also Rudin [125, Theorem 7.26] for the somewhat easier case when A = Ø.

However, this is not the end of the story, for there exist everywhere differentiable functions F such that F’ L¹. Perhaps the simplest example is F(x) = x² sin (x⁻²) (see Exercise 31). Here the only trouble is at x = 0, so for a ≤ 0 ≤ b one could consider ∫ba F’(t) dt as an improper integral, i.e., the limit of Lebesgue integrals over [a, b] \ [−, ] as → 0. However, it is not hard to construct examples in which the singularities of F′ are so complicated that F′ is not Lebesgue integrable on any interval. In this situation the Lebesgue integral is simply insufficient. However, the Henstock-Kurzweil integral (or the Denjoy or Perron integral) that was discussed in §2.8 is powerful enough to integrate such F’, and by using this integral one obtains the general fundamental theorem of calculus: If F is everywhere differentiate on [a, b], then F(b) − F(a) = ∫ba F’(t) dt.

Chapter 4

Point Set Topology

The concepts of limit, convergence, and continuity are central to all of analysis, and it is useful to have a general framework for studying them that includes the classical manifestations as special cases. One such framework, which has the advantage of not requiring many ideas beyond those occurring in analysis on Euclidean space, is that of metric spaces. However, metric spaces are not sufficiently general to describe even some very classical modes of convergence, for example, pointwise convergence of functions on . A more flexible theory can be built by taking the open sets, rather than a metric, as the primitive data, and it is this theory that we shall explore in the present chapter.

4.1 TOPOLOGICAL SPACES

Let X be a nonempty set. A topology on X is a family of subsets of X that contains Ø and X and is closed under arbitrary unions and finite intersections (i.e., if then , and if then . The pair (X, ) is called a topological space. If is understood, we shall simply refer to the topological space X. Let us examine a few examples:

If X is any nonempty set, (X) and {Ø, X} are topologies on X. They are called the discrete topology and the trivial (or indiscrete) topology, respectively.

If X is an infinite set, {U ⊂ X : U = Ø or Uc is finite} is a topology on X, called the cofinite topology.

If X is a metric space, the collection of all open sets with respect to the metric is a topology on X.

If (X, ) is a topological space and Y ⊂ X, then Y = {U Y : U } is a topology on Y, called the relative topology induced by .

We now present the basic terminology concerning topological spaces. Most of these concepts are already familiar in the context of metric spaces. Until further notice, (X, ) will be a fixed topological space.

The members of are called open sets, and their complements are called closed sets. If Y ⊂ X, the open (closed) subsets of Y in the relative topology are called relatively open (closed). We observe that, by deMorgan’s laws, the family of closed sets is closed under arbitrary intersections and finite unions.

If A ⊂ X, the union of all open sets contained in A is called the interior of A, and the intersection of all closed sets containing A is called the closure of A. We denote the interior and closure of A by Ao and , respectively. Clearly Ao is the largest open set contained in A and is the smallest closed set containing A, and we have (Ao)c = c and ()c − (Ac)o. The difference \ Ao = c is called the boundary of A and is denoted by ∂A. If = X, A is called dense in X. On the other hand, if ()o = Ø, A is called nowhere dense.

If x X (or E ⊂ X), a neighborhood of x (or E) is a set A ⊂ X such that x Ao (or E ⊂ Ao). Thus, a set A is open iff it is a neighborhood of itself. (Some authors require neighborhoods to be open sets; we do not.) A point x is called an accumulation point of A if A (U \ {x}) ≠ Ø for every neighborhood U of x. (Other terms sometimes used for the same concept are “cluster point” and “limit point.” We shall use “cluster point” to mean something a bit different below.)

4.1 Proposition. If A ⊂ X, let acc(A) be the set of accumulation points of A. Then = A acc(A), and A is closed iff acc(A) ⊂ A.

Proof. If x , then Ac is a neighborhood of x that does not intersect A, so x acc(A); thus A acc(A) ⊂ . If x A acc(A), there is an open U containing x such that U A = Ø, so that ⊂ Uc and x . Thus ⊂ A acc(A). Finally, A is closed iff A = , and this happens iff acc(A) ⊂ A.

If ₁ and ₂ are topologies on X such that ₁ ⊂ ₂, we say that ₁ is weaker (or coarser) than ₂, or that ₂ is stronger (or finer) than ₁. Clearly the trivial topology is the weakest topology on X, while the discrete topology is the strongest. If ⊂ (X), there is a unique weakest topology () on X that contains , namely the intersection of all topologies on X containing . It is called the topology generated by , and is sometimes called a subbase for ().

If is a topology on X, a neighborhood base for at x X is a family ⊂ such that

x V for all V ;

if U and x U, there exists V such that V ⊂ U.

A base for is a family ⊂ that contains a neighborhood base for at each x X. For example, if X is a metric space, the collection of open balls centered at x is a neighborhood base for the metric topology at x, and the collection of all open balls in X is a base.

4.2 Proposition. If is a topology on X and ε ⊂ , then ε is a base for iff every nonempty U is a union of members of ε.

Proof. If ε is a base, U , and x U, there exists Vx ε with x Vx ⊂ U, so U = xU Vx. Conversely, if every nonempty U is a union of members of ε, then {V ε : x V} is clearly a neighborhood base at x, so ε is a base.

4.3 Proposition. If ε ⊂ (X), in order for ε to be a base for a topology on X it is necessary and sufficient that the following two conditions be satisfied:

a. each x X is contained in some V ε;

b. if U, V ε and x U V, there exists W ε with x W ⊂ (U V).

Proof. The necessity is clear, since if U, V are open, then so is U V. To prove the sufficiency, let

Then X by condition (a) and Ø trivially, and is obviously closed under unions. If U₁, U₂ and x U₁ U₂, there exist V₁, V₂ ε with x V₁ ⊂ U₁ and x V₂ ⊂ U₂, and by condition (b) there exists W ε with x W ⊂ (V₁ V₂). Thus U₁ U₂ , so by induction is closed under finite intersections. Therefore is a topology, and ε is clearly a base for .

4.4 Proposition. If ε ⊂ (X), the topology (ε) generated by ε consists of Ø, X, and all unions of finite intersections of members of ε.

Proof. The family of finite intersections of sets in ε, together with X, satisfies the conditions of Proposition 4.3, so by Propostiion 4.2 the family of all unions of such sets, together with 0, is a topology. It is obviously contained in (ε), hence equal to (ε).

Note how the simplicity of this proposition contrasts with the corresponding result for σ-algebras (Proposition 1.23). What makes life easier here is that only finite intersections are involved.

The concept of topological space is general enough to include a great profusion of interesting examples, but — by the same token — too general to yield many interesting theorems. To build a reasonable theory one must usually restrict the class of spaces under consideration. The remainder of this section is devoted to a discussion of two types of restrictions that are commonly made, the so-called countability and separation axioms.

A topological space (X, ) satisfies the first axiom of countability, or is first countable, if there is a countable neighborhood base for at every point of X. (It is useful to observe that if X is first countable, for every x X there is a neighborhood base {Uj}∞₁ at x such that Uj ⊃ Uj+1 for all j. Indeed, if is any countable neighborhood base at x, we can take The space (X, ) satisfies the second axiom of countability, or is second countable, if has a countable base. Also, (X, ) is separable if X has a countable dense subset. Every metric space is first countable (the balls of rational radius about x are a neighborhood base at x), and a metric space is second countable iff it is separable (Exercise 5). The latter fact can be partly generalized:

4.5 Proposition. Every second countable space is separable.

Proof. If X is second countable, let ε be a countable base for the topology, and for each U ε pick a point xU U. Then the complement of the closure of {xU : U ε} is an open set that does not include any U ε; hence it is empty and {xU : U ε} is dense.

A sequence {xj} in a topological space X converges to x X (in symbols: xj → x) if for every neighborhood U of x there exists J such that xj U for all j > J. First countable spaces have the pleasant property that such things as closure and continuity can be characterized in terms of sequential convergence — which is not the case in more general spaces, as we shall see. For example:

4.6 Proposition. If X is first countable and A ⊂ X, then x iff there is a sequence {xj} in A that converges to x.

Proof. Let {Uj} be a countable neighborhood base at x with Uj ⊃ Uj+1 for all j. If x , then for all j. Pick xj Uj A; since Uk ⊂ Uj for k > j and every neighborhood of x contains some Uj, it is clear that xj → x. On the other hand, if x and {xj} is any sequence in A, then ()c is a neighborhood of x containing no xj, so xj x.

Lastly, we discuss the separation axioms. These are properties of a topological space, labeled T₀,…, T₄, that guarantee the existence of open sets that separate points or closed sets from each other. If X has the property Tj, we say that X is a Tj space or that the topology on X is Tj.

T₀: If x ≠ y, there is an open set containing x but not y or an open set containing y but not x.

T₁: If x ≠ y, there is an open set containing y but not x.

T₂: If x ≠ y, there are disjoint open sets U, V with x U and y V.

T₃: X is a T₁ space, and for any closed set A ⊂ X and any x Ac there are disjoint open sets U, V with x U and A ⊂ V.

T₄: X is a T₁ space, and for any disjoint closed sets A, B in X there are disjoint open sets U, V with A ⊂ U and B ⊂ V.

T₂, T₃, and T₄ also have other names: A T₂ space is a Hausdorff space, a T₃ space is a regular space, and a T₄ space is a normal space. (Some authors do not require regular and normal spaces to be T₁.) There is an additional useful separation condition, intermediate between T₃ and T₄, that we shall discuss in §4.2.

The following characterization of T₁ spaces is useful. It shows in particular that every normal space is regular and that every regular space is Hausdorff.

4.7 Proposition. X is a T₁ space iff{x} is closed for every x X.

Proof. If X is T₁ and x X, for each y ≠ x there is an open Uy containing y but not x; thus is open and {x} is closed. Conversely, if {x} is closed, then {x}c is an open set containing every y ≠ x.

The vast majority of topological spaces that arise in practice (and, in particular, in this book) are Hausdorff, or become Hausdorff after simple modifications. (This last phrase refers to spaces such as the space of integrable functions on a measure space, which becomes a Hausdorff space with the L¹ metric when we identify two functions that are equal a.e.) However, two classes of (usually) non-Hausdorff topologies are of sufficient importance to warrant special mention: the quotient topology on a space of equivalence classes, discussed in Exercises 28 and 29 (§4.2), and the Zariski topology on an algebraic variety. Without attempting to give the definition of an algebraic variety, we shall describe the Zariski topology on a vector space.

Let k be a field, and let k(X₁,…, Xn) be the ring of polynomials in n variables over k. Each P k(X₁,…, Xn) determines a polynomial map p : kn → k by substituting elements of k for the formal indeterminates X₁,…, Xn. The correspondence P → p is one-to-one precisely when k is infinite. The collection of all sets p⁻¹({0}) in kn, as p ranges over all polynomial maps, is closed under finite unions, since p⁻¹({0}) q⁻¹({0}) = (pq)⁻¹({0}) and it contains kn itself (take p = 0). Hence, by Propositions 4.2 and 4.3, the collection of all sets of the form being a polynomial map for each α) is the collection of closed sets for a topology on kn, called the Zariski topology. The Zariski topology is T₁ by Proposition 4.7, for if a = (a₁,…, an) kn then where pj (X₁,…, Xn) = Xj − aj. If k is finite the Zariski topology is discrete, but if k is infinite the Zariski topology is not Hausdorff; in fact, any two nonempty open sets have nonempty intersection. This is just a restatement of the fact that k(X₁,…, Xn) is an integral domain, that is, if P and Q are nonzero polynomials, then PQ is nonzero. (For n = 1, the Zariski topology is the cofinite topology.)

Other examples illustrating the separation and countability axioms will be found in the exercises.

Exercises

1. If card(X) ≥ 2, there is a topology on X that is T₀ but not T₁.

2. If X is an infinite set, the cofinite topology on X is T₁ but not T₂, and is first countable iff X is countable.

3. Every metric space is normal. (If A, B are closed sets in the metric space (X, ρ), consider the sets of points x where p(x, A) < ρ(x, B) or ρ(x, A) > ρ(x, B).)

4. Let X = , and let be the family of all subsets of of the form U (V(V ) where U, V are open in the usual sense. Then is a topology that is Hausdorff but not regular. (In view of Exercise 3, this shows that a topology stronger than a normal topology need not be normal or even regular.)

5. Every separable metric space is second countable.

6. Let ε = {(a, b] : −∞ < a < b < ∞}.

a. ε is a base for a topology on in which the members of ε are both open and closed.

b. is first countable but not second countable. (If x , every neighborhood base at x contains a set whose supremum is x.)

c. is dense in with respect to . (Thus the converse of Proposition 4.5 is false.)

7. If X is a topological space, a point x X is called a cluster point of the sequence {xj} if for every neighborhood U of x, xj U for infinitely many j. If X is first countable, x is a cluster point of {xj} iff some subsequence of {xj} converges to x.

8. If X is an infinite set with the cofinite topology and {xj} is a sequence of distinct points in X, then xj → x for every x X.

9. If X is a linearly ordered set, the topology generated by the sets {x : x < a} and {x : x > a} (a X) is called the order topology.

a. If a, b X and a < b, there exist U, V with a U, b V, and x < y for all x U and y V. The order topology is the weakest topology with this property.

b. If Y ⊂ X, the order topology on Y is never stronger than, but may be weaker than, the relative topology on Y induced by the order topology on X.

c. The order topology on is the usual topology.

10. A topological space X is called disconnected if there exist nonempty open sets U, V such that U V = Ø and U V = X; otherwise X is connected. When we speak of connected or disconnected subsets of X, we refer to the relative topology on them.

a. X is connected iff Ø and X are the only subsets of X that are both open and closed.

b. If {Eα}αA is a collection of connected subsets of X such that αA Eα ≠ Ø, then αA Eα is connected.

c. If A ⊂ X is connected, then is connected.

d. Every point x X is contained in a unique maximal connected subset of X, and this subset is closed. (It is called the connected component of x.)

11. If E₁,…, En are subsets of a topological space, the closure of .

12. Let X be a set. A Kuratowski closure operator on X is a map A A* from (X) to itself satisfying (i) Ø* = Ø, (ii) A ⊂ A* for all A, (iii) (A*)* = A* for all A, and (iv) (A B)* = A* B* for all A, B.

a. If X is a topological space, the map A is a Kuratowski closure operator. (Use Exercise 11.)

b. Conversely, given a Kuratowski closure operator, let and . Then is a topology, and for any set A ⊂ X, A* is its closure with respect to .

13. If X is a topological space, U is open in X, and A is dense in X, then .

4.2 CONTINUOUS MAPS

Topological spaces are the natural setting for the concept of continuity, which can be described in either global or local terms as follows. Let X and Y be topological spaces and f a map from X to Y. Then f is called continuous if f⁻¹(V) is open in X for every open V ⊂ Y. (Since f⁻¹(Ac) = [f⁻¹(A)]c, an equivalent condition is that f⁻¹(A) is closed in X for every closed A ⊂ Y.) If x X, f is called continuous at x if for every neighborhood V of f(x) there is a neighborhood U of x such that f(U) ⊂ V, or equivalently, if f⁻¹(V) is a neighborhood of x for every neighborhood V of f(x). Clearly, if f : X → Y and g : Y → Z are continuous (or f is continuous at x and g is continuous at f(x)), then g o f is continuous (at x). We shall denote the set of continuous maps from X to Y by C(X, Y).

4.8 Proposition. The map f : X → Y is continuous iff f is continuous at every x X.

Proof. If f is continuous and V is a neighborhood of f(x), f⁻¹(Vo) is an open set containing x, so f is continuous at x. Conversely, suppose that f is continuous at each x X. If V ⊂ Y is open, V is a neighborhood of each of its points, so f⁻¹(V) is a neighborhood of each of its points. Thus f⁻¹(V) is open, so f is continuous.

4.9 Proposition. If the topology on Y is generated by a family of sets ε, then f : X → y is continuous iff f⁻¹(V) is open in X for every V ε.

Proof. This is clear from Proposition 4.4 and the fact that the set mapping f⁻¹ commutes with unions and intersections.

If f : X → Y is bijective and f and f⁻¹ are both continuous, f is called a homeomorphism, and X and Y are said to be homeomorphic. In this case the set mapping f⁻¹ is a bijection from the open sets in Y to the open sets in X, so X and Y may be considered identical as far as their topological properties go. If f : X → Y is injective but not surjective, and f : X → f(X) is a homeomorphism when f(X) ⊂ Y is given the relative topology, f is called an embedding.

If X is any set and {fα : X → Yα}αA is a family of maps from X into some topological spaces Yα, there is a unique weakest topology on X that makes all the fα continuous; it is called the weak topology generated by . Namely, is the topology generated by sets of the form f⁻¹α(Uα) where α A and Uα is open in Yα.

The most important example of this construction is the Cartesian product of topological spaces. If {Xα}αA is any family of topological spaces, the product topology on is the weak topology generated by the coordinate maps πα : X → Xα. When we consider a Cartesian product of topological spaces, we always endow it with the product topology unless we specify otherwise. By Proposition 4.4, a base for the product topology is given by the sets of the form where n and Uαj is open in Xαj for 1 ≤ j ≤ n. These sets can also be written as ΠαA Uα where Uα = Xα if α ≠ α₁,…, αn. Notice, in particular, that if A is infinite, a product of nonempty open sets is open in iff Uα = Xα for all but finitely many α.

4.10 Proposition. If Xα is Hausdorff for each α A, then X = ΠαA Xα is Hausdorff.

Proof. If x and y are distinct points of X, we must have πα(x) ≠ πα(y) for some α. Let U and V be disjoint neighborhoods of πα(x) and πα(y) in Xα. Then π⁻¹α(U) and π⁻¹α (V) are disjoint neighborhoods of x and y in X.

4.11 Proposition. If Xα (α A) and Y are topological spaces and X = ΠαA X′α, then f : Y → X is continuous iff is continuous for each α.

Proof. If is continuous for each α, then is open in Y for each open Ua in Xa. By Proposition 4.9, f is continuous. The converse is obvious.

If the spaces Xα are all equal to some fixed space X, the product is just the set XA of mappings from A to X, and the product topology is just the topology of pointwise convergence. More precisely:

4.12 Proposition. If X is a topological space, A is a nonempty set, and {fn} is a sequence in XA, then fn → f in the product topology iff fn → f pointwise.

Proof. The sets

where k and Uj is a neighborhood of f(αj) in X for each j, form a neighborhood base for the product topology at f. If fn → f pointwise, then fn(αj) Uj for n ≥ Nj and hence fn N(U₁,…, Uk) for n ≥ max(N₁,…, Nk); therefore fn → f in the product topology. Conversely, if fn → f in the product topology, α A, and U is a neighborhood of f(α), then fn N(U) = π⁻¹α(U) for large n; hence fn(α) U for large n, and so fn(α) → f(a).

We shall be particularly interested in real- and complex-valued functions on topological spaces. If X is any set, we denote by B(X, ) (resp. B(X, )) the space of all bounded real- (resp. complex−) valued functions on X. If X is a topological space, we also have the spaces C(X, ) and C(X, ) of countinuous functions on X, and we define

In speaking of complex-valued functions we shall usually omit the ⊂ and simply write B(X), C(X), and BC(X). Since addition and multiplication are continuous from x to , C(X) and BC(X) are complex vector spaces.

If f B(X), we define the uniform norm of f to be

The function ρ(f, g) = ||f − g||u is easily seen to be a metric on B(X), and convergence with respect to this metric is simply uniform convergence on X. B(X) is obviously complete in the uniform metric: If {fn} is uniformly Cauchy, then {fn(x)} is Cauchy for each x, and if we set f(x) = limn fn(x), it is easily verified that ||fn − f||u → 0.

4.13 Proposition. If X is a topological space, BC(X) is a closed subspace of B(X) in the uniform metric; in particular, BC(X) is complete.

Proof. Suppose {fn} ⊂ BC(X) and ||fn − f||u → 0. Given > 0, choose N so large that ||fn − f||u < /3 for n > N. Given n > N and x X, since fn is continuous at x there is a neighborhood U of x such that |fn(y) − fn(x)| < /3 for y U. But then

so f is continuous at x. By Proposition 4.8, f is continuous.

For a given topological space X it may happen that C(X) consists only of constant functions. This is obviously the case, for example, if X has the trivial topology, but it can happen even when X is regular. Normal spaces, however, always have plenty of continuous functions, as the following fundamental theorems show.

4.14 Lemma. Suppose that A and B are disjoint closed subsets of the normal space X, and let Δ = {k2−n : n ≥ 1 and 0 < k < 2n} be the set of dyadic rational numbers in (0, 1). There is a family {Ur : r Δ} of open sets in X such that A ⊂ Ur ⊂ Bc for all r Δ and r ⊂ Us for r < s.

Proof. By normality, there exist disjoint open sets V, W such that A ⊂ V, B ⊂ W. Let U1/2 = V. Then since Wc is closed,

We now select Ur for r = k2−n by induction on n. Suppose that we have chosen Ur for r = k2−n when 0 < k < 2n and n ≤ N − 1. To find Ur for r = (2j + 1)2−N (0 ≤ j < 2N−1), observe that j21-N and (U(j+1)21-N)c are disjoint closed sets (where we set ₀ = A and Uc₁ = B), so as above we can choose an open Ur with

These Ur’s clearly have the desired properties.

4.15 Urysohn’s Lemma. Let X be a normal space. If A and B are disjoint closed sets in X, there exists f C(X, [0,1]) such that f = 0 on A and f = 1 on B.

Proof. Let Ur be as in Lemma 4.14 for r Δ, and set U₁ = X. For x X, define f(x) = inf{r : x Ur}. Since A ⊂ UT ⊂ Bc for 0 < r < 1, we clearly have f(x) = 0 for x A and f(x) = 1 for x B, and 0 ≤ f(x) ≤ 1 for all x X. It remains to show that f is continuous. To this end, observe that f(x) < α iff x Ur for some r < α iff so is open. Also f(x) for some for some s > α (since s ⊂ Ur for s < r) iff x sα(s)c, so f⁻¹(a, ∞)) = sα(s)c is open. Since the open half-lines generate the topology on , f is continuous by Proposition 4.9.

The proof of Urysohn’s lemma may seem somewhat opaque at first, but there is a simple geometric intuition behind it. If one pictures X as the plane ² and the sets Ur as regions bounded by curves, the curves ∂Ur form a “topographic map” of the function f.

4.16 The Tietze Extension Theorem. Let X be a normal space. If A is a closed subset of X and f C(A, [a, b]), there exists F C(X, [a, b]) such that F|A = f.

Proof. Replacing f by (f − a)/(b − a), we may assume that [a, b] = [0,1]. We claim that there is a sequence {gn} of continuous functions on X such that 0 ≤ gn ≤ 2n−1/3n on X and on A. To begin with, let B = f⁻¹([0, 1/3]) and C = f⁻¹([2/3, 1]). These are closed subsets of A, and since A itself is closed, they are closed in X. By Urysohn’s lemma there is a continuous g₁ : X → [0, 1/3] with g₁ = 0 on B and g₁ = 1/3 on C; it follows that 0 ≤ f − g₁ ≤ 2/3 on A. Having found g₁,…, gn−1, by the same reasoning we can find gn : X → [0, 2n−1/3n] such that gn = 0 on the set where and gn = 2n−1/3n on the set where f − ∑n−1₁ gj ≥ (2/3)n. Let . Since ||gn||u < 2n−1/3n, the partial sums of this series converge uniformly, so F is continuous by Proposition 4.13. Moreover, on A we have 0 ≤ f − F ≤ (2/3)n for all n, whence F = f on A.

4.17 Corollary. If X is normal, A ⊂ X is closed, and f C(A), there exists F C(X) such that F|A = f.

Proof. By considering real and imaginary parts separately, it suffices to assume that f is real-valued. Let g = f/(1 + |f|). Then g C(A, (−1, 1)), so there exists G C(X, [−1,1]) with G|A = g. Let B = G⁻¹({−1, 1}). By Urysohn’s lemma there exists h C(X, [0, 1]) with h = 1 on A, h = 0 on B. Then hG = G on A and |hG| < 1 everywhere, so F = hG/(1 − |hG|) does the job.

A topological space X is called completely regular if X is T₁ and for each closed A ⊂ X and each x A there exists f C(X, [0,1]) such that f(x) = 1 and f = 0 on A. Completely regular spaces are also called Tychonoff spaces or T3 spaces. The latter terminology is justified, for every completely regular space is T₃ (if A, x, f are as above, then f⁻¹((, ∞)) and f⁻¹ ((−∞, )) are disjoint neighborhoods of x and A), and Urysohn’s lemma shows that every T₄ space is completely regular.

Exercises

14. If X and Y are topological spaces, f : X → Y is continuous iff f() ⊂ for all A ⊂ X iff for all B ⊂ Y.

15. If X is a topological space, A ⊂ X is closed, and g C(A) satisfies g = 0 on ∂A, then the extension of g to X defined by g(x) = 0 for x Ac is continuous.

16. Let X be a topological space, Y a Hausdorff space, and f, g continuous maps from X to Y.

a. {x : f(x) = g(x)} is closed.

b. If f = g on a dense subset of X, then f = g on all of X.

17. If X is a set, a collection of real-valued functions on X, and the weak topology generated by , then is Hausdorff iff for every x, y X with x ≠ y there exists f with f(x) ≠ f(y).

18. If X and Y are topological spaces and y₀ Y, then X is homeomorphic to X x {y₀} where the latter has the relative topology as a subset of X x Y.

19. If {Xα} is a family of topological spaces, X = πα Xα (with the Product topology) is uniquely determined up to homeomorphism by the following property: There exist continuous maps πα : X → Xα such that if Y is any topological space and fα C(Y, Xα) for each α, there is a unique F C(Y, X) such that . (Thus X is the category-theoretic product of the Xα’s in the category of topological spaces.)

20. If A is a countable set and Xα is a first (resp. second) countable space for each α A, then παA (resp. second) countable.

21. If X is an infinite set with the cofinite topology, then every f C(X) is constant.

22. Let X be a topological space, (Y, ρ) a complete metric space, and {fn} a sequence in YX such that supxX ρ(fn(x), fm(x)) → 0 as m, n → ∞. There is a unique f Yx such that supxX ρ(fn(x), f(x)) → 0 as n → ∞. If each fn is continuous, so is f.

23. Give an elementary proof of the Tietze extension theorem for the case X = .

24. A Hausdorff space X is normal iff X satisfies the conclusion of Urysohn’s lemma iff X satisfies the conclusion of the Tietze extension theorem.

25. If (X, ) is completely regular, then is the weak topology generated by C(X).

26. Let X and Y be topological spaces.

a. If X is connected (see Exercise 10) and f C(X, Y), then f(X) is connected.

b. X is called arcwise connected if for all x₀, x₁ X there exists f C([0,1], X) with f(0) = x₀ and f(1) = x₁. Every arcwise connected space is connected.

c. Let X = {(s, t) ² : t = sin(s⁻¹)} {(0,0)}, with the relative topology induced from ². Then X is connected but not arcwise connected.

27. If Xα is connected for each α A (see Exercise 10), then X = ΠαA Xα is connected. (Fix x X and let Y be the connected component of x in X. Show that Y includes {y X : πα(y) = πα(x) for all but finitely many α} and that the latter set is dense in X. Use Exercises 10 and 18.)

28. Let X be a topological space equipped with an equivalence relation, X the set of equivalence classes, π : X → the map taking each x X to its equivalence class, and is open in X}.

a. is a topology on . (It is called the quotient topology.)

b. If Y is a topological space, f : → Y is continuous iff f o π is continuous.

c. is T₁ iff every equivalence class is closed.

29. If X is a topological space and G is a group of homeomorphisms from X to itself, G induces an equivalence relation on X, namely, x ~ y iff y = g(x) for some g G. Let X = ²; describe the quotient space and the quotient topology on it (as in Exercise 28) for each of the following groups of invertible linear maps. In particular, show that in (a) the quotient space is homeomorphic to [0, ∞); in (b) it is T₁ but not Hausdorff; in (c) it is T₀ but not T₁, and in (d) it is not T₀. (In fact, in (d) is uncountable, but there are only six open sets and there are points p such that

4.3 NETS

As we have hinted above, sequential convergence does not play the same central role in general topological spaces as it does in metric spaces. The reasons for this may be illustrated by the following example. Consider the space of all complex-valued functions on , with the product topology (i.e., the topology of pointwise convergence), and its subspace C(). On the one hand, by Corollary 2.9, if {fn} ⊂ C() and fn → f pointwise, then f is Borel measurable, so the set of limits of convergent sequences in C() is a proper subset of C. Nonetheless, C() is dense in . Indeed, if f , the sets

form a neighborhood base at f, and each of these sets clearly contains continuous functions.

There is, however, a generalization of the notion of sequence that works well in arbitrary topological spaces; the key idea is to use index sets more general than . The precise definitions are as follows.

A directed set is a set A equipped with a binary relation such that

α α for all α A;

if α β and β γ then α γ;

for any α, β A there exists γ A such that α γ and β < γ.

If α β, we shall also write β α. A net in a set X is a mapping α xα from a directed set A into X. We shall usually denote such a mapping by xααA, or just by xα if A is understood, and we say that xα is indexed by A.

Here are some examples of directed sets:

i. The set of positive integers , with j k iff j ≤ k.

ii. The set \ {a} (a ), with iff |x − a| ≥ |y − a|.

iii. The set of all partitions {xj}n₀ of the interval [a, b] (i.e., a = x₀ < ··· < xn = b), with iff max(xj − xj−1) ≥ max(yk − yk−1).

iv. The set of all neighborhoods of a point x in a topological space X, with U V iff U ⊃ V. (We say that is directed by reverse inclusion.)

v. The Cartesian product A x B of two directed sets, with iff and (This is always the way we make A x B into a directed set.)

Examples (i)–(iii) occur in elementary analysis: A net indexed by is just a sequence, and the nets indexed by the sets in (ii) and (iii) occur in defining limits of real variables and Riemann integrals. Example (iv) is of fundamental importance in topology, and we shall see several uses of the construction in (v).

Let X be a topological space and E a subset of X. A net xααA is eventually in E if there exists α₀ A such that xα E for , and xα is frequently in E if for every α A there exists such that xβ E. A point x X is a limit of xα (or x₀ converges to x, or xα → x) if for every neighborhood U of x, xα is eventually in U, and x is a cluster point of xα if for every neighborhod U of x, xα is frequently in U.

The next three propositions show that nets are a good substitute for sequences.

4.18 Proposition. If X is a topological space, E ⊂ X, and x X, then x is an accumulation point of E iff there is a net in E \ {x} that converges to x, and x iff there is a net in E that converges to x.

Proof. If x is an accumulation point of E, let be the set of neighborhoods of x, directed by reverse inclusion. For each U , pick xU (U \ {x}) ⊂ E. Then xU → x. Conversely, if xα E \ {x} and xα → x, then every punctured neighborhood of x contains some xα, so x is an accumulation point of E, Likewise, if xα → x where xα E, then x , and the converse follows from Proposition 4.1.

4.19 Proposition. If X and Y are topological spaces and f : X → Y, then f is continuous at x X iff for every net xα converging to x, f(xα) converges to f(x).

Proof. If f is continuous at x and V is a neighborhood of f(x), then f⁻¹ (V) is a neighborhood of x. Hence, if xα → x then xα is eventually in f⁻¹(V), so f(xα) is eventually in V, and thus f(xα) → f(x). On the other hand, if f is not continuous at x, there is a neighborhood V of f(x) such that f⁻¹(V) is not a neighborhood of x, that is, x (f⁻¹(V))o, or equivalently, x . By Proposition 4.18, there is a net xα in f⁻¹(Vc) that converges to x. But then f(xα) V, so f(xα) .

A subnet of a net xααA is a net yββB together with a map β αβ from B to A such that:

for every α₀ A there exists β₀ B such that αβ α₀ whenever β β₀;

yβ = xαβ.

Clearly if xα converges to a point x, then so does any subnet xαβ.

Warning: The name “subnet” is used because subnets perform much the same functions as subsequences, but it should not be taken too literally, as the mapping β αβ need not be injective. In particular, the index set B may well have larger cardinality than the index set A, and a subnet of a sequence need not be a subsequence.

4.20 Proposition. If xααA is a net in a topological space X, then x X is a cluster point of has a subnet that converges to x.

Proof. If is a subnet converging to x and U is a neighborhood of x, choose β₁ B such that yβ U for β β₁. Also, given α A, choose β₂ B such that αβ α for β β₂. Then there exists β B with β β₁ and β β₂, and we have αβ α and xαβ = yβ U. Thus is frequently in U, so x is a cluster point of is a cluster point of , let be the set of neighborhoods of x and make x A into a directed set by declaring that iff U ⊃ U’ and α α’. For each (U, γ) x A we can choose α(U, γ) A such that α(U, γ) γ and xα(U, γ) U. Then if we have and whence it follows that is a subnet of that converges to x.

Exercises

30. If A is a directed set, a subset B of A is called cofinal in A if for each α A there exists β B such that

a. If B is cofinal in A and is a net, the inclusion map B A makes a subnet of

b. If is a net in a topological space, then xα converges to x iff for every cofinal B ⊂ A there is a cofinal C ⊂ B such that converges to x.

31. Let be a sequence.

a. If k → nk is a map from to itself, then is a subnet of iff nk → ∞ as k → ∞, and it is a subsequence (as defined in §0.1) iff nk is strictly increasing in k.

b. There is a natural one-to-one correspondence between the subsequences of and the subnets of defined by cofinal sets as in Exercise 30.

32. A topological space X is Hausdorff iff every net in X converges to at most one point. (If X is not Hausdorff, let x and y be distinct points with no disjoint neighborhoods, and consider the directed set x x y where x, y are the families of neighborhoods of x, y.)

33. Let be a net in a topological space, and for each α A let . Then x is a cluster point of .

34. If X has the weak topology generated by a family of functions, then converges to x X iff converges to f(x) for all f . (In particular, if then xα → x iff πi(xα) → πi(x) for all i I.)

35. Let X be a set and the collection of all finite subsets of X, directed by inclusion. Let f : X → be an arbitrary function, and for A , let . Then the net zA converges in is a countable set and in which case . (Cf. Proposition 0.20.)

36. Let X be the set of Lebesgue measurable complex-valued functions on [0,1]. There is no topology on X such that a sequence converges to f with respect to iff fn → f a.e. (Use Corollary 2.32 and Exercises 30b and 31b.)

4.4 COMPACT SPACES

In §0.6 we gave three equivalent characterizations of compactness for metric spaces: the Heine-Borel property, the Bolzano-Weierstrass property, and completeness plus total boundedness. Only the first two of these make sense for general topological spaces, and it is the first one that turns out to be the most useful. Accordingly, we define a topological space X to be compact if whenever is an open cover of X — that is, a collection of open sets such that there is a finite subset B of A such that To be brief (although somewhat sylleptic, since the adjectives “open” and “finite” refer to different things), we say: X is compact if every open cover of X has a finite subcover.

A subset Y of a topological space X is called compact if it is compact in the relative topology; thus Y ⊂ X is compact iff whenever is a collection of open subsets of X with Y there is a finite B ⊂ A with . Furthermore, Y is called precompact if its closure is compact.

DeMorgan’s laws lead to the following characterization of compactness in terms of closed sets. A family of subsets of X is said to have the finite intersection property if for all finite B ⊂ A.

4.21 Proposition. A topological space X is compact iff for every family of closed sets with the finite intersection property,

Proof. Let Uα = (Fα)c. Then Uα is open, iff and {Fα} has the finite intersection property iff no finite subfamily of {Uα} covers X. The result follows.

We now list several basic facts about compact spaces.

4.22 Proposition. A closed subset of a compact space is compact.

Proof. If X is compact, F ⊂ X is closed, and {Uα}αA is a family of open sets in X with then is an open cover of X. It has a finite subcover, so by discarding Fc from the latter if necessary, we obtain a finite subcollection of that covers F.

4.23 Proposition. If F is a compact subset of a Hausdorff space X and x F, there are disjoint open sets U, V such that x U and F ⊂ V.

Proof. For each y F, choose disjoint open Uy and Vy with x Uy and is an open cover of F, so it has a finite subcover Then and have the desired properties.

4.24 Proposition. Every compact subset of a Hausdorff space is closed.

Proof. According to Proposition 4.23, if F is compact then Fc is a neighborhood of each of its points, hence is open.

We remark that in a non-Hausdorff space, compact sets need not be closed (for example, every subset of a space with the trivial topology is compact), and the intersection of compact sets need not be compact; see Exercise 37. Of course, in a Hausdorff space the intersection of any family of compact sets is compact by Propositions 4.22 and 4.24. Moreover, in an arbitrary topological space a finite union of compact sets is always compact. (If K₁,… Kn are compact and {Uα} is an open cover of n₁ Kj, choose a finite subcover of each Kj and combine them.)

4.25 Proposition. Every compact Hausdorff space is normal.

Proof. Suppose that X is compact Hausdorff and E, F are disjoint closed subsets of X. By Proposition 4.23, for each x E there exist disjoint open sets Ux, Vx with x Ux, F ⊂ Vx. By Proposition 4.22, E is compact, and {Ux}xE is an open cover of E, so there is a finite subcover {Uxj}n₁. Let and Then U and V are disjoint open sets with E ⊂ U and F ⊂ V.

4.26 Proposition. If X is compact and f : X → Y is continuous, then f(X) is compact.

Proof. Let {Vα} be an open cover of f(X) in Y. Then {f⁻¹(Vα)} is an open cover of X, so it has a finite subcover {f⁻¹(Vαj)}, and {Vαj} is then a finite subcover of f(X).

4.27 Corollary. If X is compact, then C(X) = BC(X).

4.28 Proposition. If X is compact and Y is Hausdorff, then any continuous bijection f : X → Y is a homeomorphism.

Proof. If E ⊂ X is closed, then E is compact, hence f(E) is compact, hence f(E) is closed, by Propositions 4.22, 4.26, and 4.24. This means that f⁻¹ is continuous, so f is a homeomorphism.

We now show that a version of the Bolzano-Weierstrass property holds for compact topological spaces. As one might suspect, it is merely necessary to replace sequences by nets.

4.29 Theorem. If X is a topological space, the following are equivalent:

a. X is compact.

b. Every net in X has a cluster point.

c. Every net in X has a convergent subnet.

Proof. The equivalence of (b) and (c) follows from Proposition 4.20. If X is compact and is a net in X, let Since for any α, β A there exists γ A with γ α and γ β, the family has the finite intersection property, so by Proposition 4.21, If and U is a neighborhood of x, then U intersects each Eα, which means that is frequently in U, so x is a cluster point of On the other hand, if X is not compact, let {Uβ}βB be an open cover of X with no finite subcover. Let be the collection of finite subsets of B, directed by inclusion, and for each A let xA be a point in Then is a net with no cluster point. Indeed, if x X, choose β B with x Uβ. If A and then xA Uβ, so x is not a cluster point of

We conclude by mentioning two other useful concepts related to compactness. A topological space X is called countably compact if every countable open cover of X has a finite subcover, and sequentially compact if every sequence in X has a convergent subsequence. Of course, every compact space is countably compact, and for metric spaces compactness and sequential compactness are equivalent. However, in general there is no relation between compactness and sequential compactness. See Exercises 39–43 for further results and examples.

Exercises

37. Let 0’ denote a point that is is not an element of (−1,1), and let X = (−1,1) {0’}. Let be the topology on X generated by the sets (−1, a), (a, 1), [(−1, b) \ {0}]{0’}, and [(c, 1)\{0}]{0’} where −1 < a < 1, 0 < b < 1, and −1 < c < 0. (One should picture X as (−1, 1) with the point 0 split in two.)

a. Define f, g : (−1,1) → X by f(x) = x for all x, g(x) = x for x ≠ 0, and g(0) = 0’. Then f and g are homeomorphisms onto their ranges.

b. X is T₁ but not Hausdorff, although each point of X has a neighborhood that is homeomorphic to (−1,1) (and hence is Hausdorff).

c. The sets are compact but not closed in X, and their intersection is not compact.

38. Suppose that (X, ) is a compact Hausdorff space and is another topology on X. If is strictly stronger than , then (X, ) is Hausdorff but not compact. If is strictly weaker than , then (X, ) is compact but not Hausdorff.

39. Every sequentially compact space is countably compact.

40. If X is countably compact, then every sequence in X has a cluster point. If X is also first countable, then X is sequentially compact.

41. A T₁ space X is countably compact iff every infinite subset of X has an accumulation point.

42. The set of countable ordinals (§0.4) with the order topology (Exercise 9) is sequentially compact and first countable but not compact. (To prove sequential compactness, use Proposition 0.19.)

43. For x [0,1), let be the base-2 decimal expansion of x. (If x is a dyadic rational, choose the expansion such that an(x) = 0 for n large.) Then the sequence in {0, 1}[0,1) has no pointwise convergent subsequence. (Hence {0, 1}[0,1), with the product topology arising from the discrete topology on {0,1}, is not sequentially compact. It is, however, compact, as we shall show in §4.6.)

44. If X is countably compact and f : X → Y is continuous, then f (X) is countably compact.

45. If X is normal, then X is countably compact iff C(X) = BC(X). (Use Exercises 40 and 44. If is a sequence in X with no cluster point, then {xn : n } is closed, and Corollary 4.17 applies.)

4.5 LOCALLY COMPACT HAUSDORFF SPACES

A topological space is called locally compact if every point has a compact neighborhood. We shall be mainly concerned with locally compact Hausdorff spaces, which we call LCH spaces for short.

4.30 Proposition. If X is an LCH space, U ⊂ X is open, and x U, there is a compact neighborhood N of x such that N ⊂ U.

Proof. We may assume is compact; otherwise, replace U by U Fo where F is a compact neighborhood of x. By Proposition 4.23, there are disjoint relatively open sets V, W in with x V and ∂U ⊂ W. Then V is open in X since V U, and is a closed and hence compact subset of U \ W. Thus we may take N = .

4.31 Proposition. If X is an LCH space and K ⊂ U ⊂ X where K is compact and U is open, there exists a precompact open V such that K ⊂ V ⊂ ⊂ U.

Proof. By Proposition 4.30, for each x K we can choose a compact neighborhood Nx of x with Nx ⊂ U. Then {Nox}xK is an open cover of K, so there is a finite subcover Let then K ⊂ V and ⊂ n₁ Nxj is compact and contained in U.

4.32 Urysohn’s Lemma, Locally Compact Version. If X is an LCH space and K ⊂ U ⊂ X where K is compact and U is open, there exists f C(X, [0, 1]) such that f = 1 on K and f = 0 outside a compact subset of U.

Proof. Let V be as in Proposition 4.31. Then is normal by Proposition 4.25, so by Urysohn’s lemma 4.15 there exists f C(, [0,1]) such that f = 1 on K and f = 0 on ∂V. We extend f to X by setting f = 0 on c. Suppose that E ⊂ [0,1] is closed. If 0 E we have , and if 0 E we have since . In either case, f⁻¹(E) is closed, so f is continuous.

4.33 Corollary. Every LCH space is completely regular.

4.34 Tietze Extension Theorem, Locally Compact Version. Suppose that X is an LCH space and K ⊂ X is compact. If f C(K), there exists F C(X) such that F|K = f. Moreover, F may be taken to vanish outside a compact set.

The proof is similar to that of Theorem 4.32; details are left to the reader (Exercise 46).

The preceding results show that LCH spaces have a rich supply of continuous functions that vanish outside compact sets. Let us introduce some terminology: If X is a topological space and f C(X), the support of f, denoted by supp(f), is the smallest closed set outside of which/vanishes, that is, the closure of {x : f(x) ≠ 0}. If supp(f) is compact, we say that f is compactly supported, and we define

Moreover, if f C(X), we say that f vanishes at infinity if for every > 0 the set is compact, and we define

Clearly Cc(X) ⊂ C₀(X). Moreover, C₀(X) ⊂ BC(X), because for f C₀(X) the image of the set is compact, and |f| < on its complement.

4.35 Proposition. If X is an LCH space, C₀(X) is the closure of Cc(X) in the uniform metric.

Proof. If {fn} is a sequence in Cc(X) that converges uniformly to f C(X), for each > 0 there exists n such that Then |f(x)| < if x supp(fn), so f C₀(X). Conversely, if f C₀(X), for n let Then Kn is compact, so by Theorem 4.32 there exists gn Cc(X) with 0 ≤ gn ≤ 1 and gn = 1 on Kn. Let fn = gnf. Then fn Cc(X) and uniformly.

If X is a noncompact LCH space, it is possible to make X into a compact space by adding a single point “at infinity” in such a way that the functions in C₀(X) are precisely those continuous functions f such that f(x) → 0 as x approaches the point at infinity. More precisely, let ∞ denote a point that is not an element of X, let X* = X {∞}, and let be the collection of all subsets of X* such that either (i) U is an open subset of X, or (ii) ∞ U and Uc is a compact subset of X.

4.36 Proposition. If X, X*, and are as above, then (X*, ) is a compact Hausdorff space, and the inclusion map i : X → X* is an embedding. Moreover, if f C(X), then f extends continuously to X* iff f = g + c where g C₀(X) and c is a constant, in which case the continuous extension is given by f(∞) = c.

The proof is straightforward and is left to the reader (Exercise 47). The space X* is called the one point compactification or Alexandroff compactification of X.

If X is a topological space, the space X of all complex-valued functions on X can be topologized in various ways. One way, of course, is the product topology, that is, the topology of pointwise convergence. Another is the topology of uniform convergence, which is generated by the sets

The proof of Proposition 4.13 shows that C(X) is a closed subspace of X in the topology of uniform convergence. Intermediate between these two topologies is the topology of uniform convergence on compact sets, which is generated by the sets

We shall now examine this topology in the case where X is an LCH space.

4.37 Lemma. If X is an LCH space and E ⊂ X, then E is closed iff E K is closed for every compact K ⊂ X.

Proof. If E is closed, then is closed by Propositions 4.22 and 4.24. If is not closed, pick x \E and let K be a compact neighborhood of x. Then x is an accumulation point of E K but is not in E K, so by Proposition 4.1 E K is not closed.

4.38 Proposition. If X is an LCH space, C(X) is a closed subspace of X in the topology of uniform convergence on compact sets.

Proof. If f is in the closure of C(X), then f is a uniform limit of continuous functions on each compact K ⊂ X, so f|K is continuous. If E ⊂ is closed, f⁻¹(E) K = (f|K)⁻¹(E) is thus closed for each compact K, so by Lemma 4.37 f⁻¹(E) is closed, whence f is continuous.

A topological space X is called σ-compact if it is a countable union of compact sets. To appreciate the significance of the next two propositions, see Exercise 54.

4.39 Proposition. If X is a σ-compact LCH space, there is a sequence {Un} of precompact open sets such that for all n and X .

Proof. Suppose where each Kn is compact. Every compact subset of X has a precompact open neighborhood by Proposition 4.31. Thus we may take U₁ to be a precompact open neighborhood of K₁, and then, proceeding inductively, take Un to be a precompact open neighborhood of .

4.40 Proposition. If X is a σ-compact LCH space and {Un} is as in Proposition 4.39, then for each f X the sets

form a neighborhood base for f in the topology of uniform convergence on compact sets. Hence this topology is first countable, and fj → f uniformly on compact sets iff fj → f uniformly on each n.

Proof. These assertions follow easily from the observation that if K ⊂ X is compact, then {Un}∞₁ is an open cover of K and hence for some n. Details are left to the reader (Exercise 48).

We close this section with a construction that is useful in a number of situations. If X is a topological space and E ⊂ X, a partition of unity on E is a collection {hα}αA of functions in C(X, [0,1]) such that

each x X has a neighborhood on which only finitely many hα’s are nonzero;

∑αA hα(z) = 1 for x E.

A partition of unity {hα} is subordinate to an open cover of E if for each α there exists with supp(hα) ⊂ U.

4.41 Proposition. Let X be an LCH space, K a compact subset of X, and {Uj}n₁ an open cover of K. There is a partition of unity on K subordinate to consisting of compactly supported functions.

Proof. By Proposition 4.30, each x K has a compact neighborhood Nx such that Nx ⊂ Uj for some j. Since is an open cover of K, there exist x₁,…, xm such that . Let Fj be the union of those Nxk’s that are subsets of Uj. Then Fj is a compact subset of Uj, so by Urysohn’s lemma there exist g₁,…, gn Cc(X, [0,1]) with gj = 1 on Fj and supp(gj) ⊂ Uj. Since the Fj’s cover K we have on K, so by Urysohn again there exists f Cc(X, [0,1]) with f = 1 on K and supp. Let gn+1 = 1 − f, so that 0 everywhere, and for j = 1,…, n let . Then supp(hj) = supp(gj) ⊂ Uj and on K.

A generalization of this result may be found in Exercise 57.

Exercises

46. Prove Theorem 4.34.

47. Prove Proposition 4.36. Also, show that if X is Hausdorff but not locally compact, Proposition 4.36 remains valid except that X* is not Hausdorff.

48. Complete the proof of Proposition 4.40.

49. Let X be a compact Hausdorff space and E ⊂ X.

a. If E is open, then E is locally compact in the relative topology.

b. If E is dense in X and locally compact in the relative topology, then E is open. (Use Exercise 13.)

c. E is locally compact in the relative topology iff E is relatively open in .

50. Let U be an open subset of a compact Hausdorff space X and U* its one-point compactification (see Exercise 49a). If ϕ : X → U* is defined by ϕ(x) = x if x U and ϕ(x) = ∞ if x Uc, then ϕ is continuous.

51. If X and Y are topological spaces, ϕ C(X, Y) is called proper if ϕ⁻¹(K) is compact in X for every compact K ⊂ Y. Suppose that X and Y are LCH spaces and X* and Y* are their one-point compactifications. If ϕ C(X, Y), then ϕ is proper iff ϕ extends continuously to a map from X* to Y* by setting ϕ(∞X) = ∞Y.

52. The one-point compactification of n is homeomorphic to the n-sphere .

53. Lemma 4.37 remains true if the assumption that X is locally compact is replaced by the assumption that X is first countable.

54. Let have the relative topology induced from .

a. is not locally compact.

b. is σ-compact (it is a countable union of singleton sets), but uniform convergence on singletons (i.e., pointwise convergence) does not imply uniform convergence on compact subsets of .

55. Every open set in a second countable LCH space is σ-compact.

56. Define by Φ(t) = t/(t + 1) for t [0, ∞) and Φ(∞) = 1.

a. Φ is strictly increasing and Φ(t + s) ≤ Φ(t) + Φ(s).

b. If (Y, ρ) is a metric space, then Φ o ρ is a bounded metric on Y that defines the same topology as ρ.

c. If X is a topological space, the function is a metric on X whose associated topology is the topology of uniform convergence.

d. If X is a σ-compact LCH space and is as in Proposition 4.39, the function

is a metric on X whose associated topology is the topology of uniform convergence on compact sets.

57. An open cover of a topological space X is called locally finite if each x X has a neighborhood that intersects only finitely many members of . If , are open covers of X, is a refinement of if for each V there exists U with V U. X is called paracompact if every open cover of X has a locally finite refinement.

a. If X is a σ-compact LCH space, then X is paracompact. In fact, every open cover has locally finite refinements such that α is compact and α ⊂ Vα for all α. (Let {Un}∞₁ be as in Proposition 4.39. For each n, {E (Un+2 \ n−1) : E } is an open cover of n+1 \ Un. Choose a finite subcover to obtain {Vα} and mimic the beginning of the proof of Proposition 4.41 to obtain {Wα}.)

b. If X is a σ-compact LCH space, for any open cover of X there is a partition of unity on X subordinate to and consisting of compactly supported functions.

4.6 TWO COMPACTNESS THEOREMS

The geometric objects on which one does analysis (Euclidean spaces, manifolds, etc.) tend to be compact or locally compact. However, in infinite-dimensional spaces such as spaces of functions, compactness is a rather rare phenomenon and is to be greatly prized when it is available. Almost all compactness results in such situations are obtained via two basic theorems, Tychonoff’s theorem and the Arzelà-Ascoli theorem, which we present in this section.

Tychonoff’s theorem has to do with compactness of Cartesian products. To prepare for it, we introduce some notation. Recall that an element is, strictly speaking, a mapping from A into namely, x(α) Xα is the αth coordinate of x, which we generally denote by πα(x). If B ⊂ A, there is a natural map namely, πB(x) is the restriction of the map x to B. (In particular, π{α} is essentially identical to πα, and we shall not distinguish between them.) If and , we shall say that q is an extension of p if q extends p as a mapping, that is, if B ⊂ C and p(α) = q(α) for α B.

4.42 Tychonoff’s Theorem. If {Xα}αA is any family of compact topological spaces, then (with the product topology) is compact.

Proof. By Theorem 4.29, it is enough to show that any net xiiI in X has a cluster point. We shall do this by examining cluster points of the nets πB(xi) in the subproducts of X. To wit, let

is nonempty, because each Xα is compact and so has cluster points when B = {α}. Moreover, is partially ordered by extension; that is, p ≤ q if q is an extension of p as defined above.

Suppose that {pl : l L} is a linearly ordered subset of , where Let B* = lL Bl, and let p* be the unique element of that extends every pl. We claim that p* . Indeed, from the definition of the product topology, any neighborhood of p* contains a set of the form where each Uα is open in Xα and Uα = Xα for all but finitely many α, say α₁,…, αn. Each of these αj’s belongs to some Bl, so by linearity of the ordering they all belong to a single Bl. But then is a neighborhood of pl, so πBl (xi) is frequently in hence πB* (xi) is frequently in is a cluster point of πB*(xi). Therefore p* is an upper bound for {pl} in .

By Zorn’s lemma, then, has a maximal element . We claim that = A. If not, pick γ A \ . By Proposition 4.20 there is a subnet that converges to , and since Xγ is compact, there is a subnet that converges to some pγ Xγ. Let q be the unique element of that extends both and pγ; then the net converges to q and hence q is a cluster point of contradicting the maximality of . Therefore is a cluster point of , and we are done.

We now turn to the Arzelà-Ascoli theorem, which has to do with compactness in spaces of continuous mappings. There are several variants of this result; the theorems below are two of the most useful ones. See also Exercise 61.

If X is a topological space and is called equicontinuous at x X if for every > 0 there is a neighborhood U of x such that |f(y) − f(x)| < for all y U and all f , and is called equicontinuous if it is equicontinuous at each x X. Also, is said to be pointwise bounded if is a bounded subset of for each x X.

4.43 Arzelà-Ascoli Theorem I. Let X be a compact Hausdorff space. If is an equicontinuous, pointwise bounded subset of C(X), then is totally bounded in the uniform metric, and the closure of in C(X) is compact.

Proof. Suppose > 0. Since is equicontinuous, for each x X there is an open neighborhood Ux of x such that for all y Ux and all f . Since X is compact, we can choose x₁,…, xn X such that . Then by pointwise boundedness, is a bounded subset of , so there is a finite set that is -dense in it — that is, each f(xj) is at a distance less than from some zk. Let A = {x₁,…, xn} and B = {z₁,…, zm}; then the set BA of functions from A to B is finite. For each ϕ BA, let

Then clearly and we claim that each ϕ has diameter at most , so we obtain a finite -dense subset of by picking one f from each nonempty . To prove the claim, suppose . Since and on A, we have on A. If x X, we have x Uxj. for some j, and then

This shows that is totally bounded. Since the closure of a totally bounded set is totally bounded and C(X) is complete, the theorem is proved.

4.44 Arzelà-Ascoli Theorem II. Let X be a σ-compact LCH space. If {fn} is an equicontinuous, pointwise bounded sequence in C(X), there exist f C(X) and a subsequence of {fn} that converges to f uniformly on compact sets.

Proof. By Proposition 4.39 there is a sequence {Uk} of precompact open sets such that and . By Theorem 4.43 there is a subsequence that is uniformly Cauchy on ₁: we denote it by . Proceeding inductively, for k we obtain a subsequence that is uniformly Cauchy on k. Let ; then {gk} is a subsequence of and hence is uniformly Cauchy on each k. Let f = lim gk. Then f C(X) and gk → f uniformly on compact sets by Propositions 4.38 and 4.40.

Exercises

58. If is a family of topological spaces of which infinitely many are noncompact, then every closed compact subset of is nowhere dense.

59. The product of finitely many locally compact spaces is locally compact.

60. The product of countably many sequentially compact spaces is sequentially compact. (Use the “diagonal trick” as in the proof of Theorem 4.44.)

61. Theorem 4.43 remains valid for maps from a compact Hausdorff space X into a complete metric space Y provided the hypothesis of pointwise boundedness is replaced by pointwise total boundedness. (Make this statement precise and then prove it.)

62. Rephrase Theorem 4.44 in a form similar to Theorem 4.43 by using the metric in Exercise 56d.

63. Let . For . Then , and is precompact in C([0, 1]).

64. Let (X, ρ) be a metric space. A function f C(X) is called Hölder continuous of exponent α (α > 0) if the quantity

is finite. If X is compact, and Nα(f) ≤ 1} is compact in C(X).

65. Let U be an open subset of , and let {fn} be a sequence of holomorphic functions on U. If {fn} is uniformly bounded on compact subsets of U, there is a subsequence that converges uniformly to a holomorphic function on compact subsets of U. (Use the Cauchy integral formula to obtain equicontinuity.)

4.7 THE STONE-WEIERSTRASS THEOREM

In this section we prove a far-reaching generalization of the well-known theorem of Weierstrass to the effect that any continuous function on a compact interval [a, b] is the uniform limit of polynomials on [a, b]. Throughout this section, X will denote a compact Hausdorff space, and we equip the space C(X) with the uniform metric.

A subset of C(X, ) or C(X) is said to separate points if for every x, y X with x ≠ y there exists f such that f(x) ≠ f(y). is called an algebra if it is a real (resp. complex) vector subspace of C(X, ) (resp. C(X)) such that fg whenever f, g . If ⊂ C(X, ), is called a lattice if max(f, g) and min(f, g) are in whenever f, g . Since the algebra and lattice operations are continuous, one easily sees that if is an algebra or a lattice, so is its closure in the uniform metric.

4.45 The Stone-Weierstrass Theorem. Let X be a compact Hausdorff space. If is a closed subalgebra of C(X, ) that separates points, then either = C(X, ) or = {f C(X, ) : f(x₀) = 0} for some x₀ X. The first alternative holds iff contains the constant functions.

The proof will require several lemmas. The first one, in effect, proves the theorem when X consists of two points, and the second one is a special case of the classical Weierstrass theorem for X = [−1, 1]. After these two we return to the general case.

4.46 Lemma. Consider ² as an algebra under coordinatewise addition and multiplication. Then the only subalgebras of ² are ², {(0, 0)}, and the linear spans of (1, 0), (0, 1), and (1,1).

Proof. The subspaces of ² listed above are evidently subalgebras. If ² is a nonzero algebra and (0, 0) ≠ (a, b) , then (a², b²) . If a ≠ 0, b ≠ 0, and a ≠ b, then (a, b) and (a², b²) are linearly independent, so = ². The other possibilities — a ≠ 0 = b, a = 0 ≠ b, and a = b ≠ 0 for all nonzero (a, b) — give the other three subalgebras.

4.47 Lemma. For any > 0 there is a polynomial P on such that P(0) = 0 and ||x| − P(x)| < for x [−1, 1].

Proof. Consider the Maclaurin series for (1 − t)1/2:

By the ratio test, this series converges for |t| < 1; a proof that its sum is actually (1 − t)1/2 is outlined in Exercise 66. Moreover, by the monotone convergence theorem (applied to counting measure on ),

It follows from the finiteness of that the series converges absolutely and uniformly on [−1, 1], and its sum is (1 − t)1/2 there. Therefore, given > 0, by taking a suitable partial sum of this series we obtain a polynomial Q such that for . Setting t = 1 − x² and R(x) = Q(1 − x²), we obtain a polynomial R such that for x [−1, 1]. In particular, , so if we set P(x) = R(z) − R(0), P is a polynomial such that P(0) = 0 and ||x| − P(x)| < for x [−1,1].

4.48 Lemma., If is a dosed subalgebra of C(X, ), then |f| whenever f and is a lattice.

Proof. If f and f ≠ 0, let Then h maps X into [−1, 1], so if > 0 and P is as in Lemma 4.47, we have . Since P(0) = 0, P has no constant term, so P o h since is an algebra. Since is closed and is arbitrary, we have |h| and hence This proves the first assertion, and the second one follows because

4.49 Lemma. Suppose is a closed lattice in C(X, ) and f C(X, ). If for every x, y X there exists gxy such that gxy(x) = f(x) and gxy(y) − f(y), then f .

Proof. Given > 0, for each and . These sets are open and contain x and y. Fix y; then {Uxy : x X} covers X, so there is a finite subcover Let gy = max then f < gy + on , which is open and contains y. Thus is another open cover of X, so there is a finite subcover Let . Since is a lattice, g and since is closed and is arbitrary, f .

Proof of Theorem 4.45. Given x ≠ y X, let . Then xy is a subalgebra of ² as in Lemma 4.46 because f → (f(x), f(y)) is an algebra homomorphism. If xy = ² for all x, y, then Lemmas 4.48 and 4.49 imply that = C(X, ). Otherwise, there exist x, y for which xy is a proper subalgebra of ². It cannot be {(0, 0)} or the linear span of (1, 1) because separates points, so by Lemma 4.46 xy is the linear span of (1, 0) or (0, 1). In either case there exists x₀ X such that f(x₀) = 0 for all f . There is only one such x₀ since separates points, so if neither x nor y is x₀, we have xy = ². Lemmas 4.48 and 4.49 now imply that = {f C(X, ) : f(x₀) = 0}. Finally, if contains constant functions, there is no x₀ such that f(x₀) = 0 for all f , so must equal C(X, ).

We have stated the Stone-Weierstrass theorem in the form that is most natural for the proof. However, in applications one is typically dealing with a subalgebra of C(X, ) that is not closed, and one applies the theorem to The resulting restatement of the theorem is as follows:

4.50 Corollary. Suppose is a subalgebra of C(X, ) that separates points. If there exists x₀ X such that f(x₀) = 0 for all f , then is dense in {f C(X, ) : f(x₀) = 0}. Otherwise, is dense in C(X, ).

The classical Weierstrass approximation theorem is the special case of this corollary where X is a compact subset of n and is the algebra of polynomials on n (restricted to X); here contains the constant functions, so the conclusion is that it is dense in C(X, ).

The Stone-Weierstrass theorem, as it stands, is false for complex-valued functions. For example, the algebra of polynomials in one complex variable is not dense in C(K) for most compact subsets K of . (In particular, if Ko ≠ Ø, any uniform limit of polynomials on K must be holomorphic on Ko.) Here we shall give a simple proof that the function cannot be approximated uniformly by polynomials on the unit circle {eit : t [0, 2π]}. If P(z) = ∑n₀ ajzj, then

Thus, abbreviating f(eit) and P(eit) by f and P, since |f| = 1 on the unit circle we have

Therefore, ||f − P||u ≥ 1 for any polynomial P.

There is, however, a complex version of the Stone-Weierstrass theorem.

4.51 The Complex Stone-Weierstrass Theorem. Let X be a compact Hausdorff space. If is a closed complex subalgebra of C(X) that separates points and is closed under complex conjugation, then either = C(X) or = {f C(X) : f(x₀) = 0} for some x₀ X.

Proof. Since Re and Im the set of real and imaginary parts of functions in is a subalgebra of C(X, ) to which the Stone-Weierstrass theorem applies. Since , the desired result follows.

There is also a version of the Stone-Weierstrass theorem for noncompact LCH spaces. We state this result for real functions; the corresponding analogue of Theorem 4.51 for complex functions is an immediate consequence.

4.52 Theorem. Let X be a noncompact LCH space. If is a closed subalgebra of C₀(X, ) (= C₀(X) C(X, )) that separates points, then either = C₀(X, ) or = {f C₀(X, ) : f(x₀) = 0} for some x₀ X.

The proof is outlined in Exercise 67.

Exercises

66. Let be the Maclaurin series for (1 − t)1/2.

a. The series converges absolutely and uniformly on compact subsets of (−1, 1), as does the termwise differentiated series . Thus, if f(t) = , then .

b. By explicit calculation, , from which it follows that (1 − t)−1/2f(t) is constant. Since f(0) = 1, f(t) = (1 − t)1/2.

67. Prove Theorem 4.52. (If there exists x₀ X such that f(x₀) = 0 for all f , let Y be the one-point compactification of X \ {x₀}; otherwise let Y be the one-point compactification of X. Apply Proposition 4.36 and the Stone-Weierstrass theorem on Y.)

68. Let X and Y be compact Hausdorff spaces. The algebra generated by functions of the form f(x, y) = g(x)h(y), where g C(X) and h C(Y), is dense in C(X x Y).

69. Let A be a nonempty set, and let X = [0, 1]A. The algebra generated by the coordinate maps πα : X → [0, 1] (α A) and the constant function 1 is dense in C(X).

70. Let X be a compact Hausdorff space. An ideal in C(X, ) is a subalgebra of C(X, ) such that if f and g C(X, ) then fg .

a. If is an ideal in C(X, ), let h() = {x X : f(x) = 0 for all f }. Then h() is a closed subset of X, called the hull of .

b. If E ⊂ X, let k(E) = {f C(X, ) : f(x) = 0 for all x E}. Then k(E) is a closed ideal in C(X, ), called the kernel of E.

c. If E X, then h(k(E)) = .

d. If is an ideal in C(X, ), then. (Hint: k(h()) may be identified with a subalgebra of C₀(U, ) where U = X \ h().)

e. The closed subsets of X are in one-to-one correspondence with the closed ideals of C(X, ).

71. (This is a variation on the theme of Exercise 70; it does not use the Stone-Weierstrass theorem.) Let X be a compact Hausdorff space, and let M be the set of all nonzero algebra homomorphisms from C(X, ) to . Each x X defines an element .

a. If ϕ M, then {f C(X, ) : ϕ(f) = 0} is a maximal proper ideal in C(X, ).

b. If is a proper ideal in C(X, ), there exists x₀ X such that f(x₀) = 0 for all f . (Suppose not; construct an f with f > 0 everywhere and conclude that 1 . This requires no deep theorems.)

c. The map x is a bijection from X to M.

d. If M is equipped with the topology of pointwise convergence, then the map x is a homeomorphism from X to M. (Since M is defined purely algebraically, it follows that the topological structure of X is completely determined by the algebraic structure of C(X, ).)

4.8 EMBEDDINGS IN CUBES

We now present a technique for embedding topological spaces in products of intervals and discuss some of its applications. (These results will not be used elsewhere in this book.) Throughout this section we shall denote the unit interval [0, 1] by I, and if A is any nonempty set, we shall call the product space IA a cube.

If X is a topological space and ⊂ C(X, I), we say that separates points and closed sets if for every closed E X and every x Ec there exists f such that If separates points and closed sets, there is another family with the following slightly stronger property: For every closed set E ⊂ X and every x Ec there exists g with g(x) = 1 and g = 0 on E. (Indeed, if f satisfies f(x) , take where ϕ C(I, I), ϕ(f(x)) = 1, and ϕ = 0 on It follows that a T₁ space X admits a family that separates points and closed sets iff X is completely regular.

Each nonempty canonically induces a map by the formula πf(e(x)) = f(x), where is the coordinate map. We call e the map from X into the cube associated to . (Evidently this construction can be generalized to target spaces other than I; see Exercise 19.)

4.53 Proposition. Let X be a topological space, ⊂ C(X, I), and e : be the map associated to Then

a. e is continuous.

b. If separates points, then e is injective.

c. If X is T₁ and separates points and closed sets, then e is an embedding.

Proof. (a) follows from Proposition 4.11, and (b) is obvious. Next, observe that if separates points and closed sets and X is T₁, then e is injective by (b) and Proposition 4.7. To prove the continuity of the inverse, suppose that U is open in X. If x U, choose f with and let

Then V is open in I and e(x) V e(X) ⊂ e(U). Thus e(U) is a neighborhood of e(x) in e(X) at every x U, so e(U) is open in e(X). It follows that e⁻¹ is continuous.

4.54 Corollary. Every compact Hausdorff space is homeomorphic to a closed subset of a cube.

Proof. By Proposition 4.25 and Urysohn’s lemma, we can take = C(X, I).

4.55 Corollary. A topological space is completely regular iff it is homeomorphic to a subset of a compact Hausdorff space.

Proof. Proposition 4.53, with = C(X, I), gives the “only if” implication; the converse is left to the reader (Exercise 72).

A compactification of a topological space X is a pair (Y, ϕ) where Y is a compact Hausdorff space and ϕ is a homeomorphism from X onto a dense subset of Y. (Frequently one identifies X with its image ϕ(X) Y and then speaks simply of “the compactification Y of X.”) For example, ([−1, 1], tanh) is a compactification of , and the one-point compactification (X*, i) of an LCH space X is a compactification in the present sense, where i : X → X* is the inclusion map.

Suppose X is completely regular. According to Proposition 4.53, if ⊂ C(X, I) separates points and closed sets, e : X → I is the associated embedding, and Y is the closure of e(X) in I, then (Y, e) is a compactification of X. It has the property that if we identify X with its image e(X), every f has a continuous extension to Y, which is unique since X is dense in Y. Indeed, the identification of X with e(X) turns f into the coordinate map πf|e(X), which extends to πf|Y. Moreover, if f and g are bounded continuous functions on X that extend continuously to Y, obviously so are f + g and fg, and if {fn} is a uniformly convergent sequence of functions on X that extend continuously to Y, their extensions converge uniformly on Y since X is dense in Y, so f = lim fn also extends continuously. We have proved:

4.56 Proposition. Suppose that separates points and closed sets. Let (Y, e) be the compactification of X associated to , and let be the smallest closed subalgebra of BC(X) that contains . Then every f has a continuous extension to Y.

This result has a converse: see Exercise 73.

If X is a completely regular space, the compactification of X associated to = C(X, I) is called the Stone-Cech compactification of X and is denoted by (βX, e), or simply by βX if we identify X with e(X). Every f BC(X) extends continuously to βX; in fact, a much more general result holds:

4.57 Theorem. If X is a completely regular space, Y is a compact Hausdorff space, and ϕ C(X, Y), then ϕ has a unique continuous extension that is, there is a unique such that is a compactification of X, then is surjective; if also every f BC(X) extends continuously to Y (i.e., f = g o ϕ for some g C(Y)), then is a homeomorphism.

Proof. Let = C(X, I) and , and let (βY, i) be the Stone-ech compactification of Y. (That is, i : Y → is the embedding associated to , and βY = i(Y); βY is homeomorphic to Y since Y is compact.) Given ϕ C(X, Y), define by . The map Φ is continuous by Proposition 4.11, and

that is, . It follows that Φ(e(X)) = i(ϕ(X)) ⊂ βY and hence that . The situation is summarized in the following commutative diagram:

Let . Then , and uniqueness of is clear since e(X) is dense in βX; thus the first assertion is proved. If (Y, ϕ) is a compactification of X, then ϕ(X) is dense in Y; but then is dense in Y and also compact, so that = Y. Finally, if every f BC(X) is of the form g o ϕ for some g C(Y), then Φ is injective; hence is bijective and therefore, by Proposition 4.28, a homeomorphism.

This theorem shows that βX is the “largest” compactification of a completely regular space X, in the sense that every other compactification is a continuous image of it. At the other end of the scale, if X is locally compact, then separates points and closed sets by Urysohn’s lemma. A glance at the construction of the compactification (Y, e) associated to this shows that Y consists of e(X) together with the single point of I all of whose coordinates are zero. It is then easy to verify that Y is homeomorphic to the one-point compactification of X constructed in §4.5.

As a final application of the embedding e : X → I, we give a partial answer to the question: When is a topological space metrizable, that is, when is its topology defined by a metric? A necessary condition for X to be metrizable is that X be normal (Exercise 3). On the other hand:

4.58 The Urysohn Metrization Theorem. Every second countable normal space is metrizable.

Since every subset of a metrizable space is metrizable (with the same metric), this theorem is an immediate consequence of Proposition 4.53 and the following two facts, whose proofs are outlined in Exercises 76 and 77:

If X is normal and second countable, there is a countable family that separates points and closed sets.

If is countable, I is metrizable.

Exercises

72. Every subset of a completely regular space is completely regular in the relative topology.

73. If X is a completely regular space, a subalgebra of BC(X) is called completely regular if (i) it is closed and contains the constant functions, and (ii) C(X, I) separates points and closed sets.

a. If (Y, e) is a Hausdorff compactification of is a completely regular subalgebra of BC(X).

b. If (Y, e) and (Y’, e′) are Hausdorff compactifications of X such that Y = Y, there is a homeomorphism ϕ : Y → Y’ such that . (Adapt the proof of Theorem 4.57, which deals with the case Y = βX.)

c. If (Y, e) is the compactification of X associated to ⊂ C(X, I), then Y is the smallest closed subalgebra of BC(X) that contains . (Use Exercise 69.)

d. The Hausdorff compactifications of X are in one-to-one correspondence with the completely regular subalgebras of BC(X).

74. Consider (with the discrete topology) as a subset of its Stone-ech compactification β.

a. If A and B are disjoint subsets of , their closures in β are disjoint. (Hint: χA C(, I).)

b. No sequence in converges in β unless it is eventually constant (so β is emphatically not sequentially compact).

75. Suppose X is a completely regular space. The set M of nonzero algebra homomorphisms from BC(X, ) to , equipped with the topology of pointwise convergence, is homeomorphic to βX. (See Exercise 71. This realization of βX is the natural one from the point of view of Banach algebra theory.)

76. If X is normal and second countable, there is a countable family that separates points and closed sets. (Let be a countable base for the topology. Consider the set of pairs (U, V) x such that U ⊂ V, and use Urysohn’s lemma.)

77. Let be a countable family of metric spaces whose metrics take values in [0,1]. (The latter restriction can always be satisfied; see Exercise 56b.) Let . If x, y X, say x = (x₁, x₂,…) and y = (y₁, y₂,…), define . Then ρ is a metric that defines the product topology on X.

4.9 NOTES AND REFERENCES

The germ of the concept of topological space is clearly present in Riemann’s lecture [113] on the foundations of geometry, delivered in 1854, but another half century passed before the mathematical world was ready to consider abstract spaces in a systematic way. The first attempt to construct an abstract framework for the study of limits and continuity was made in 1906 by Fréchet [52], who introduced metric spaces as well as a more general class of quasi-topological spaces whose properties were defined in terms of sequential convergence. A few years later Hausdorff [68] devised axioms for neighborhoods of points that amount to the definition of a Hausdorff space, and he deduced from them many of the basic results of general topology. The usefulness of his point of view was quickly recognized, and it became the foundation for the further development of the subject.

There are several good books to which the reader may refer for a more comprehensive treatment of point set topology, including Bourbaki [20], Dugundji [34], Engelking [38], Kelley [83], and Nagata [102]. Engelking [38] contains extensive references and historical notes.

§4.2: Urysohn’s lemma and the Tietze extension theorem were both first proved in Urysohn [152]. Special cases of the latter had previously been obtained by several authors, including Tietze (see [152] for references). Examples of completely regular spaces that are not normal and regular spaces that are not completely regular, which are all rather complicated, were first constructed by Tychonoff [151]. Particularly noteworthy is the existence of a regular space that admits no nonconstant continuous functions, a result due to Hewitt [73]. Examples may also be found in the books cited above.

§4.3: The theory of nets is sometimes called the Moore-Smith theory of convergence, after its originators [10.1]. Another general theory of convergence, invented by H. Cartan and publicized by Bourbaki, is based on the notion of filters. A filter in a set X is a family with the following properties:

If F and E ⊃ F, then E .

If E and F , then E F .

.

If X is a topological space, a filter in X converges to x X if every neighborhood of x belongs to . Filters and nets are related as follows. If xααA is a net in X, its derived filter is the collection of all E ⊂ X such that xα is eventually in E. On the other hand, if is a filter, then is a directed set under reverse inclusion, and a net xFF indexed by is said to be associated to if xF F for all F . It is then easy to verify that a net xα converges to x iff its derived filter converges to x, and a filter converges to x iff all of its associated nets converge to x. See Bourbaki [20] or Dugundji [34] for more information.

§4.4: The usage of the term “compact” is not completely standardized. In many older works the terms “compact” and “bicompact” were used to mean countably compact and compact, respectively, and some authors use “compact” and “quasi-compact” to mean compact Hausdorff and compact, respectively. Synonyms for “precompact” that are frequently found in the literature are “conditionally compact” and “relatively compact”; the latter one is infelicitous because it suggests compactness in the relative topology, which is quite different.

§4.6: Tychonoff [151] proved that [0, 1]A is compact for any set A; together with Corollary 4.54, which is in the same paper, this easily implies that any product of compact Hausdorff spaces is compact. The Tychonoff theorem in full generality is due to ech [23]. The proof we have presented, which is simpler and more elegant than the older ones, is due to Chernoff [24].

The axiom of choice, usually in the form of Zorn’s lemma, is an essential ingredient in all the proofs of Tychonoff’s theorem. It is an intriguing fact, discovered by Kelley [82], that Tychonoff’s theorem in turn implies the axiom of choice. Here is the proof:

Suppose that {Xα}αA is a nonempty collection of nonempty sets. Pick a point ω that is not an element of any Xα, set X*α = Xα {ω}, and define a topology on X*α by declaring the open sets to be Ø, Xα, {ω}, and X*α. Evidently X*α is compact, so Tychonoff’s theorem implies that is compact. Let Fα = π⁻¹α(Xα). The sets Fα are closed, and by the axiom of choice for finite collections of sets — which is provable from the other standard axioms of set theory — they have the finite intersection property. Indeed, given a finite set B ⊂ A, pick xβ Xβ for β B; then contains the point x X such that πβ(x) = xβ for β B and πα(x) = ω for α B. By Proposition 4.21, , which is precisely ΠαA Xα is nonempty.

By elaboration of this argument, one can deduce the axiom of choice from the special case of Tychonoff’s theorem that XA is compact for any A if X is compact; see Ward [156].

The original results of Arzelà and Ascoli had to do with functions on ; see Arzelà [6]. Other versions of the Arzelà-Ascoli theorem, pertaining to the compactness of subsets of C(X, Y) under various hypotheses on X and Y, can be found in the books cited above and in Royden [121].

§4.7. The Stone-Weierstrass theorem first appeared in the middle of a lengthy and difficult paper of Stone [144]. Later Stone [145] wrote a much-simplified exposition of the theorem and some of its applications, which still makes good reading.

§4.8. The history of this material begins with Urysohn [153], where the metrization theorem is proved, essentially by the method we have outlined. The technique of embedding spaces in cubes is implicit in this paper, but it was first developed explicitly in Tychonoff [151]. The Stone-ech compactification, in turn, is implicit in the latter paper, but it was first described explicitly and investigated by Stone [144] and ech [23].

It is not hard to show that every second countable regular space is normal (see Kelley [82, Lemma 4.1]; consequently, the hypothesis of normality in the Urysohn metrization theorem can be replaced by regularity. Necessary and sufficient conditions are known for an arbitrary topological space to be metrizable, but they are not as readily verifiable as the conditions in Urysohn’s theorem. See the books cited above.

Occasionally the term “compactification” is used to mean a continuous injection ϕ : X → Y from a topological space X onto a dense subset of a compact space Y without the requirement that it be an embedding. Such “compactifications” arise from subalgebras of C(X) that separate points but are not completely regular in the sense of Exercise 73. An example is provided by the algebra of “uniformly almost periodic” functions on , which is the algebra generated by the functions fλ(x) = eiλx, λ ; the associated “compactification” of is known as the Bohr compactification. See Folland [47, §4.7].

Chapter 5

Elements of Functional Analysis

“Functional analysis” is the traditional name for the study of infinite-dimensional vector spaces over or and the linear maps between them. What distinguishes this from mere linear algebra is the importance of topological considerations. On finite-dimensional vector spaces there is only one reasonable topology, and linear maps are automatically continuous, but in infinite dimensions things are not so simple. (As we have already observed, if {fn} is a sequence of functions on , there are many things one can mean by the statement “fn → f.”) As our aim in this chapter is only to give a brief introduction to the subject, we shall restrict attention — except in §5.4 — to topologies defined by norms on vector spaces.

5.1 NORMED VECTOR SPACES

Let K denote either or , and let be a vector space over K. We denote the zero element of simply by 0, relying on context to distinguish it from the scalar 0 K. By a subspace we shall always mean a vector subspace. If x , we denote by Kx the one-dimensional subspace spanned by x. Also, if and are subspaces of , + denotes the subspace {x + y : x , y }.

A seminorm on is a function x ||x|| from to [0, ∞) such that

||x + y|| ≤ ||x|| + ||y|| for all x, y (the triangle inequality),

||λx|| = |λ|||x|| for all x and λ K.

The second property clearly implies that ||0|| = 0. A seminorm such that ||x|| = 0 only when x = 0 is called a norm, and a vector space equipped with a norm is called a normed vector space (or normed linear space).

If is a normed vector space, the function ρ(x, y) = ||x − y|| is a metric on , since

The topology it defines is called the norm topology on . Two norms ||·||₁ and ||·||₂ on are called equivalent if there exist C₁, C₂ > 0 such that

Equivalent norms define equivalent metrics and hence the same topology and the same Cauchy sequences.

A normed vector space that is complete with respect to the norm metric is called a Banach space. (Every normed vector space can be embedded in a Banach space as a dense subspace. One way to do this is to mimic the construction of from via Cauchy sequences; we shall present a simpler way in §5.2.) The following is a useful criterion for completeness of a normed vector space. If {xn} is a sequence in , the series ∑∞₁xn is said to converge to x if ∑N₁ xn → x as N → ∞, and it is called absolutely convergent if ∑∞₁ ||xn|| < ∞.

5.1 Theorem. A normed vector space is complete iff every absolutely convergent series in converges.

Proof. If is complete and ∑∞₁ ||xn|| < ∞, let SN = ∑N₁ xn. Then for N > M we have

so the sequence {SN} is Cauchy and hence convergent. Conversely, suppose that every absolutely convergent series converges, and let {xn} be a Cauchy sequence. We can choose n₁ < n₂ < ··· such that ||xn − xm|| < 2−j for m, n > nj. Let y₁ = xn₁ and yj = xnj − xnj−1 for j > 1. Then ∑k₁ yj = xnk, and

so lim xnk = ∑∞₁ yj exists. But since {xn} is Cauchy, it is easily verified that {xn} converges to the same limit as {xnk}.

We have already seen some examples of Banach spaces. First, if X is a topological space, B(X) and BC(X) are Banach spaces with the uniform norm ||f||u = supxX |f(x)|. Second, if (X, , μ) is a measure space, L¹(μ) is a Banach space with the L¹ norm ||f||₁ = ∫|f| dμ. (Observe that ||·||₁ is only a seminorm if we think of L¹(μ) as consisting of individual functions, but it becomes a norm if we identify functions that are equal a.e.) That L¹(μ) is complete follows from Theorems 2.25 and 5.1. Indeed, if ∑∞₁||fn||₁ < ∞, Theorem 2.25 shows that f = ∑∞₁ fn exists a.e., and

More examples will be found in Exercises 8–11 and in subsequent sections.

If and are normed vector spaces, x becomes a normed vector space when equipped with the product norm

(Here, of course, ||x|| refers to the norm on while ||y|| refers to the norm on .) Sometimes other norms equivalent to this one, such as ||(x, y)|| = ||x|| + ||y|| or ||(x, y)|| = (||x|| + ||y||²)1/2, are used instead.

A related construction is that of quotient spaces. If is a vector subspace of the vector space , it defines an equivalence relation on as follows: x ~ y iff x − y M. The equivalence class of x is denoted by x + , and the set of equivalence classes, or quotient space, is denoted by /. / is a vector space with vector operations (x + ) + (y + ) = (x+y) + and λ(x+) = (λx) + . If is a normed vector space and is closed, / inherits a norm from called the quotient norm, namely

See Exercise 12 for a more detailed discussion.

A linear map T : → between two normed vector spaces is called bounded if there exists C ≥ 0 such that

This is different from the notion of boundedness for functions on a set, according to which T would be bounded if ||Tx|| ≤ C for all x. Clearly no nonzero linear map can satisfy the latter condition, since T(λx) = λTx for all scalars λ. The present definition means that T is bounded on bounded subsets of .

5.2 Proposition. If and are normed vector spaces and T : → is a linear map, the following are equivalent:

a. T is continuous.

b. T is continuous at 0.

c. T is bounded.

Proof. That (a) implies (b) is trivial. If T is continuous at 0 , there is a neighborhood U of 0 such that T(U) ⊂ {y : ||y|| ≤ 1}, and U must contain a ball B = {x : ||x|| < δ} about 0; thus ||Tx|| ≤ 1 when ||x|| ≤ δ. Since T commutes with scalar multiplication, it follows that ||Tx|| ≤ aδ⁻¹ whenever ||x|| ≤ a, that is, ||Tx|| ≤ δ⁻¹||x||. This shows that (b) implies (c). Finally, if ||Tx|| ≤ C||x|| for all x, then ||Tx₁ − Tx₂|| = ||T(x₁ − x₂)|| ≤ whenever ||x₁ − x₂|| ≤ C⁻¹, so that T is continuous.

If and are normed vector spaces, we denote the space of all bounded linear maps from to by L(, ). It is easily verified that L(, ) is a vector space and that the function T → ||T|| defined by

(5.3) equation

is a norm on L(, ), called the operator norm (Exercise 2). We always assume L(, ) to be equipped with this norm unless we specify otherwise.

5.4 Proposition. If is complete, so is L(, ).

Proof. Let {Tn} be a Cauchy sequence in L(, ). If x , then {Tnx} is Cauchy in because ||Tnx − Tmx|| ≤ ||Tn − Tm|| ||x||. Define T : → by Tx = limTnx. We leave it to the reader (Exercise 3) to verify that T L(, ) (in fact, ||T|| = lim ||Tn||) and that ||Tn − T|| → 0.

Another useful property of the operator norm is the following. If T L(, ) and S L(, ), then

so that ST L(, ) and ||ST|| < ||S|| ||T||. In particular, L(, ) is an algebra. If is complete, L(, ) is in fact a Banach algebra: a Banach space that is also an algebra, such that the norm of a product is at most the product of the norms. (Another example of a Banach algebra is BC(X), where X is a topological space, with pointwise multiplication and the uniform norm.)

If T L(, ), T is said to be invertible, or an isomorphism, if T is bijective and T⁻¹ is bounded (in other words, ||Tx|| ≥ C||x|| for some C > 0). T is called an isometry if ||Tx|| = ||x|| for all x X. An isometry is injective but not necessarily surjective; it is, however, an isomorphism onto its range.

Exercises

1. If is a normed vector space over K (= or ), then addition and scalar multiplication are continuous from x and K x to . Moreover, the norm is continuous from to [0, ∞); in fact, |||x|| − ||y||| ≤ ||x − y||.

2. L(, ) is a vector space and the function ||·|| defined by (5.3) is a norm on it. In particular, the three expressions on the right of (5.3) are always equal.

3. Complete the proof of Proposition 5.4.

4. If , are normed vector spaces, the map (T, x) → Tx is continuous from L(, ) x to . (That is, if Tn → T and xn → x then Tnxn → Tx.)

5. If is a normed vector space, the closure of any subspace of is a subspace.

6. Suppose that is a finite-dimensional vector space. Let e₁,…, en be a basis for , and define ||∑n₁ ajej||₁ = ∑n₁ |aj|.

a. ||·||₁ is a norm on .

b. The map (a₁i,…, an) → ∑n₁ ajej is continuous from Kn with the usual Euclidean topology to with the topology defined by ||·||₁.

c. {x : ||x||₁ = 1} is compact in the topology defined by ||·||₁.

d. All norms on are equivalent. (Compare any norm to ||·||₁.)

7. Let be a Banach space.

a. If T L(, ) and ||I − T|| < 1 where I is the identity operator, then T is invertible; in fact, the series ∑∞₀(I − T)n converges in L(, ) to T⁻¹.

b. If T L(, ) is invertible and ||S − T|| < ||T⁻¹||⁻¹. then S is invertible. Thus the set of invertible operators is open in L(, ).

8. Let (X, ) be a measurable space, and let M(X) be the space of complex measures on (X, ). Then ||μ|| = |μ|(X) is a norm on M(X) that makes M(X) into a Banach space. (Use Theorem 5.1.)

9. Let Ck([0, 1]) be the space of functions on [0, 1] possessing continuous derivatives up to order k on [0, 1], including one-sided derivatives at the endpoints.

a. If f C([0, 1]), then f Ck([0, 1]) iff f is k times continuously differentiable on (0, 1) and and exist for j ≤ k. (The mean value theorem is useful.)

b. ||f|| = ∑k₀||f(j)||u is a norm on Ck([0, 1]) that makes Ck([0, 1]) into a Banach space. (Use induction on k. The essential point is that if {fn} ⊂ C¹([0, 1]), fn fn → f uniformly, and f’n → g uniformly, then f C¹([0, 1]) and f’ = g. The easy way to prove this is to show that f(x) − f(0) = ∫x₀ g(t) dt.)

10. Let L¹k([0, 1]) be the space of all f Ck−1([0, 1]) such that f(k-1) is absolutely continuous on [0, 1] (and hence f(k) exists a.e. and is in L¹([0, 1])). Then ||f|| = ∑k₀ ∫¹₀|f(j)(x)|dx is a norm on L¹k([0, 1]) that makes L¹k([0, 1]) into a Banach space. (See Exercise 9 and its hint.)

11. If 0 < α ≤ 1, let λα([0, 1]) be the space of Hölder continuous functions of exponent α on [0, 1]. That is, f Λα([0, 1]) iff ||f||Λα < , where

a. ||·||Λα is a norm that makes Λα([0,1]) into a Banach space.

b. Let λα([0,1]) be the set of all f Λα([0,1]) such that

If α < 1, λα([0,1]) is an infinite-dimensional closed subspace of λα([0,1]). If α = 1, λα([0,1]) contains only constant functions.

12. Let be a normed vector space and a proper closed subspace of .

a. ||x + || = inf {||x + y|| : y } is a norm on /.

b. For any > 0 there exists x such that ||x|| = 1 and ||x + || ≥ 1 − .

c. The projection map π(x) = x + from to / has norm 1.

d. If is complete, so is /. (Use Theorem 5.1.)

e. The topology defined by the quotient norm is the quotient topology as defined in Exercise 28 in §4.2.

13. If ||·|| is a seminorm on the vector space , let = {x : ||x|| = 0}. Then is a subspace, and the map x + → ||x|| is a norm on /.

14. If is a normed vector space and is a nonclosed subspace, then ||x + ||, as defined in Exercise 12, is a seminorm on /. If one divides by its nullspace as in Exercise 13, the resulting quotient space is isometrically isomorphic to . (Cf. Exercise 5.)

15. Suppose that and are normed vector spaces and T L(, ). Let (T) = {x : Tx = 0}.

a. (T) is a closed subspace of .

b. There is a unique S L(/(T), ) such that T = S π where π : → /(T) is the projection (see Exercise 12). Moreover, ||S|| = ||T||.

16. The purpose of this exercise is to develop a theory of integration for functions with values in a separable Banach space. Let (X, , μ) be a measure space, a separable Banach space, and L the space of all (, )-measurable maps from X to , and F the set of maps f : X → of the form f(x) = ∑n₁ χEj (x)yj where n , yj , Ej , and μ(Ej) < ∞. If f L, since y → ||y|| is continuous (Exercise 1), x ||f(x)|| is (, )-measurable, and we define ||f|| = ∫||f(x)||dμ(x). Finally, let L¹ = {f L : ||f||₁ < ∞}.

a. L is a vector space, F and L¹ are subspaces of it, F L¹, and ||·||₁ is a seminorm on L¹ that becomes a norm if we identify two functions that are equal a.e.

b. Let {yn}∞₁ be a countable dense set in . Given > 0, let Bn = {y : ||y − yn|| < ||yn||}. Then ∪∞₁ Bn ⊃ \ {0}.

c. If f L¹, there is a sequence {hn} ⊂ F with hn → f a.e. and ||hn − f||₁ → 0. (With notation as in (b), let Anj = and Enj = f⁻¹ (Anj), and consider gj = ∑∞n=1 yn\χEnj.)

d. There is a unique linear map ∫ : L¹ → such that ∫ yχE = μ(E)y for y and E (μ(E) < ∞), and ||∫ f|| ≤ ||f||₁.

e. The dominated convergence theorem: If {fn} is a sequence in L¹ such that fn → f a.e. and there exists g L¹ such that ||fn(x)|| ≤ g(x) for all n and a.e. x, then ∫ fn → ∫ f.

f. If is a separable Banach space, T L(, ), and f L¹, then and

5.2 LINEAR FUNCTIONALS

Let be a vector space over K, where K = or . A linear map from to K is called a linear functional on . If is a normed vector space, the space L(, K) of bounded linear functionals on is called the dual space of and is denoted by *. According to Proposition 5.4, * is a Banach space with the operator norm.

If is a vector space over , it is also a vector space over , and we can consider both real and complex linear functionals on , that is, maps f : → that are linear over and maps f : → that are linear over . The relationship between the two is as follows:

5.5 Proposition. Let be a vector space over If f is a complex linear functional on and u = Re f, then u is a real linear functional, and f(x) = u(x) − iu(ix) for all x . Conversely, if u is a real linear functional on and f : → is defined by f(x) = u(x) − iu(ix), then f is complex linear. In this case, if is normed, we have ||u|| = ||f||.

Proof. If f is complex linear and u = Re f, u is clearly real linear and Im f(x) = − Re[if(x)] = −u(ix), so f(x) = u(x) − iu(ix). On the other hand, if u is real linear and f(x) = u(x) − iu(ix), then f is clearly linear over , and f(ix) = u(ix) − iu(−x) = u(ix) + iu(x) = if(x), so f is also linear over . Finally, if is normed, since |u(x)| = |Re f(x)| ≤ |f(x)| we have ||u|| ≤ ||f||. On the other hand, if f(x) ≠ 0, let α = . Then |f(x)| = αf(x) = f(αx) = u(αx) (since f(αx) is real), so |f(x)| ≤ ||u|| ||αx|| = ||u|| ||x||, whence ||f|| ≤ ||u||.

It is not obvious that there are any nonzero bounded linear functionals on an arbitrary normed vector space. The fact that such functionals exist in great abundance is one of the fundamental theorems of functional analysis. We shall now present this result in a more general form that has other important applications.

If is a real vector space, a sublinear functional on is a map p : → such that

For example, every seminorm is a sublinear functional.

5.6 The Hahn-Banach Theorem. Let be a real vector space, p a sublinear functional on , a subspace of , and f a linear functional on such that f(x) ≤ p(x) for all x . Then there exists a linear functional F on such that F(x) ≤ p(x) for all x and F| = f.

Proof. We begin by showing that if x \ , f can be extended to a linear functional g on + x satisfying g(y) ≤ p(y) there. If y₁, y₂ , we have

or

Hence

Let α be any number satisfying

and define g : + x → by g(y + λx) = f(y) + λα. Then g is clearly linear, and g| = f, so that g(y) ≤ p(y) for y . Moreover, if λ > 0 and y ,

whereas if λ = −μ, < 0,

Thus g(z) ≤ p(z) for all z + x.

Evidently the same reasoning can be applied to any linear extension F of f satisfying F ≤ p on its domain, and it shows that the domain of a maximal linear extension satisfying F ≤ p must be the whole space . But the family of all linear extensions F of f satisfying F ≤ p is partially ordered by inclusion (maps from subspaces of to being regarded as subsets of x ). Since the union of any increasing family of subspaces of is again a subspace, one easily sees that the union of a linearly ordered subfamily of lies in . The proof is therefore completed by invoking Zorn’s lemma.

If p is a seminorm and f : → is linear, the inequality f ≤ p is equivalent to the inequality |f| ≤ p, because |f(x)| = ±f(x) = f(±x) and p(−x) = p(x). In this situation the Hahn-Banach theorem also applies to complex linear functionals:

5.7 The Complex Hahn-Banach Theorem. Let be a complex vector space, p a seminorm on , a subspace of , and f a complex linear functional on such that |f(x)| ≤ p(x) for x . Then there exists a complex linear functional F on such that |F(x)| ≤ p(x) for all x and F| = f.

Proof. Let u = Re f. By Theorem 5.6 there is a real linear extension U of u to such that |U(x)| ≤ p(x) for all x . Let F(x) = U(x) − iU(ix) as in Proposition 5.5. Then F is a complex linear extension of f, and as in the proof of Proposition 5.5, if α = we have |F(x)| = αF(x) = F(αx) = U(αx) ≤ p(αx) = p(x).

From now on until §5.5, all of our results apply equally to real or complex vector spaces, but for the sake of definiteness we shall assume that the scalar field is . The principal applications of the Hahn-Banach theorem to normed vector spaces are summarized in the following theorem.

5.8 Theorem. Let be a normed vector space.

a. If is a closed subspace of and x \ , there exists f * such that f(x) ≠ 0 and f| = 0. In fact, if δ = infy ||x − y||, f can be taken to satisfy ||f|| = 1 and f(x) = δ.

b. If x ≠ 0 , there exists f * such that ||f|| = 1 and f(x) = ||z||.

c. The bounded linear functionals on separate points.

d. If x , define x : * → by (f) = f(x). Then the map x → is a linear isometry from into ** (the dual of *).

Proof. To prove (a), define f on + x by f(y + λx) = λδ (y , λ ). Then f(x) = δ, f| = 0, and for #LD ≠ 0, |f(y + λx)| = |λ|δ ≤ |λ| ||λ⁻¹y + x|| = ||y + λx||. Thus the Hahn-Banach theorem can be applied, with p(x) = ||x|| and replaced by + x. (b) is the special case of (a) with = {0}, and (c) follows immediately: if x ≠ y, there exists f * with f(x − y) ≠ 0, i.e., f(x) ≠ f(y). As for (d), obviously is a linear functional on * and the map x → is linear. Moroever, |(f)| = |f(x)| ≤ ||f|| ||x||, so |||| ≤ ||x||. On the other hand, (b) implies that |||| ≥ ||x||.

With notation as in Theorem 5.8d, let = { : x }. Since ** is always complete, the closure of in ** is a Banach space, and the map x → embeds into as a dense subspace. is called the completion of . In particular, if is itself a Banach space then .

If is finite-dimensional, then of course , since these spaces have the same dimension. For infinite-dimensional Banach spaces it may or may not happen that if it does, is called reflexive. The examples of Banach spaces we have examined so far are not reflexive except in trivial cases where they turn out to be finite-dimensional. We shall prove some cases of this assertion and present examples of reflexive Banach spaces in later sections.

Usually we shall identify with x and thus regard ** as a superspace of ; reflexivity then means that ** = .

Exercises

17. A linear functional f on a normed vector space is bounded iff f⁻¹({0}) is closed. (Use Exercise 12b.)

18. Let be a normed vector space.

a. If is a closed subspace and x \ then + x is closed. (Use Theorem 5.8a.)

b. Every finite-dimensional subspace of is closed.

19. Let be an infinite-dimensional normed vector space.

a. There is a sequence {xj} in such that ||xj|| = 1 for all j and ||xj − xk|| ≥ for j ≠ k. (Construct xj inductively, using Exercises 12b and 18.)

b. is not locally compact.

20. If is a finite-dimensional subspace of a normed vector space , there is a closed subspace such that ⊂ = {0} and + = .

21. If and are normed vector spaces, define α : * x * → ( x )* by α(f, g) (x, y) = f(x) + g(y). Then α is an isomorphism which is isometric if we use the norm ||(x, y)|| = max(||x||, ||y||) on x , the corresponding operator norm on ( x )*, and the norm ||(f, g)|| = ||f|| + ||g|| on * x *.

22. Suppose that and are normed vector spaces and T L(, ).

a. Define . Then T† L(*, *) and ||T†|| = ||T||. T† is called the adjoint or transpose of T.

b. Applying the construction in (a) twice, one obtains T†† L(**, **). If and are identified with their natural images and in ** and **, then T††| = T.

c. T† is injective iff the range of T is dense in .

d. If the range of T† is dense in *, then T is injective; the converse is true if is reflexive.

23. Suppose that is a Banach space. If is a closed subspace of and is a closed subspace of *, let ⁰ = {f * : f| = 0} and ⊥ = {x : f(x) = 0 for all f }. (Thus, if we identify with its image in **, ⊥ = ⁰ ∩ .)

a. ⁰ and ⊥ are closed subspaces of * and , respectively.

b. (⁰)⊥ = and (⊥)⁰ ⊃ . If is reflexive, (⊥)⁰ = .

c. Let π : → / be the natural projection, and define α : (/)* → * by α(f) = f π. Then α is an isometric isomorphism from (/)* onto ⁰, where / has the quotient norm.

d. Define β : * → * by β(f) = f|; then β induces a map : */⁰ → * as in Exercise 15, and is an isometric isomorphism.

24. Suppose that is a Banach space.

a. Let be the natural images of , * in **, ***, and let . Then and .

b. is reflexive iff * is reflexive.

25. If is a Banach space and * is separable, then is separable. (Let {fn}∞₁ be a countable dense subset of *. For each n choose xn with ||xn|| = 1 and . Then the linear combinations of {xn}∞₁ are dense in .) Note: Separability of does not imply separability of *.

26. Let be a real vector space and let P be a subset of such that (i) if x, y P, then x + y P, (ii) if x P and λ ≥ 0, then λx P, (iii) if x P and −x P, then x = 0. (Example: If is a space of real-valued functions, P can be the set of nonnegative functions in .)

a. The relation ≤ defined by x ≤ y iff y − x P is a partial ordering on .

b. (The Krein Extension Theorem) Suppose that is a subspace of such that for each x there exists y with x ≤ y. If f is a linear functional on such that f(x) ≥ 0 for x ∩ P, there is a linear functional F on such that F(x) ≥ 0 for x P and F| = f. (Consider p(x) = inf {f(y) : y and x ≤ y}.)

5.3 THE BAIRE CATEGORY THEOREM AND ITS CONSEQUENCES

In this section we present an important theorem about complete metric spaces and use it to obtain some fundamental results concerning linear maps between Banach spaces.

5.9 The Baire Category Theorem. Let X be a complete metric space.

a. If {Un}∞₁ is a sequence of open dense subsets of X, then ∩∞₁ Un is dense in X.

b. X is not a countable union of nowhere dense sets.

Proof. For part (a), we must show that if W is a nonempty open set in X, then W intersects ∩∞₁ Un. Since U₁ ∩ W is open and nonempty, it contains a ball B(r₀, x₀), and we can assume that 0 < r₀ < 1. For n > 0, we choose xn X and rn (0, ∞) inductively as follows: Having chosen xj and rj for j < n, we observe that Un ∩ B(rn−1, xn−1) is open and nonempty, so we can choose xn, rn so that 0 < rn < 2−n and ⊂ Un ∩ B(rn−1, xn−1). Then if n, m ≥ N, we see that xn, xm B(rN, xN), and since rn → 0, the sequence {xn} is Cauchy. As X is complete, x = lim xn exists. Since xn B(rN, xN) for n ≥ N we have

for all N, and the proof is complete.

As for (b), if {En} is a sequence of nowhere dense sets in X, then is a sequence of open dense sets. Since , we have .

We remark that since the conclusions of the Baire category theorem are purely topological, it suffices for X to be homeomorphic to a complete metric space. For example, the theorem applies to X = (0, 1), which is not complete with the usual metric but is homeomorphic to .

The name of this theorem comes from Baire’s terminology for sets: If X is a topological space, a set E ⊂ X is of the first category, according to Baire, if E is a countable union of nowhere dense sets; otherwise E is of the second category. Thus Baire’s theorem asserts that every complete metric space is of the second category in itself. A more modern and more descriptive synonym for “of the first category” is meager. The complement of a meager set is called residual.

The Baire category theorem is often used to prove existence results: One shows that objects having a certain property exist by showing that the set of objects not having the property (within a suitable complete metric space) is meager. For example, one can prove the existence of nowhere differentiable continuous functions in this way; see Exercise 42.

We turn to the applications of the Baire category theorem in the theory of linear maps. Some terminology: If X and Y are topological spaces, a map f : X → Y is called open if f(U) is open in Y whenever U is open in X. If X and Y are metric spaces, this amounts to requiring that if B is a ball centered at x X, then f(B) contains a ball centered at f(x). Specializing still further, if X and Y are normed linear spaces and f is linear, then f commutes with translations and dilations; it follows that f is open iff f(B) contains a ball centered at 0 in Y when B is the ball of radius 1 about 0 in X.

5.10 The Open Mapping Theorem. Let and be Banach spaces. If T L(, ) is surjective, then T is open.

Proof. Let Br denote the (open) ball of radius r about 0 in . By the preceding remarks, it will suffice to show that T(B₁) contains a ball about 0 in . Since = ∪∞₁ Bn and T is surjective, we have = ∪∞₁ T(Bn). But is complete and the map y → ny is a homeomorphism of that maps T(B₁) to T(Bn), so Baire’s theorem implies that T(B₁) cannot be nowhere dense. That is, there exist y₀ and r > 0 such that the ball B(4r, y₀) is contained in . Pick y₁ = Tx₁ T(B₁) such that ||y₁ − y₀|| < 2r; then B(2r, y₁) ⊂ B(4r, y₀) ⊂ , so if ||y|| < 2r,

Dividing both sides by 2, we conclude that there exists r > 0 such that if ||y|| < r then . If we could replace by T(B₁), perhaps shrinking r at the same time, the proof would be complete; we now proceed to accomplish this.

Since T commutes with dilations, it follows that if ||y|| < r2−n, then y , Suppose ||y|| < r/2; we can find x₁ B1/2 such that ||y − Tx₁|| < r/4, and proceeding inductively, we can find xn B2−n such that ||y − ⊂n₁ Txj|| < r2−n−1. Since is complete, by Theorem 5.1 the series ∑∞₁ xn converges, say to x. But then ||x|| < ⊂∞₁ 2−n = 1 and = Tx. In other words, T(B₁) contains all y with ||y|| < r/2, so we are done.

5.11 Corollary. If and are Banach spaces and T L(, ) is bijective, then T is an isomorphism; that is, T⁻¹ L(, ).

Proof. If T is bijective, continuity of T⁻¹ is equivalent to the openness of T.

For the next results we need some more terminology. If and are normed vector spaces and T is a linear map from to , we define the graph of T to be

which is a subspace of x . (From a strict set-theoretic point of view, of course, T and Γ(T) are identical; the distinction is a psychological one.) We say that T is closed if Γ(T) is a closed subspace of x . Clearly, if T is continuous, then T is closed, and if and are complete the converse is also true:

5.12 The Closed Graph Theorem. If and are Banach spaces and T : → is a closed linear map, then T is bounded.

Proof. Let π₁ and π₂ be the projections of Γ(T) onto and , that is, π₁(x, Tx) = x and π₂(x, Tx) = Tx. Obviously π₁ L(Γ(T), ) and π₂ L(Γ(T)). Since and are complete, so is x , and hence so is Γ(T) since T is closed. The map π₁ is a bijection from Γ(T) to , so by Corollary 5.11, π⁻¹₁ is bounded. But then T = π₂ π⁻¹₁ is bounded.

Continuity of a linear map T : → means that if xn → x then Txn → Tx, whereas closedness means that if xn → x and Txn → y then y = Tx. Thus the significance of the closed graph theorem is that in verifying that Txn → Tx when xn → x, we may assume that Txn converges to something, and we need only to show that the limit is the right thing. This frequently saves a lot of trouble.

The completeness of and was used in a crucial way in proving the open mapping theorem and hence also in proving the closed graph theorem. In fact, the conclusions of both of these theorems may fail if either or is incomplete; see Exercises 29–31.

Our final result in this section is a theorem of almost magical power that allows one to deduce uniform estimates from pointwise estimates in certain situations.

5.13 The Uniform Boundedness Principle. Suppose that and are normed vector spaces and is a subset of L(, ).

a. If supT ||Tx|| < ∞ for all x in some nonmeager subset of , then supT||T|| < ∞.

b. If is a Banach space and supT ||Tx|| < ∞ for all x , then supT ||T|| < ∞.

Proof. Let

Then the En’s are closed, so under the hypothesis of (a) some En must contain a nontrivial closed ball . But then E2n ⊃ , for if ||x|| ≤ r, then x + x₀ En and hence

In other words, ||Tx|| ≤ 2n whenever T and ||x|| ≤ r, so supT ||T|| ≤ 2n/r. This proves (a), and (b) follows by the Baire category theorem.

Exercises

27. There exist meager subsets of whose complements have Lebesgue measure zero.

28. The Baire category theorem remains true if X is assumed to be an LCH space rather than a complete metric space. (The proof is similar; the substitute for completeness is Proposition 4.21.)

29. Let = L¹(μ) where μ is counting measure on , and let = {f : ∑∞₁ n|f(n)| < ∞}, equipped with the L¹ norm.

a. is a proper dense subspace of ; hence is not complete.

b. Define T : → by T f(n) = nf(n). Then T is closed but not bounded.

c. Let S = T⁻¹. Then S : → is bounded and surjective but not open.

30. Let = C([0,1]) and = C¹([0,1]), both equipped with the uniform norm.

a. is not complete.

b. The map (d/dx) : → is closed (see Exercise 9) but not bounded.

31. Let , be Banach spaces and let S : → be an unbounded linear map (for the existence of which, see §5.6). Let Γ(S) be the graph of S, a subspace of x .

a. Γ(S) is not complete.

b. Define T : → Γ(S) by Tx = (x, Sx). Then T is closed but not bounded.

c. T⁻¹ : Γ(S) → is bounded and surjective but not open.

32. Let ||·||₁ and ||·||₂ be norms on the vector space such that ||·||₁ ≤ ||·||₂. If is complete with respect to both norms, then the norms are equivalent.

33. There is no slowest rate of decay of the terms of an absolutely convergent series; that is, there is no sequence {an} of positive numbers such that ∑ an|cn| < ∞ iff {cn} is bounded. (The set of bounded sequences is the space B() of bounded functions on , and the set of absolutely summable sequences is L¹(μ) where μ is counting measure on . If such an {an} exists, consider T : B() → L¹(μ) defined by T f(n) = anf(n). The set of f such that f(n) = 0 for all but finitely many n is dense in L¹(μ) but not in B().)

34. With reference to Exercises 9 and 10, show that the inclusion map of L¹k([0,1]) into Ck−1([0,1]) is continuous (a) by using the closed graph theorem, and (b) by direct calculation. (This is to illustrate the use of the closed graph theorem as a labor-saving device.)

35. Let and be Banach spaces, T L(, ), (T) = {x : Tx = 0}, and = range(T). Then /(T) is isomorphic to iff is closed. (See Exercise 15.)

36. Let be a separable Banach space and let μ be counting measure on . Suppose that {xn}∞₁ is a countable dense subset of the unit ball of , and define T : L¹(μ) → by T f = ∑∞₁ f(n)xn.

a. T is bounded.

b. T is surjective.

c. is isomorphic to a quotient space of L¹(μ). (Use Exercise 35.)

37. Let and be Banach spaces. If T : → is a linear map such that f T * for every f *, then T is bounded.

38. Let and be Banach spaces, and let {Tn} be a sequence in L(, ) such that lim Tnx exists for every x X. Let Tx = lim Tnx; then T L(, ).

39. Let , , be Banach spaces and let B : x → be a separately continuous bilinear map; that is, B(x, ·) L(, ) for each x and B(·, y) L(, ) for each y . Then B is jointly continuous, that is, continuous from x to . (Reduce the problem to proving that ||B(x, y)|| ≤ C||x|| ||y|| for some C > 0.)

40. (The Principle of Condensation of Singularities) Let and be Banach spaces and {Tjk : j, k } ⊂ L(, ). Suppose that for each k there exists x such that sup{||Tjkx|| : j } = ∞. Then there is an x (indeed, a residual set of x’s) such that sup{||Tjkx|| : j } = ∞ for all k.

41. Let be a vector space of countably infinite dimension (that is, every element is a finite linear combination of members of a countably infinite linearly independent set). There is no norm on with respect to which is complete. (Given a norm on , apply Exercise 18b and the Baire category theorem.)

42. Let En be the set of all f C([0, 1]) for which there exists x₀ [0,1] (depending on f) such that |f(x) − f(x₀)| ≤ n|x − x₀| for all x [0,1].

a. En is nowhere dense in C([0,1]). (Any real f C((0,1]) can be uniformly approximated by a piecewise linear function g whose linear pieces, finite in number, have slope ±2n. If ||h − g||u is sufficiently small, then h En.)

b. The set of nowhere differentiate functions is residual in C([0,1]).

5.4 TOPOLOGICAL VECTOR SPACES

It is frequently useful to consider topologies on vector spaces other than those defined by norms, the only crucial requirement being that the topology should be well behaved with respect to the vector operations. Precisely, a topological vector space is a vector space over the field K (= or ) which is endowed with a topology such that the maps (x, y) → x + y and (λ, x) → λx are continuous from x and K x to . A topological vector space is called locally convex if there is a base for the topology consisting of convex sets (that is, sets A such that if x, y A then tx + (1 − t)y A for 0 < t < 1). Most topological vector spaces that arise in practice are locally convex and Hausdorff.

The most common way of defining locally convex topologies on vector spaces is in terms of seminorms. Namely, if we are given a family of seminorms on , the “balls” that they define can be used to generate a topology in the same way that the balls defined by a norm generate the topology on a normed vector space. The precise result is as follows:

5.14 Theorem. Let {pα}αA be a family of seminorms on the vector space . If x , α A, and > 0, let

and let be the topology generated by the sets Uxα.

a. For each x , the finite intersections of the sets Uxα (α A, > 0) form a neighborhood base at x.

b. If xiiI is a net in , then xi → x iff pα (xi − x) → 0 for all α A.

c. (, ) is a locally convex topological vector space.

Proof. (a) If x . By the triangle inequality, we have . Thus the assertion follows from Proposition 4.4.

(b) In view of (a), it suffices to observe that pα(xi − x) → 0 iff xi is eventually in Uxα for every > 0.

(c) The continuity of the vector operations follows easily from Proposition 4.19 and part (b). Indeed, if xi → x and yi → y, then

so xi + yi → x + y. If also λi → λ, then eventually |λi| ≤ C = |λ| + 1, so

and it follows that λixi → λx. Moreover, the sets Uxα are convex, for if y, z Uxα, then

The local convexity of the topology therefore follows from (a).

In this context there is an analogue of Proposition 5.2:

5.15 Proposition. Suppose and are vector spaces with topologies defined, respectively, by the families {pα}αA and {qβ}βB of seminorms, and T : → is a linear map. Then T is continuous iff for each β B there exist α₁,…, αk A and C > 0 such that qβ(Tx) ≤ C ∑k₁ pαj (x).

Proof. If the latter condition holds and xi is a net converging to x , by Theorem 5.14b we have pα(xi − x) → 0 for all α, hence qβ(Txi − Tx) → 0 for all β, hence Txi → Tx. By Proposition 4.19, T is continuous. Conversely, if T is continuous, for every β B there is a neighborhood U of 0 in such that qβ(Tx) < 1 for x U. By Theorem 5.14a we may assume that U = ∩k₁ Uxαjj. Let = min(₁,…, k); then qβ(Tx) < 1 whenever pαj (x) < for all j. Now, given x , there are two possibilities. If pαj (x) > 0 for some j, let y = x/∑k₁ pαj(x). Then pαj(y) < for all j, so

On the other hand, if pαj (x) = 0 for all j, then pαja(rx) = 0 for all j and all r > 0, hence rqβ(Tx) = qβ(T(rx)) < 1 for all r > 0, hence qβ(Tx) = 0. Thus qβ(Tx) ≤ ⁻¹ ∑k₁ pαj(x) in this case too, and we are done.

The proof of the following proposition is left to the reader (Exercise 43).

5.16 Proposition. Let be a vector space equipped with the topology defined by a family {pα}αA of seminorms.

a. is Hausdorff iff for each x ≠ 0 there exists α A such that pα(x) ≠ 0.

b. If is Hausdorff and A is countable, then is metrizable with a translation-invariant metric (i.e., ρ(x, y) = ρ(x + z, y + z) for all x, y, z ).

If has the topology defined by the seminorms {pα}αA, by Proposition 5.15 a linear functional f on is continuous iff |f(x)| ≤ C ∑k₁ pαj (x) for some C > 0 and α₁,…, αk A. Since a finite sum of seminorms is again a seminorm, the Hahn-Banach theorem guarantees the existence of lots of continuous linear functionals on — enough to separate points, if is Hausdorff. The set of all such functionals is denoted, as before, by *. There are various ways of making * into a topological vector space, but we shall not consider this question systematically. The simplest way is to impose the weakest topology that makes all the evaluation maps f → f(x) (x ) continuous, an idea that we shall discuss further below.

In a topological vector space χ the notion of Cauchy sequence or Cauchy net makes sense. Namely, a net xiiI in is called Cauchy if the net xi − xj(i,j)IxI converges to zero. (Here I × I is directed in the usual way: (i, j) (i’, j’) iff i i’ and j j’.) Naturally, is called complete if every Cauchy net converges. Completeness is of most interest when is first countable, in which case it is equivalent to the condition that every Cauchy sequence converges (Exercise 44). More particularly, if is Hausdorff and its topology is defined by a countable family of seminorms, then this topology is first countable by Theorem 5.14a; indeed, it is given by a translation-invariant metric ρ by Proposition 5.16b, and a sequence is Cauchy according to the definition just given iff it is Cauchy with respect to ρ. A complete Hausdorff topological vector space whose topology is defined by a countable family of seminorms is called a Fréchet space.

Let us now consider some interesting examples of topological vector spaces whose topologies are defined by families of seminorms rather than by single norms. We have already met a couple of them in previous chapters:

Let X be an LCH space. On X, the topology of uniform convergence on compact sets is defined by the seminorms pK(f) = supxK |f(x)| as K ranges over compact subsets of X. If X is σ-compact and {Un} are as in Propositions 4.39 and 4.40, this topology is defined by the seminorms . In this case, X is easily seen to be complete, so it is a Fréchet space; by Proposition 4.38, so is C(X).

The space L¹loc(n), defined in §3.4, is a Fréchet space with the topology defined by the seminorms pk(f) = ∫|x|≤k|f(x)|dx. (Completeness follows easily from the completeness of L¹.) An obvious generalization of this construction yields a locally convex topological vector space L¹loc(X, μ) where X is any LCH space and μ is a Borel measure on X that is finite on compact sets.

Another class of topological vector spaces arises naturally in connection with the theory of differential equations. One often wishes to study the operator d/dx, or more complicated operators constructed from it, acting on various spaces of functions. Unfortunately, it is virtually impossible to define norms on most infinite-dimensional functions spaces so that d/dx becomes a bounded operator. Here is one precise result along these lines: There is no norm on the space C∞([0,1]) of infinitely differentiable functions on [0,1] with respect to which d/dx is bounded. Indeed, if fλ(x) = eλx, then (d/dx)fλ = λfλ, so ||d/dx|| ≤ |λ| for all λ no matter what norm is used on C∞([0,1]).

In view of this difficulty, three courses of action are available. First, one can consider differentiation as an unbounded operator from to where is a suitable Banach space and is a dense subspace of , as in Exercise 30. Second, one can consider differentiation as a bounded linear map from one Banach space to a different one , such as = Ck([0,1]) and = Ck−1([0,1]) in Exercise 9. Finally, one can consider differentiation as a continuous operator on a locally convex space whose topology is not given by a norm. All of these points of view have their uses, but it is the last one that concerns us here. It is easy to construct families of seminorms on spaces of smooth functions such that differentiation becomes continuous almost by definition. For example, the seminorms (k = 0,1,2,…) make C∞([0,1]) into a Fréchet space (the completeness is proved as in Exercise 9), and d/dx is continuous on this space by Proposition 5.15 since pk(f’) = pk+1(f). Other examples are considered in Exercise 45 and in Chapter 9.

One of the most useful procedures for constructing topologies on vector spaces is by requiring the continuity of certain linear maps. Namely, suppose that is a vector space, is a normed linear space, and {Tα}αA is a collection of linear maps from to . Then the weak topology generated by {Tα} makes into a locally convex topological vector space. Indeed, is just the topology ’ defined by the seminorms pα(x) = ||Tαx|| according to Theorem 5.14. ( is generated by sets of the form {x : ||Tαx − y₀|| < } with y₀ , whereas is generated by sets of the form {x : ||Tαx − Tαx₀|| < } with x₀ . If the Tα’s are surjective, these are obviously the same; the general case is left as Exercise 46.) The topology on C∞([0,1]) in the preceding paragraph is an example of this construction, with = C([0,1]) and Tkf = f(k). We now present some more.

First, let be a normed vector space. The weak topology generated by * is known simply as the weak topology on , and convergence with respect to this topology is known as weak convergence. Thus, if xα is a net in , xα → x weakly iff f(xα) → f(x) for all f *. When is infinite-dimensional, the weak topology is always weaker than the norm topology; see Exercise 49.

Next, let be a normed vector space, * its dual space. The weak topology on * as defined above is the topology generated by **; of more interest is the topology generated by (considered as a subspace of **), which is called the weak* topology (read “weak star topology”) on *. * is a space of functions on , and the weak* topology is simply the topology of pointwise convergence: fα → f iff fα(x) → f(x) for all x . The weak* topology is even weaker than the weak topology on *; the two coincide precisely when is reflexive.

Finally, let and be Banach spaces. The topology on L(, ) generated by the evaluation maps T Tx (x ) is called the strong operator topology on L(, ), and the topology generated by the linear functionals T → f(Tx) (x , f *) is called the weak operator topology on L(, ). Again, these topologies are best understood in terms of convergence: Tα → T strongly iff Tαx → Tx in the norm topology of for each x , whereas Tα → T weakly iff Tαx → Tx in the weak topology of for each x . Thus the strong operator topology is stronger than the weak operator topology but weaker than the norm topology on L(, ).

The following result concerning strong convergence is almost trivial but extremely useful:

5.17 Proposition. Suppose {Tn}∞₁ ⊂ L(, ), supn ||Tn|| < ∞, and T L(, ). If ||Tnx − Tx|| → 0 for all x in a dense subset D of , then Tn → T strongly.

Proof. Let C = sup{||T||, ||T₁||, ||T₂||,…}. Given x and > 0, choose x’ D such that ||x − x’|| < /3C. If n is large enough so that ||Tnx’ − Tx’|| < /3, we have

so that Tnx → Tx.

Our final result in this section is a compactness theorem that is one of the main reasons for the usefulness of the weak* topology on a dual space. The idea of the proof is similar to the techniques discussed in §4.8.

5.18 Alaoglu’s Theorem. If is a normed vector space, the closed unit ball B* = {f * : IIf|| ≤ 1} in * is compact in the weak* topology.

Proof. For each x let x = {z : |z| ≤ ||x||}, and let = Πx x. Then is compact by Tychonoff’s theorem. The elements of are precisely those complex-valued functions ϕ on such that |ϕ(x)| < ||x|| for all x , and B* consists of those elements of that are linear. Moreover, the relative topologies that B* inherits from the product topology on and the weak* topology on * both coincide with the topology of pointwise convergence, so it suffices to see that B* is closed in . But this is easy: If fα is a net in B* that converges to f , for any x, y and a, b we have

so that f B*.

Warning: Alaoglu’s theorem does not imply that * is locally compact in the weak* topology; see Exercise 49b.

Exercises

43. Prove Proposition 5.16. (For part (b), proceed as in Exercise 56d in §4.5.)

44. If is a first countable topological vector space and every Cauchy sequence in converges, then every Cauchy net in χ converges.

45. The space C∞() of all infinitely differentiable functions on has a Fréchet space topology with respect to which fn → f iff f(k)n → f(k) uniformly on compact sets for all k ≥ 0.

46. If is a vector space, a normed linear space, the weak topology on generated by a family of linear maps {Tα : → }, and the topology defined by the seminorms {x → ||Tαx||}, then .

47. Suppose that and are Banach spaces.

a. If {Tn}∞₁ ⊂ L(, ) and Tn → T weakly (or strongly), then supn ||Tn|| < ∞.

b. Every weakly convergent sequence in , and every weak*-convergent sequence in *, is bounded (with respect to the norm).

48. Suppose that is a Banach space.

a. The norm-closed unit ball B = {x : ||x|| ≤ 1} is also weakly closed. (Use Theorem 5.8d.)

b. If E ⊂ is bounded (with respect to the norm), so is its weak closure.

c. If F ⊂ * is bounded (with respect to the norm), so is its weak* closure.

d. Every weak*-Cauchy sequence in * converges. (Use Exercise 38.)

49. Suppose that is an infinite-dimensional Banach space.

a. Every nonempty weakly open set in , and every nonempty weak*-open set in *, is unbounded (with respect to the norm).

b. Every bounded subset of is nowhere dense in the weak topology, and every bounded subset of * is nowhere dense in the weak* topology. (Use Exercise 48b,c.)

c. is meager in itself with respect to the weak topology, and * is meager in itself with respect to the weak* topology.

d. The weak* topology on * is not defined by any translation-invariant metric. (Use Exercise 48d.)

50. If is a separable normed linear space, the weak* topology on the closed unit ball in * is second countable and hence metrizable. (But cf. Exercise 49d.)

51. A vector subspace of a normed vector space is norm-closed iff it is weakly closed. (However, a norm-closed subspace of * need not be weak*-closed unless is reflexive; see Exercise 52d.)

52. Let be a Banach space and let f₁,…, fn be linearly independent elements of *.

a. Define T : → n by Tx = (f₁(x),…, fn(x)). If = {x : Tx = 0} and is the linear span of f₁,…, fn, then = ⁰ in the notation of Exercise 23 and hence * is isomorphic to (/)*.

b. If F **, for any > 0 there exists x such that F(fj) = fj(x) for j = 1,…, n and ||x|| ≤ (1 + )||F||, (F| can be identified with an element of (/)** and hence with an element of / since the latter is finite-dimensional.)

c. If is considered as a subspace of **, the relative topology on induced by the weak* topology on ** is the weak topology on .

d. In the weak* topology on **, is dense in ** and the closed unit ball in is dense in the closed unit ball in **.

e. is reflexive iff its closed unit ball is weakly compact.

53. Suppose that is a Banach space and {Tn}, {Sn} are sequences in L(, ) such that Tn → T strongly and Sn → S strongly.

a. If {xn} ⊂ and ||xn − x|| → 0, then ||Tnxn − Tx|| → 0. (Use Exercise 47a.)

b. TnSn → TS strongly.

5.5 HILBERT SPACES

The most important Banach spaces, and the ones on which the most refined analysis can be done, are the Hilbert spaces, which are a direct generalization of finite-dimensional Euclidean spaces. Before defining them, we need to introduce a few concepts.

Let be a complex vector space. An inner product (or scalar product) on is a map (x, y) → x, y from x → such that:

i. ax + by, z = ax, z + by, z for all x, y, z and a, b .

ii. y, x = for all x, y .

iii. x, x (0, ∞) for all nonzero x .

We observe that (i) and (ii) imply that

(One can also define inner products on real vector spaces: x, y is then real, a and b are assumed real in (i), and (ii) becomes y, x = x, y.)

A complex vector space equipped with an inner product is called a pre-Hilbert space. If is a pre-Hilbert space, for x we define

5.19 The Schwarz Inequality. |x, y| ≤ ||x|| ||y|| for all x, y , with equality iff x and y are linearly dependent.

Proof. If x, y = 0, the result is obvious. If x, y ≠ 0 (and in particular y ≠ 0), let α = sgnx, y and z = αy, so that x, z = z, x = |x, y| and ||z|| = ||y||. Then for t we have

The expression on the right is a quadratic function of t whose absolute minimum occurs at t = ||y||⁻²|x, y|. Setting t equal to this value, we obtain

with equality iff x − tz = x − αty = 0, from which the desired result is immediate.

5.20 Proposition. The function x → ||x|| is a norm on .

Proof. That ||x|| = 0 iff x = 0 and that ||λx|| = |λ| ||x|| are obvious from the definition. As for the triangle inequality, we have

so by the Schwarz inequality,

as desired.

A pre-Hilbert space that is complete with respect to the norm ||x|| = √x, x is called a Hilbert space. (One can also consider real Hilbert spaces with real inner products. However, Hilbert spaces are usually assumed to be complex unless otherwise specified.)

Example: Let (X, , μ) be a measure space, and let L²(μ) be the set of all measurable functions f : X → such that ∫ |f|²dμ < ∞ (where, as usual, we identify two functions that are equal a.e.). From the inequality , valid for all a, b ≥ 0, we see that if f, g L²(μ) then , so that . It follows easily that the formula

defines an inner product on L²(μ). In fact, L²(μ) is a Hilbert space for any measure μ. We shall prove completeness in Theorem 6.6; for the present we shall take this result for granted.

An important special case of this construction is obtained by taking μ to be counting measure on (A, (A)), where A is any nonempty set; in this situation L²(μ) is usually denoted by l²(A). Thus, l²(A) is the set of functions f : A → such that the sum ∑αA |f(α)|² (as defined in §0.5) is finite. The completeness of l²(A) is rather easy to prove directly (Exercise 54).

For the remainder of this section, will denote a Hilbert space.

5.21 Proposition. If xn → x and yn → y, then xn, yn → x, y.

Proof. By the Schwarz inequality,

which tends to zero since ||yn|| → ||y||.

5.22 The Parallelogram Law. For all x, y

(“The sum of the squares of the diagonals of a parallelogram is the sum of the squares of the four sides.”)

Proof. Add the two formulas ||x ± y||² = ||x||² ± 2 Rex, y + ||y||².

If x, y , we say that x is orthogonal to y and write x ⊥ y if x, y = 0. If E ⊂ , we define

It is immediate from Proposition 5.21 and the linearity of the inner product in its first argument that E⊥ is a closed subspace of .

5.23 The Pythagorean Theorem. If x₁,…, xn and xj ⊥ xk for j ≠ k,

Proof. ||∑ xj||² = ∑ xj, ∑ xj = ∑j,k xj, xk. The terms with k ≠ j are all zero, leaving only ∑xj, xj = ∑ ||xj||².

5.24 Theorem. If is a closed subspace of then = ⊕ ⊥; that is, each x can be expressed uniquely as x = y + z where y and z ⊥. Moreover, y and z are the unique elements of and ⊥ whose distance to x is minimal.

Proof. Given x , let δ = inf{||x − y||: y }, and let {yn} be a sequence in such that ||x − yn|| → δ. By the paralellogram law,

so since (yn + ym) ,

As m, n → ∞ this last quantity tends to zero, so {yn} is a Cauchy sequence. Let y = lim yn and z = x − y. Then y since is closed, and ||x − y|| = δ.

We claim that z ⊥. Indeed, if u , after multiplying u by a nonzero scalar we may assume that z, u is real. Then the function

is real for t , and it has a minimum (namely, δ²) at t = 0 because z + tu = x − (y − tu) and y − tu . Thus 2z, u = f’(0) = 0, so z ⊥. Moreover, if z’ is another element of ⊥, by the Pythagorean theorem (since x − z = y ) we have

with equality iff z = z’. The same reasoning shows that y is the unique element of closest to x.

Finally, if x = y’ + z’ with y’ and z’ ⊥, then y − y’ = z’ − z ∩ ⊥, so y − y’ and z’ − z are orthogonal to themselves and hence are zero.

If y , the Schwarz inequality shows that the formula fy(x) = x, y defines a bounded linear functional on such that ||fy|| = ||y||. Thus, the map y → fy is a conjugate-linear isometry of into *. It is a fundamental fact that this map is surjective:

5.25 Theorem. If f *, there is a unique y such that f(x) = x, y for all x .

Proof. Uniqueness is easy: If x, y = x, y’ for all x, by taking x = y − y’ we conclude that ||y − y’||² = 0 and hence y = y’. If f is the zero functional, then obviously y = 0. Otherwise, let = {x : f(x) = 0}. Then is a proper closed subspace of , so ⊥ ≠ {0} by Theorem 5.24. Pick z ⊥ with ||z|| = 1. If u = f(x)z − f(z)x then u , so

Hence f(x) = x, y where .

Thus, Hilbert spaces are reflexive in a very strong sense: Not only is naturally isomorphic to **, it is naturally isomorphic (via a conjugate-linear map) to *.

A subset {uα}αA of is called orthonormal if ||uα|| = 1 for all α and uα ⊥ uβ whenever α ≠ β. If {xn}∞₁ is a linearly independent sequence in , there is a standard inductive procedure, called the Gram-Schmidt process, for converting {xn} into an orthonormal sequence {un} such that the linear span of {xn}N₁ coincides with the linear span of {un}N₁ for all N. Namely, the first step is to set u₁ = x₁ / ||x₁||. Having defined u₁, …, uN−1, we set . Then vN is nonzero because xN is not in the linear span of x₁,…, xN−1 and hence of u₁,…, un−1, and vN, um) = xN, um − xN, um = 0 for all m < N. We can therefore take uN = vN/||vN||.

5.26 Bessel’s Inequality. If {uα}αA is an orthonormal set in , then for any x ,

In particular, {α : x, uα ≠ 0} is countable.

Proof. It suffices to show that ∑αF |x, uα|² ≤ ||x||² for any finite F ⊂ A. But

where the Pythagorean theorem was used in the third line.

5.27 Theorem. If {uα}αA is an orthonormal set in , the following are equivalent:

a. (Completeness) If x, uα = 0 for all α, then x = 0.

b. (Parseval’s Identity) ||x||² = ∑αA |x, uα|² for all x .

c. For each x , x = ∑αA x, uαuα, where the sum on the right has only countably many nonzero terms and converges in the norm topology no matter how these terms are ordered.

Proof. (a) implies (c): If x , let α₁, α₂,… be any enumeration of the α’s for which x, uα ≠ 0. By Bessel’s inequality the series ∑|x, uαj|² converges, so by the Pythagorean theorem,

The series ∑x, uαjuαj therefore converges since is complete. If y = x − ∑x, uαjuαj, then clearly y, uα = 0 for all α, so by (a), y = 0.

(c) implies (b): With notation as above, as in the proof of Bessel’s inequality we have

Finally, that (b) implies (a) is obvious.

An orthonormal set having the properties (a–c) in Theorem 5.27 is called an orthonormal basis for . For example, let = l²(A). For each α A, define eα l²(A) by eα(β) = 1 if β = α, eα(β) = 0 otherwise. The set {eα}αA is clearly orthonormal, and for any f l²(A) we have f, eα = f(α), from which it follows that {eα} is an orthonormal basis.

5.28 Proposition. Every Hilbert space has an orthonormal basis.

Proof. A routine application of Zorn’s lemma shows that the collection of orthonormal sets, ordered by inclusion, has a maximal element; and maximality is equivalent to property (a) in Theorem 5.27.

5.29 Proposition. A Hilbert space is separable iff it has a countable orthonormal basis, in which case every orthonormal basis for is countable.

Proof. If {xn} is a countable dense set in , by discarding recursively any xn that is in the linear span of x₁,…, xn−1 we obtain a linearly independent sequence {yn} whose linear span is dense in . Application of the Gram-Schmidt process to {yn} yields an orthonormal sequence {un} whose linear span is dense in and which is therefore a basis. Conversely, if {un} is a countable orthonormal basis, the finite linear combinations of the un’s with coefficients in a countable dense subset of form a countable dense set in . Moreover, if {vα}αA is another orthonormal basis, for each n the set An = {α A : un, vα ≠ 0} is countable. By completeness of {un}, A = ∪∞₁ An, so A is countable.

Most Hilbert spaces that arise in practice are separable. We discuss some examples in Exercises 60–62.

If ₁ and ₂ are Hilbert spaces with inner products (·, ·)₁ and (·, ·)₂, a unitary map from ₁ to ₂ is an invertible linear map U : ₁ → ₂ that preserves inner products:

By taking y = x, we see that every unitary map is an isometry: ||Ux||₂ = ||z||₁. Conversely, every surjective isometry is unitary (Exercise 55). Unitary maps are the true “isomorphisms” in the category of Hilbert spaces; they preserve not only the linear structure and the topology but also the norm and the inner product. From the point of view of this abstract structure, every Hilbert space looks like an l² space:

5.30 Proposition. Let {eα}αA be an orthonormal basis for . Then the correspondence x defined by (α) = x, uα) is a unitary map from to l²(A).

Proof. The map x → x is clearly linear, and it is an isometry from to l²(A) by the Parseval identity ||x||² = ∑|(α)|². If f l²(A) then ∑|f(α)|² < ∞, so the Pythagorean theorem shows that the partial sums of the series ∑ f(α)uα (of which only countably many terms are nonzero) are Cauchy; hence x = ∑ f(α)uα exists in and = f. By Exercise 55b, x → is unitary.

Exercises

54. For any nonempty set A, l²(A) is complete.

55. Let be a Hilbert space.

a. (The polarization identity) For any x, y ,

(Completeness is not needed here.)

b. If ’ is another Hilbert space, a linear map from to is unitary iff it is isometric and surjective.

56. If E is a subset of a Hilbert space , (E⊥)⊥ is the smallest closed subspace of containing E.

57. Suppose that is a Hilbert space and T L(, ).

a. There is a unique T* L(, ), called the adjoint of T, such that Tx, y = x, T*y for all x, y . (Cf. Exercise 22. We have T* = V⁻¹T†V where V is the conjugate-linear isomorphism from to in Theorem 5.25, (Vy) (x) = x y.)

b. ||T*|| = ||T||, ||T*T|| = ||T||², (aS + bT)* = , (ST)* = T*S*, and T** = T.

c. Let and denote range and nullspace; then (T)⊥ = (T*) and (T)⊥ = .

d. T is unitary iff T is invertible and T⁻¹ = T*.

58. Let be a closed subspace of the Hilbert space , and for x let Px be the element of such that x − Px ⊥ as in Theorem 5.24.

a. P L(, ), and in the notation of Exercise 57 we have P* = P, P² = P, (P) = , and (P) = ⊥. P is called the orthogonal projection onto .

b. Conversely, suppose that P L(, ) satisfies P² = P* = P. Then (P) is closed and P is the orthogonal projection onto (P).

c. If {uα} is an orthonormal basis for , then Px = ∑x, uαuα.

59. Every closed convex set K in a Hilbert space has a unique element of minimal norm. (If 0 K, the result is trivial; otherwise, adapt the proof of Theorem 5.24.)

60. Let (X, , μ) be a measure space. If E , we identify L²(E, μ) with the subspace of L²(X, μ) consisting of functions that vanish outside E. If {En} is a disjoint sequence in with X = ∪∞₁ En, than {L²(En, μ)} is a sequence of mutually orthogonal subspaces of L²(X, μ), and every f L²(X, μ) can be written uniquely as f = ∑∞₁ fn (the series converging in norm) where fn L²(En, μ). If L²(En, μ) is separable for every n, so is L²(X, μ).

61. Let (X, , μ) and (Y; , μ) be σ-finite measure spaces such that L²(μ) and L²(v) are separable. If {fm} and {gn} are orthonormal bases for L²(μ) and L²(v) and hmn(x, y) = fm(x)gn(y), then {hmn} is an orthonormal basis for L²(μ × v).

62. In this exercise the measure defining the L² spaces is Lebesgue measure.

a. C([0,1]) is dense in L²([0,1]). (Adapt the proof of Theorem 2.26.)

b. The set of polynomials is dense in L²([0,1]).

c. L²([0,1]) is separable.

d. L²() is separable. (Use Exercise 60.)

e. L²(n) is separable. (Use Exercise 61.)

63. Let be an infinite-dimensional Hilbert space.

a. Every orthonormal sequence in converges weakly to 0.

b. The unit sphere S = {x : ||x|| = 1} is weakly dense in the unit ball B = {x : ||x|| ≤ 1}. (In fact, every x B is the weak limit of a sequence in S.)

64. Let be a separable infinite-dimensional Hilbert space with orthonormal basis {un}∞₁.

a. For k , define Lk L(, ) by Lk(∑∞₁ anun) = ∑∞k anun–k. Then Lk → 0 in the strong operator topology but not in the norm topology.

b. For k , define Rk L(, ) by Rk(∑∞₁ anun) = ∑∞₁ anun+k. Then Rk → 0 in the weak operator topology but not in the strong operator topology.

c. RkLk → 0 in the strong operator topology, but LkRk = I for all k. (Use Exercise 53b.)

65. l²(A) is unitarily isomorphic to l²(B) iff card(A) = card(B).

66. Let be a closed subspace of L²([0,1], m) that is contained in C([0,1]).

a. There exists C > 0 such that ||f||u ≤ C||f||L² for all f . (Use the closed graph theorem.)

b. For each x [0,1] there exists gx such that f(x) = f, gx for all f , and ||gx||L² ≤ C.

c. The dimension of is at most C². (Hint: If {fj} is an orthonormal sequence in , ∑|fj(x)|² ≤ C² for all x [0,1].)

67. (The Mean Ergodic Theorem). Let U be a unitary operator on the Hilbert space , = {x : Ux = x}, P the orthogonal projection onto (Exercise 58), and Sn = n⁻¹ ∑n−1₀ Uj. Then Sn → P in the strong operator topology. (If x , then Snx = x; if x = y − Uy for some y, then Snx → 0. By Exercise 57d, = {x : U*x = x}. Apply Exercise 57c with T = I − U.)

5.6 NOTES AND REFERENCES

Functional analysis is a vast subject of which we have barely scratched the surface here. For the reader who wishes to learn more, Reed and Simon [1121 and Rudin [126] are good places to start; one should also familiarize oneself with the treatises of Dunford and Schwartz [35] and Yosida [163].

Functional analysis has roots in a number of classical problems, particularly in the theory of differential and integral equations. The study of particular infinite-dimensional function spaces began in earnest around 1907 with work of F. Riesz, Fréchet, Schmidt, Helly, and others, and the notion of an abstract normed vector space appeared in papers by several authors about 1920. The research of the succeeding decade culminated in Banach’s classic book [9], which marked the emergence of functional analysis as an established discipline. Detailed historical accounts can be found in Dieudonné [33] and in the notes in Dunford and Schwartz [35].

§5.1: The integral for vector-valued functions developed in Exercise 16 is called the Bochner integral. The hypothesis that is separable can be dropped, but the functions in L¹ must then be required to have separable range (after modification on a null set). A more detailed account can be found in Cohn [27] or Yosida [163].

Another approach to vector-valued integrals is as follows. Suppose that (X, , μ) is a measure space and is a topological vector space on which the continuous linear functionals separate points, A function f : X → is called weakly integrable if (i) for all ϕ *, and (ii) there exists y (necessarily unique) such that ∫ ϕ f dμ = ϕ(y) for all ϕ *. In this case we set ∫ f dμ = y. If is a separable Banach space, this notion of integral coincides with the Bochner integral. See Yosida [163] and Rudin [126].

§5.3: The open mapping and closed graph theorems are due to Banach [9]. See Grabiner [58] for an interesting comment on the relation between the proofs of the open mapping theorem and the Tietze extension theorem.

The uniform boundedness principle, as we have stated it, is due to Banach and Steinhaus [10]; however, the second part of the theorem — that if is a Banach space and supT||Tx|| < ∞ for all x , then supT||T|| < ∞ — had been proved previously by what Dieudonné [33] calls the “method of the gliding hump.” This rather pretty (and elementary) argument has been largely neglected in recent years, but a modern exposition of it can be found in Hennefeld [71].

It is simple to construct examples of unbounded linear maps T : → from one normed vector space to another when is incomplete (see Exercises 29 and 30), but virtually impossible to do so when is complete without using the axiom of choice. The standard method is as follows: Start with an unbounded T : ₀ → where ₀ is incomplete, and let be the completion of ₀. Pick a basis {uα}αA for ₀ (meaning that every x ₀ is a finite linear combination of the uα’s), and extend it to a basis {uα}αB (B ⊃ A) for . (This is where the axiom of choice comes in.) Let be the linear span of {uα}αB\A, so that each x can be written uniquely as x = x₀ + x₁ where x₁ ₀ and x₁ . Then T can be extended to by setting T(x₀ + x₁) = Tx₀.

§5.4: Treves [150] contains a readable account of the general theory of topological vector spaces, with many concrete examples.

Alaoglu’s theorem, which was first announced in Alaoglu [3] and proved in detail in Alaoglu [4], supersedes a number of earlier results dealing with special cases. It was discovered independently by Bourbaki [19].

§5.5: The space envisaged by Hilbert himself was l²(); the notion of an abstract Hilbert space was introduced by von Neumann [154] in his work on the mathematics of quantum mechanics. Theorem 5.25 is originally due to F. Riesz [115] in the setting of L² spaces. It is one of several representation theorems for linear functionals on various spaces that bear his name, the others being Theorems 6.15,7.2, and 7.17. To avoid confusion, we reserve the name “Riesz representation theorem” for the latter two, which are closely related.

In the literature of quantum physics, scalar products are customarily denoted by x|y and are taken to be linear in the second variable and conjugate-linear in the first.

Chapter 6

Lp Spaces

Lp spaces are a class of Banach spaces of functions whose norms are defined in terms of integrals and which generalize the L¹ spaces discussed in Chapter 2. They furnish interesting examples of the general theory of Chapter 5 and play a central role in modern analysis.

6.1 BASIC THEORY OF Lp SPACES

In this chapter we shall be working on a fixed measure space (X, , μ). If f is a measurable function on X and 0 < p < ∞, we define

(allowing the possibility that ||f||p = ∞), and we define

We abbreviate Lp(X, , μ) by Lp(μ), Lp(X), or simply Lp when this will cause no confusion. As we have done with L¹, we consider two functions to define the same element of Lp when they are equal almost everywhere.

If A is any nonempty set, we define lp(A) to be Lp(μ) where μ is counting measure on (A, (A)), and we denote lp() simply by lp.

Lp is a vector space, for if f, g Lp, then

so that f + g Lp. Our notation suggests that ||·||p is a norm on Lp. Indeed, it is obvious that ||f||p = 0 iff f = 0 a.e. and ||cf||p = |c| ||f||p, so the only question is the triangle inequality. It turns out that the latter is valid precisely when p ≥ 1, so our attention will be focused almost exclusively on this case.

Before proceeding further, however, let us see why the triangle inequality fails for p < 1. Suppose a > 0, b > 0, and 0 < p < 1. For t > 0 we have tp−1 > (a + t)p−1, and by integrating from 0 to b we obtain ap + bp > (a + b)p. Thus, if E and F are disjoint sets of positive finite measure in X and we set a = μ(E)1/p and b = μ(F)1/p, we see that

The cornerstone of the theory of Lp spaces is Holder’s inequality, which we now derive.

6.1 Lemma. If a ≥ 0, b ≥ 0, and 0 < λ < 1, then

with equality iff a = b.

Proof. The result is obvious if b = 0; otherwise, dividing both sides by b and setting t = a/b, we are reduced to showing that tλ ≤ λt + (1 − λ) with equality iff t = 1. But by elementary calculus, tλ − λt is strictly increasing for t < 1 and strictly decreasing for t > 1, so its maximum value, namely 1 − λ, occurs at t = 1.

6.2 Hölder’s Inequality. Suppose 1 < p < ∞ and p⁻¹ + q⁻¹ = 1 (that is, q = p/(p − 1)| If f and g are measurable functions on X, then

(6.3) equation

In particular, if f Lp and g Lq, then fg L¹, and in this case equality holds in (6.3) iff α|f|p = β|g|q a.e. for some constants α, β with (α, β) ≠ (0,0).

Proof. The result is trivial if ||f||p = 0 or ||g||q = 0 (since then f = 0 or g = 0 a.e.), or if ||f||p = ∞ or ||p||g = ∞. Moreover, we observe that if (6.3) holds for a particular f and g, then it also holds for all scalar multiples of f and g, for if f and g are replaced by af and bg, both sides of (6.3) change by a factor of |ab|. It therefore suffices to prove that (6.3) holds when ||f||p = ||g||q = 1 with equality iff |f|p = |g|q a.e. To this end, we apply Lemma 6.1 with α = |f(x)|p, b = |g(x)|q, and λ = p⁻¹ to obtain

(6.4) equation

Integration of both sides yields

Equality holds here iff it holds a.e. in (6.4), and by Lemma 6.1 this happens precisely when |f|p = |g|q a.e.

The condition p⁻¹ + q⁻¹ = 1 occurring in Hölder’s inequality turns up frequently in Lp theory. If 1 < p < ∞, the number q = p/(p − 1) such that p⁻¹ + q⁻¹ = 1 is called the conjugate exponent to p.

6.5 Minkowski’s Inequality. If 1 ≤ p < ∞ and f, g Lp, then

Proof. The result is obvious if p = 1 or if f + g = 0 a.e. Otherwise, we observe that

and apply Hölder’s inequality, noting that (p − 1)q = p when q is the conjugate exponent to p:

Therefore,

This result shows that, for p ≥ 1, Lp is a normed vector space, More is true:

6.6 Theorem. For 1 ≤ p < ∞, Lp is a Banach space.

Proof. We use Theorem 5.1. Suppose {fk} ⊂ Lp and ∑∞₁ ||fk||p = B < ∞. Let Gn = ∑n₁|fk| and G = ∑∞₁|fk|. Then ||Gn||p ≤ ∑n₁ ||fk||p ≤ B for all n, so by the monotone convergence theorem, ∫ Gp = lim ∫ Gpn ≤ Bp. Hence G Lp, and in particular G(x) < ∞ a.e., which implies that the series ∑∞₁ fk converges a.e. Denoting its sum by F, we have |F| ≤ G and hence F Lp; moreover, |F − ∑n₁ fk|p ≤ (2G)p L₁, so by the dominated convergence theorem,

Thus the series ∑∞₁ fk converges in the Lp norm.

6.7 Proposition. For 1 ≤ p ≤ ∞, the set of simple functions f = ∑n₁ ajχE, where μ(Ej) < ∞ for all j, is dense in Lp.

Proof. Clearly such functions are in Lp. If f Lp, choose a sequence {fn} of simple functions such that fn → f a.e. and |fn| ≤ |f|, according to Theorem 2.10. Then fn Lp and (fn − f|p ≤ 2p|f|p L¹, so by the dominated convergence theorem, ||fn − f||p → 0. Moreover, if fn = ∑ ajχEj where the Ej are disjoint and the aj are nonzero, we must have μ(Ej) < ∞ since ∑ |aj|pμ(Ej) = ∫ |fn|p < ∞.

To complete the picture of Lp spaces, we introduce a space corresponding to the limiting value p = ∞. If f is a measurable function on X, we define

with the convention that inf Ø = ∞. We observe that the infimum is actually attained, for

and if the sets on the right are null, so is the one on the left. ||f||∞ is called the essential supremum of |f| and is sometimes written

We now define

with the usual convention that two functions that are equal a.e. define the same element of L∞. Thus f L∞ iff there is a bounded measurable function g such that f = g a.e.; we can take g = fχE where E = {x : |f(x)| ≤ ||f||∞}.

Two remarks: First, for fixed X and , L∞(X, , μ) depends on μ only insofar as μ determines which sets have measure zero; if μ and v are mutually absolutely continuous, then L∞(μ) = L∞(v). Second, if μ is not semifinite, for some purposes it is appropriate to adopt a slightly different definition of L∞. This point will be explored in Exercises 23–25.

The results we have proved for 1 ≤ p < ∞ extend easily to the case p = ∞, as follows:

6.8 Theorem.

a. If f and g are measurable functions on X, then ||fg||₁ ≤ ||f||₁||g||∞. If f L¹ and g L∞, ||fg||₁ = ||f||₁||g||∞ iff |g(x)| = ||g||∞ a.e. on the set where f(x) ≠ 0.

b. ||·||∞ is a norm on L∞.

c. ||fn − f||∞ → 0 iff there exists E such that μ(Ec) = 0 and fn → f uniformly on E.

d. L∞ is a Banach space.

e. The simple functions are dense in L∞.

The proof is left to the reader (Exercise 2).

In view of Theorem 6.8a and the formal equality 1⁻¹ + ∞⁻¹ = 1, it is natural to regard 1 and ∞ as conjugate exponents of each other, and we do so henceforth.

Theorem 6.8c shows that ||·||∞ is closely related to, but usually not identical with, the uniform norm ||·||u. However, if we are dealing with Lebesgue measure, or more generally any Borel measure that assigns positive values to all open sets, then ||f||∞ = ||f||u whenever f is continuous, since {x : |f(x)| > a} is open. In this situation we may use the notations ||f||∞ and ||f||u interchangeably, and we may regard the space of bounded continuous functions as a (closed!) subspace of L∞.

In general we have Lp Lq for all p ≠ q; to see what is at issue, it is instructive to consider the following simple examples on (0, ∞) with Lebesgue measure. Let fa(x) = x−a, where a > 0. Elementary calculus shows that faχ(0,1) Lp iff p < a⁻¹, and faχ(1,∞) Lp iff p > a⁻¹. Thus we see two reasons why a function f may fail to be in Lp: either |f|p blows up too rapidly near some point, or it fails to decay sufficiently rapidly at infinity. In the first situation the behavior of |f|p becomes worse as p increases, while in the second it becomes better. In other words, if p < q, functions in Lp can be locally more singular than functions in Lq, whereas functions in Lq can be globally more spread out than functions in Lp. These somewhat imprecisely expressed ideas are actually a rather accurate guide to the general situation, concerning which we now give four precise results. The last two show that inclusions Lp ⊂ Lq can be obtained under conditions on the measure space that disallow one of the types of bad behavior described above; for a more general result, see Exercise 5.

6.9 Proposition. If 0 < p < q < r ≤ ∞, then Lp Lp + Lr; that is, each f Lq is the sum of a Junction in Lp and a function in Lr.

Proof. If f Lq, let E = {x : |f(x)| > 1} and set g = fχE and h = fχEc. Then |g|p = |f|pχE ≤ |f|qχE, so g Lp, and |h|r = |f|rχEc ≤ |f|qχEc, so h Lr. (For r = ∞, obviously ||h||∞ ≤ 1.)

6.10 Proposition. If 0 < p < q < r ≤ ∞, then Lp Lr Lq and ||f||q ≤ ||f||pλ||f||r1-λ, where λ (0,1) is defined by

Proof. If r = ∞, we have and λ = p/q, so

If r < ∞, we use Hölder’s inequality, taking the pair of conjugate exponents to be p/λq and r/(1 − λ)q:

Taking qth roots, we are done.

6.11 Proposition. If A is any set and 0 < p < q ≤ ∞, then lp(A) ∞ lq(A) and ||f||q ≤ ||f||p||.

Proof. Obviously ||f||∞p = supα |f(α)|p ≤ Σα |f(α)|p, so that ||f||∞ ≤ ||f||p. The case q < ∞ then follows from Proposition 6.10: if λ = p/q,

6.12 Proposition. If μ(X) < ∞ and 0 < p < q ≤ ∞, then Lp(μ) Lq(μ) and .

Proof. If q = ∞, this is obvious:

If q < ∞, we use Hölder’s inequality with the conjugate exponents q/p and q/(q − p):

We conclude this section with a few remarks about the significance of the Lp spaces. The three most obviously important ones are L¹, L², and L∞. With L¹ we are already familiar; L² is special because it is a Hilbert space; and the topology on L∞ is closely related to the topology of uniform convergence. Unfortunately, L¹ and L∞ are pathological in many respects, and it is more fruitful to deal with the intermediate Lp spaces. One manifestation of this is the duality theory in §6.2; another is the fact that many operators of interest in Fourier analysis and differential equations are bounded on Lp for 1 < p < ∞ but not on L¹ or L∞, (Some examples are mentioned in §9.4.)

Exercises

1. When does equality hold in Minkowski’s inequality? (The answer is different for p = 1 and for 1 < p < ∞. What about p = ∞?)

2. Prove Theorem 6.8.

3. If 1 ≤ p < r ≤ ∞, Lp Lr is a Banach space with norm ||f|| = ||f||p + ||f||r, and if p < q < r, the inclusion map Lp Lr → Lq is continuous.

4. If 1 ≤ p < r ≤ ∞, Lp + Lr is a Banach space with norm ||f|| = inf{||g||p + ||h||r : f = g + h}, and if p < q < r, the inclusion map Lq → Lp + Lr is continuous.

5. Suppose 0 < p < q < ∞. Then iff X contains sets of arbitrarily small positive measure, and Lq ∞ Lp iff X contains sets of arbitrarily large finite measure. (For the “if implication: In the first case there is a disjoint sequence {En} with 0 μ(En) < 2−n, and in the second case there is a disjoint sequence {En} with 1 ≤ μ(En) < ∞. Consider f = ΣanχEn for suitable constants an.) What about the case q = ∞?

6. Suppose 0 < p₀ < p₁ < ∞. Find examples of functions f on (0, ∞) (with Lebesgue measure), such that f Lp iff (a) p₀ < p < p₁, (b) p₀ ≤ p ≤ p₁, (c) p = p₀. (Consider functions of the form f(x) = x−a | log x|b.)

7. If f Lp L∞ for some p < ∞, so that f Lq for all q > p, then ||f||∞ = limq→∞ ||f||q.

8. Suppose μ(X) = 1 and f Lp for some p > 0, so that f Lq for 0 < q < p.

a. log ||f||, ≥ ∫ log |f|. (Use Exercise 42d in §3.5, with F(t) = et.)

b. (∫|f|q − 1)/q ≥ log ||f||q, and (∫|f|q − 1)/q → ∫ log |f| as q → 0.

c. limq→0 ||f||q = exp(∫ log|f|).

9. Suppose 1 ≤ p < ∞. If ||fn − f||p → 0, then fn → f in measure, and hence some subsequence converges to f a.e. On the other hand, if fn → f in measure and |fn| ≤ g Lp for all n, then ||fn − f||p → 0.

10. Suppose 1 ≤ p < ∞. If fn, f Lp and fn → f a.e., then ||fn − f||p → 0 iff ||fn||p → ||f||p. (Use Exercise 20 in §2.3.)

11. If f is a measurable function on X, define the essential range Rf of f to be the set of all z such that {x : |f(x) − z| < } has positive measure for all > 0.

a. Rf is closed.

b. If f L∞, then Rf is compact and ||f||∞ = max{|z| : z Rf}.

12. If p ≠ 2, the Lp norm does not arise from an inner product on Lp, except in trivial cases when dim(Lp) ≤ 1. (Show that the parallelogram law fails.)

13. Lp(n, m) is separable for 1 ≤ p < ∞. However, L∞(n, m) is not separable. (There is an uncountable set such that ||f − g||∞ ≥ 1 for all f, g F with f ≠ g.)

14. If g L∞, the operator T defined by T f = fg is bounded on Lp for 1 ≤ p ≤ ∞. Its operator norm is at most ||g||∞, with equality if μ is semifinite.

15. (The Vitali Convergence Theorem) Suppose 1 ≤ p < ∞ and {fn}₁∞ ⊂ Lp. In order for {fn} to be Cauchy in the Lp norm it is necessary and sufficient for the following three conditions to hold: (i) {fn} is Cauchy in measure; (ii) the sequence {|fn|p} is uniformly integrable (see Exercise 11 in §3.2); and (iii) for every > 0 there exists E ⊂ X such that μ(E) < ∞ and ∫Ec |fn|p < for all n. (To prove the sufficiency: Given > 0, let E be as in (iii), and let Amn = {x E : |fm(x) − fn(x)| ≥ }. Then the integrals of |fn − fm|p over E\Amn, Amn, and Ec are small when m and n are large — for three different reasons.)

16. If 0 < p < 1, the formula ρ(f, g) = ∫|f − g|p defines a metric on Lp that makes Lp into a complete topological vector space. (The proof of Theorem 6.6 still works for p < 1 if ||f||p is replaced by ∫ |f|p, as it uses only the triangle inequality and not the homogeneity of the norm.)

6.2 THE DUAL OF Lp

Suppose that p and q are conjugate exponents. Hölder’s inequality shows that each g Lq defines a bounded linear functional ϕg on Lp by

and the operator norm of ϕg is at most ||g||q. (If p = 2 and we are thinking of L² as a Hilbert space, it is more appropriate to define ϕg(f) = . The same convention can be used for p ≠ 2 without changing the results below in an essential way.) In fact, the map g → ϕg is almost always an isometry from Lq into (Lp)*.

6.13 Proposition. Suppose that p and q are conjugate exponents and 1 ≤ q < ∞. If g Lq, then

If μ is semifinite, this result holds also for q = ∞.

Proof. Hölder’s inequality says that ||ϕg|| ≤ ||g||q, and equality is trivial if g = 0 (a.e.). If g ≠ 0 and q < ∞, let

Then

so

(If q = 1, then , ||f||∞ = 1, and ∫ fg = ||g||₁.) If q = ∞, for > 0 let A = {x : |g(x)| > ||g||∞ − }. Then μ(A) > 0, so if μ is semifinite there exists B ⊂ A with 0 < μ(B) < ∞. Let f = ; then ||f||₁ = 1, so

Since is arbitrary, ||ϕg|| = ||g||∞.

Conversely, if f → ∫ fg is a bounded linear functional on Lp, then g Lq in almost all cases. In fact, we have the following stronger result.

6.14 Theorem. Let p and q be conjugate exponents. Suppose that g is a measurable function on X such that fg L¹ for all f in the space Σ of simple functions that vanish outside a set of finite measure, and the quantity

is finite. Also, suppose either that Sg = {x : g(x) ≠ 0} is σ−finite or that μ, is semifinite. Then g Lq and Mq(g) = ||g||q.

Proof. First, we remark that if f is a bounded measurable function that vanishes outside a set E of finite measure and ||f||p = 1, then |∫ fg| ≤ Mq(g). Indeed, by Theorem 2.10 there is a sequence {fn} of simple functions such that |fn| ≤ |f| (in particular, fn vanishes outside E) and fn → f a.e. Since |fn| ≤ ||f||∞χE and χEg L¹, by the dominated convergence theorem we have |∫ fg| = lim |∫ fng| ≤ Mq(g).

Now suppose that q < ∞. We may assume that Sg is σ-finite, as this condition automatically holds when μ is semifinite; see Exercise 17. Let {En} be an increasing sequence of sets of finite measure such that Sg = ∪₁∞ En. Let {ϕn} be a sequence of simple functions such that ϕn → g pointwise and |ϕn| ≤ |g|, and let gn = ϕnχEn. Then gn → g pointwise, |gn| ≤ |g|, and gn vanishes outside En. Let

Then as in the proof of Proposition 6.13 we have ||fn||p = 1, and by Fatou’s lemma,

(For the last estimate we used the remark at the beginning of the proof.) On the other hand, Hölder’s inequality gives Mq(g) ≤ ||g||q, so the proof is complete for the case q < ∞.

Now suppose q = ∞. Given > 0, let A = {x : |g(x)| ≥ M∞(g) + }. If μ(A) were positive, we could choose B ⊂ A with 0 < μ(B) < ∞ (either because μ is semifinite or because A ⊂ Sg). Setting , we would then have ||f||₁ = 1, and ∫ fg = μ(B)⁻¹ ∫B |g| ≥ M∞(g) + . But this is impossible by the remark at the beginning of the proof. Hence ||g||∞ ≤ M∞(g), and the reverse inequality is obvious.

The last and deepest part of the description of (Lp)* is the fact that the map g → ϕg is, in almost all cases, a surjection.

6.15 Theorem. Let p and q be conjugate exponents. If 1 < p < ∞, for each ϕ (Lp)* there exists g Lq such that ϕ(f) = ∫ fg for all f Lp, and hence Lq is isometrically isomorphic to (Lp)*. The same conclusion holds for p = 1 provided μ is σ−finite.

Proof. First let us suppose that μ is finite, so that all simple functions are in Lp. If ϕ (Lp)* and E is a measurable set, let v(E) = ϕ(χE). For any disjoint sequence {Ej}, if E ∪₁∞ Ej we have χE = Σ₁∞ χEj where the series converges in the Lp norm:

(It is at this point that we need the assumption that p < ∞.) Hence, since ϕ is linear and continuous,

so that v is a complex measure. Also, if μ(E) = 0, then χE = 0 as an element of Lp, so v(E) = 0; that is, v μ. By the Radon-Nikodym theorem there exists g L¹ (μ) such that ϕ(χE) = v(E) = ∫Eg dμ for all E and hence χ(f) = ∫ fg dμ for all simple functions f. Moreover, |∫ fg| ≤ ||ϕ|| ||f||p, so g Lq by Theorem 6.14. Once we know this, it follows from Proposition 6.7 that ϕ(f) = ∫ fg for all f Lp.

Now suppose that μ is σ-finite. Let {En} be an increasing sequence of sets such that 0 < μ{En) < ∞ and X = ∪₁∞ En, and let us agree to identify Lp(En) and Lq(En) with the subspaces of Lp(X) and Lq(X) consisting of functions that vanish outside En. The preceding argument shows that for each n there exists gn Lq(En) such that ϕ(f) = ∫ fgn for all f Lp(En), and ||gn||q = ||ϕ|Lp(En)|| ≤ ||ϕ||. The function gn is unique modulo alterations on nullsets, so gn = gm a.e. on En for n < m, and we can define g a.e. on X by setting g = gn on En. By the monotone convergence theorem, ||g||q = lim ||gn||q ≤ ||ϕ||. so g Lq. Moreover, if f Lp, then by the dominated convergence theorem, fχEn → f in the Lp norm and hence ϕ(f) = lim ϕ(fχEn) = lim ∫En fg = ∫ fg.

Finally, suppose that μ is arbitrary and p > 1, so that q < ∞. As above, for each σ-finite set E ⊂ X there is an a.e.-unique gE Lq(E) such that ϕ(f) = ∫ fgE for all f Lp(E) and ||gE||q ≤ ||ϕ||. If F is σ-finite and F ⊃ E, then gF = gE a.e. on E, so ||gF||q ≥ ||gE||q. Let M be the supremum of ||gE||q as E ranges over all σ-finite sets, noting that M ≤ ||ϕ||. Choose a sequence {En} so that ||gEn||q → M, and set F = ∪₁∞ En. Then F is σ-finite and ||gF||q ≥ ||gEn||q for all n, whence ||gF||q = M. Now, if A is a σ-finite set containing F, we have

and thus gA\F = 0 and gA = gF a.e. (Here we use the fact that q < ∞.) But if f Lp, then A = F ∪ {x: f(x) ≠ 0} is σ-finite, so ϕ(f) = ∫ fgA = ∫ fgF. Thus we may take g = gF, and the proof is complete.

6.16 Corollary. If 1 < p < ∞, Lp is reflexive.

We conclude with some remarks on the exceptional cases p = 1 and p = ∞. For any measure μ, the correspondence g ∞ ϕg maps L∞ into (L¹)*, but in general it is neither injective nor surjective. Injectivity fails when μ is not semifinite. Indeed, if E ⊂ X is a set of infinite measure that contains no subsets: of positive finite measure, and f L¹, then {x : f(x) ≠ 0} is σ-finite and hence intersects E in a null set. It follows that ϕχE = 0 although χ ≠ 0 in L∞. This problem, however, can be remedied by redefining L∞; see Exercises 23–25. The failure of surjectivity is more subtle and is best illustrated by an example; see also Exercise 19.

Let X be an uncountable set, μ = counting measure on (X, P(X)), M = the σ-algebra of countable or co-countable sets, and μ₀ = the restriction of μ to M. Every f L¹(μ) vanishes outside a countable set, and it follows that L¹(μ) = L¹(μ₀). On the other hand, L∞(μ) consists of all bounded functions on X, whereas L∞(μ₀) consists of those bounded functions that are constant except on a countable set. With this in mind, it is easy to see that the dual of Ll(μ₀) is L∞(μ) and not the smaller space L∞(μ₀).

As for the case p = ∞: the map g → ϕg is always an isometric injection of L¹ into (L∞)* by Proposition 6.13, but it is almost never a surjection. We shall say more about this in §6.6; for the present, we give a specific example. (Another example can be found in Exercise 19.)

Let X = [0, 1], μ = Lebesgue measure. The map f f(0) is a bounded linear functional on C(X), which we regard as a subspace of L∞. By the Hahn-Banach theorem there exists ϕ (L∞)* such that ϕ(f) = f(0) for all f C(X). To see that ϕ cannot be given by integration against an L¹ function, consider the functions fn C(X) defined by fn(x) = max(1 − nx, 0). Then ϕ(fn) = fn(0) = 1 for all n, but fn(x) → 0 for all x > 0, so by the dominated convergence theorem, ∫ fng → 0 for all g L¹.

Exercises

17. With notation as in Theorem 6.14, if μ is semifinite, q < ∞, and Mq(g) < ∞, then {x : |g(x)| > } has finite measure for all > 0 and hence Sg is σ-finite.

18. The self-duality of L² follows from Hilbert space theory (Theorem 5.25), and this fact can be used to prove the Lebesgue-Radon-Nikodym theorem by the following argument due to von Neumann. Suppose that μ, v are positive finite measures on (X, M) (the σ-finite case follows easily as in §3.2), and let; λ = μ + v.

a. The map f f dv is a bounded linear functional on L²(λ), so ∫ f dv = ∫ fg dλ for some g L²(λ). Equivalently, ∫ f(1 − g)dv = ∫ fg dμ for f L²(λ).

b. 0 ≤ g ≤ 1 λ-a.e., so we may assume 0 ≤ g ≤ 1 everywhere.

c. Let A = {x : g(x) < 1}, B = {x : g(x) = 1}, and set va(E) = v(A E), vs(E) = v(B E). Then vs μ and va μ; in fact, dva = g(1 − g)⁻¹χA dμ.

19. Define ϕn (l∞)* by ϕn(f) = n⁻¹ Σ₁n f(j). Then the sequence {ϕn} has a weak* cluster point ϕ, and ϕ is an element of (l∞)* that does not arise from an element of l¹.

20. Suppose supn ||fn||p < ∞ and fn → f a.e.

a. If 1 < p < ∞, then fn → f weakly in Lp. (Given g Lq, where q is conjugate to p, and > 0, there exist (i) δ > 0 such that ∫E |g|q < whenever μ(E) < δ, (ii) A ⊂ X such that μ(A) < ∞ and ⊂X/A |g|q < , and (iii) B ⊂ A such that μ(A \ B) < δ and fn → f uniformly on B.)

b. The result of (a) is false in general for p = 1. (Find counterexamples in L¹(, m) and l¹.) It is, however, true for p = ∞ if μ, is σ-finite and weak convergence is replaced by weak* convergence.

21. If 1 < p < ∞, fn → f weakly in lp(A) iff supn ||fn||p < ∞ and fn → f pointwise.

22. Let X = [0, 1], with Lebesgue measure.

a. Let fn(x) = cos 2πnx. Then fn → 0 weakly in L² (see Exercise 63 in §5.5), but fn 0 a.e. or in measure.

b. Let fn(x) = nχ(0,1/n). Then fn → 0 a.e. and in measure, but fn 0 weakly in Lp for any p.

23. Let (X, M, μ) be a measure space. A set E M is called locally null if μ(E F) = 0 for every F M such that μ(F) < ∞. If f : X → is a measurable function, define

and let ∞ be the space of all measurable f such that ||f||* < ∞. We consider f, g to be identical if {x : f(x) ≠ g(x)} is locally null.

a. If E is locally null, then μ(E) is either 0 or ∞. If μ is semifinite, then every locally null set is null.

b. ||·||* is a norm on that makes into a Banach space. If μ is semifinite, then .

24. If ∞ (see Exercise 23), then ||g||* = sup{|∫ fg| : ||f||₁ = 1}, so the map is an isometry from into (L¹)*. Conversely, if as in Theorem 6.14, then g , and .

25. Suppose μ is decomposable (see Exercise 15 in §3.2). Then every ϕ (L¹)* is of the form ϕ(f) = ∫ fg for some g L∞, and hence (see Exercises 23 and 24). (If F is a decomposition of μ and f L¹, there exists {Ej} such that fχEj where the series converges in L¹.)

6.3 SOME USEFUL INEQUALITIES

Estimates and inequalities lie at the heart of the applications of Lp spaces in analysis. The most basic of these are the Hölder and Minkowski inequalities. In this section we present a few additional important results in this area. The first one is almost a triviality, but it is sufficiently useful to warrant special mention.

6.17 Chebyshev’s Inequality. If f Lp (0 < p < ∞), then for any α > 0,

Proof. Let Eα = {x: |f(x)| > α}. Then

The next result is a rather general theorem about boundedness of integral operators on Lp spaces.

6.18 Theorem. Let and be σ−finite measure spaces, and let K be an −measurable function on X x Y. Suppose that there exists C > 0 such that C for a.e. y Y and |K(x, y)|dv(y) ≤ C for a.e. x X, and that 1 ≤ p ≤ ∞. If f Lp(u), then the integral

converges absolutely for a.e. x X, the function T f thus defined is in Lp(μ), and ||T f||p ≤ C||f||p.

Proof. Suppose that 1 < p < ∞. Let q be the conjugate exponent to p. By applying Hölder’s inequality to the product

we have

for a.e. x X. Hence, by Tonelli’s theorem,

Since the last integral is finite, Fubini’s theorem implies that K(x, ·)f L¹(v) for a.e. x, so that T f is well defined a.e., and

Taking pth roots, we are done.

For p = 1 the proof is similar but easier and requires only the hypothesis ≤ C; for p = ∞ the proof is trivial and requires only the hypothesis ∫ |K(x, y)| dv(y) ≤ C. Details are left to the reader (Exercise 26).

Minkowski’s inequality states that the Lp norm of a sum is at most the sum of the Lp norms. There is a generalization of this result in which sums are replaced by integrals:

6.19 Minkowski’s Inequality for Integrals. Suppose that and are σ-finite measure spaces, and let f be an −measurable function on X × Y.

a. If f ≤ 0 and 1 ≤ p < ∞, then

b. If 1 ≤ p ≤ ∞, f(·, y) Lp(μ) for a.e. y, and the function y ||f(·, y)||p is in L¹(v), then f(x, ·) L¹(v) for a.e. x, the function x f(x, y) dv(y) is in Lp(μ), and

Proof. If p = 1, (a) is merely Tonelli’s theorem. If 1 < p < ∞, let q be the conjugate exponent to p and suppose g Lq(μ). Then by Tonelli’s theorem and Hölder’s inequality,

Assertion (a) therefore follows from Theorem 6.14. When p < ∞, (b) follows from (a) (with f replaced by |f|) and Fubini’s theorem; when p = ∞, it is a simple consequence of the monotonicity of the integral.

Our final result is a theorem concerning integral operators on (0, ∞) with Lebesgue measure.

6.20 Theorem. Let K be a Lebesgue measurable function on (0, ∞) x (0, ∞) such that K(λx, λy) = λ⁻¹K(x, y) for all λ 0 and ∞ |K(x, 1)|x−1/p dx = C < ∞ for some p [1, ∞], and let q be the conjugate exponent to p. For f Lp and g Lq, let

Then T f and S g are defined a.e., and ||T f||p ≤ C||f||p and ||S g||q ≤ C||g||q.

Proof. Setting z = x/y, we have

where fz(y) = f(yz); moreover,

Therefore, by Minkowski’s inequality for integrals, T f exists a.e. and

Finally, setting u = y⁻¹, we have

so the same reasoning shows that S g is defined a.e. and that ||S g||q ≤ C||g||q.

6.21 Corollary. Let

Then for 1 < p ≤ ∞ and 1 ≤ q < ∞,

Proof. Let K(x, y) = y⁻¹χE(x, y) where E = {(x, y) : x < y}. Then dx = p/(p − 1) = q, where q is the conjugate exponent to p, so Theorem 6.20 yields the result.

Corollary 6.21 is a special case of Hardy’s inequalities; the general result is in Exercise 29.

Exercises

26. Complete the proof of Theorem 6.18 for the cases p = 1 and p = ∞.

27. (Hubert’s Inequality) The operator T f(x) = dy satisfies ||T f||p ≤ Cp||f||p for 1 < p < ∞, where Cp = dx. (For those who know about contour integrals: Show that Cp = π csc(π/p).)

28. Let Iα be the αth fractional integral operator as in Exercise 61 of §2.6, and let Jαf(x) = x−αf(x).

a. Jα is bounded on Lp(0, ∞) for 1 < p ≤ ∞; more precisely,

b. There exists f L¹(0, ∞) such that J₁f L¹(0, ∞).

29. Suppose that 1 ≤ p < ∞, r > 0, and h is a nonnegative measurable function on (0, ∞). Then:

(Apply Theorem 6.20 with and g(x) = xδh(x) for suitable β, γ, δ)

30. Suppose that K is a nonnegative measurable function on (0, ∞) such that for 0 < s < 1.

a. If 1 < p < ∞, p⁻¹ + q⁻¹ = 1, and f, g are nonnegative measurable functions on (0, ∞), then (with ∫ = )

b. The operator T f(x) = ∞ K(xy)f(y) dy is bounded on L²((0, ∞)) with norm . (Interesting special case: If K(x) = e−x, then T is the Laplace transform and ϕ(s) = Γ(s).)

31. (A Generalized Hölder Inequality) Suppose that 1 pj ≤ ∞ and Σ₁n pj¹ = r⁻¹ ≤ 1. If fj Lpj for j = 1, …, n, then Lr and (First do the case n = 2.)

32. Suppose that (X, , μ) and (Y, , v) are σ-finite measure spaces and K L²(μ x v). If f L²(v), the integral T f(x) = ∫ K(x, y)f(y) dv (y) converges absolutely for a.e. x X; moreover, T f L²(μ) and ||T f||₂ ≤ ||K||₂ ||f||₂.

33. Given 1 < p < ∞, let T f(x) = dt. If p⁻¹ + q⁻¹ = 1, then T is a bounded linear map from Lq((0, ∞)) to C₀((0, ∞)).

34. If f is absolutely continuous on [, 1] for 0 , then limx→0 f(x) exists (and is finite) if p > 2, |f(x)|/| log x|1/2 → 0 as x → 0 if p = 2, and |f(x)|/x1-(2/p) → 0 as x → 0 if p < 2.

6.4 DISTRIBUTION FUNCTIONS AND WEAK LP

If f is a measurable function on , we define its distribution function λf: (0, ∞) → [0, ∞] by

(This is closely related, but not identical, to the “distribution functions” discussed in §1.5 and §10.1.) We compile the basic properties of λf in a proposition:

6.22 Proposition.

a. λf is decreasing and right continuous.

b. |f| ≤ |g|, then λf ≤ λg.

c. |fn| increases to |f|, then λfn increases to λf.

d. f = g + h, then

Proof. Let E(α, f) = {x : |f(x)| > α}. The function λf is decreasing since E(α, f) ⊃ E(β, f) if α < β, and it is right continuous since E(α, f) is the increasing union of {E(α + n⁻¹, f)}₁∞. If |f| ≤ |g|, then E(α, f) ⊂ E(α, g), so λf ≤ λg. If |fn| increases to |f|, then E(α, f) is the increasing union of {E(α, fn)}, so λfn increases to λf. Finally, if f = g + h, then E(α, f) ⊂ E(α, g) ∪ E(α, h), which implies that

Suppose that λf(α) < for all α > 0. In view of Proposition 6.22a, λf defines a negative Borel measure v on (0, ∞) such that v((a, b]) = λf(b) − λf(a) whenever 0 < a < b. (Our construction of Borel measures on in §1.5 works equally well on (0, ∞).) We can therefore consider the Lebesgue-Stieltjes integrals ∫ ϕdλf = ∫ ϕdv of functions ϕ on (0, ∞). The following result shows that the integrals of functions of |f| on X can be reduced to such Lebesgue-Stieltjes integrals.

6.23 Proposition. If λf(α) < ∞ for all α > 0 and ϕ is a nonnegative Borel measurable function on (0, ∞), then

Proof. If v is the negative measure determined by λf, we have

It follows that v(E) = μ(|f|⁻¹ (E)) for all Borel sets E ⊂ (0, ∞), by the uniqueness of extensions (Theorem 1.14). But this means that ∫X ϕ ∞ |f|dμ = − when ϕ is the characteristic function of a Borel set, and hence when ϕ is simple. The general case then follows by virtue of Theorem 2.10 and the monotone convergence theorem.

The case of this result in which we are most interested is ϕ(α) = αp, which gives

A more useful form of this equation is obtained by integrating the right side by parts (Theorem 3.36) to obtain . The validity of this calculation is not clear unless we know that as α → 0 and α θ ∞; nonetheless, the conclusion is correct.

6.24 Proposition. If 0 < p < ∞, then

Proof. If λf(α) = ∞ for some α > 0, then both integrals are infinite. If not, and f is simple, then λf is bounded as α → 0 and vanishes for a sufficiently large, so the integration by parts described above works. (It is also easy to verify the formula directly in this case.) For the general case, let {gn} be a sequence of simple functions that increases to |f|; then the desired result is true for gn, and it follows for f by Proposition 6.22c and the monotone convergence theorem.

A variant of the Lp spaces that turns up rather often is the following. If f is a measurable function on X and 0 < p < ∞, we define

and we define weak Lp to be the set of all f such that [f]p < ∞. [·]p is not a norm; it is easily checked that [cf]p = |c|[f]p, but the triangle inequality fails. However, weak Lp is a topological vector space; see Exercise 35.

The relationship between Lp and weak Lp is as follows. On the one hand,

(This is just a restatement of Chebyshev’s inequality.) On the other hand, if we replace λf(α) by ([f]p/α)p in the integral dα, which equals ||f||pp, we obtain a constant times , which is divergent at both 0 and ∞ — but just barely. One needs only slightly stronger estimates on λf near 0 and ∞ to obtain f Lp. (See also Exercise 36.) The standard example of a function that is in weak Lp but not in Lp is f(x) = x−1/p on (0, ∞) (with Lebesgue measure).

Frequently it is convenient to express a function as the sum of a “small” part and a “big” part. The following is a way of doing this that gives a simple formula for the distribution functions.

6.25 Proposition. If f is a measurable function and A > 0, let E(A) = {x : |f(x)| > A}, and set

Then

The proof is left to the reader (Exercise 37).

Exercises

35. For any measurable f and g we have [cf]p = |c|[f]p and [f + g]p ≤ 2([f]pp + [g]pp1/p; hence weak Lp is a vector space. Moreover, the “balls” {g : [g − f]p < r} (r > 0, f weak Lp) generate a topology on weak Lp that makes weak Lp into a topological vector space.

36. If f weak Lp and μ({x : f(x) ≠ 0}) < ∞, then f Lq for all q < p. On the other hand, if f (weak Lp) L∞, then f Lq for all q > p.

37. Prove Proposition 6.25.

38. f Lp iff .

39. If f Lp, then limα→0 αp λf(α) = limα→∞ αp λf(α) = 0. (First suppose f is simple.)

40. If f is a measurable function on X, its decreasing rearrangement is the function f* : (0, ∞) → [0, ∞] defined by

a. f* is decreasing. If f*(t) < ∞ then λf (f*(t)) ≤ t, and if λf(α) < ∞ then f*(λf(α)) ≤ α.

b. λf = λf*, where λf* is defined with respect to Lebesgue measure on (0, ∞).

c. If λf (α) < ∞ for all α > 0 and limα→∞ λf (α) = 0 (so that f*(t) < ∞ for all t > 0), and (ϕ ≥ 0 is a Borel measurable function on (0, ∞), then dt. In particular, ||f||p = ||f*||p for 0 < p < ∞.

d. If 0 < p < ∞, [f]p = supt>0 t1/p f*(t).

e. The name “rearrangement” for f* comes from the case where f is a nonnegative function on (0, ∞). To see why it is appropriate, pick a step function on (0, ∞) assuming four or five different values and draw the graphs of f and f*.

6.5 INTERPOLATION OF LP SPACES

If 1 ≤ p < q < r ≤ ∞ then (), and it is natural to ask whether a linear operator T on Lp + Lr that is bounded on both Lp and Lr is also bounded on Lq. The answer is affirmative, and this result can be generalized in various ways. The two fundamental theorems on this question are the Riesz-Thorin and Marcinkiewicz interpolation theorems, which we present in this section. We begin with the Riesz-Thorin theorem, whose proof is based on the following result from complex function theory.

6.26 The Three Lines Lemma. Let ϕ be a bounded continuous function on the strip 0 ≤ Re z ≤ 1 that is holomorphic on the interior of the strip. If|ϕ(z)| ≤ M₀ for Re z = 0 and |ϕ(z)| ≤ M₁ for Re z = 1, then |ϕ(z)| ≤ M₀1−t M₁t for Re z = t, 0 < t < 1.

Proof. For > 0 let . Then ϕ satisfies the hypotheses of the lemma with M₀ and M₁ replaced by 1, and also |ϕ(z)| → 0 as |Im z| → ∞. Thus |ϕ(z)| ≤ 1 on the boundary of the rectangle 0 ≤ Re z ≤ 1, –A ≤ Im z ≤ A provided that A is large, and the maximum modulus principle therefore implies that |ϕ(z)| ≤ 1 on the strip 0 ≤ Re z ≤ 1. Letting → 0, we obtain the desired result:

6.27 The Riesz-Thorin Interpolation Theorem. Suppose that and are measure spaces and p₀, p₁, q₀, q₁ [1, ∞]. If q₀ = q₁ = ∞, suppose also that v is semifinite. For 0 < t < 1, define pt and qt by

If T is a linear map from Lp0 (μ) + Lp0 (μ) into Lq0 (v) + Lq1 (v) such that ||T f||q0 ≤ M₀||f||p0 for f Lp0(μ) and ||T f||q1 ≤ M₁||f||p1 for f Lp1(μ), then ||T f||qt ≤ M₀1−t M₁t||f||pt for f Lpt(μ), 0 < t < 1.

Proof. To begin with, we observe that the case p₀ = p₁ follows from Proposition 6.10: If p = p₀ = p₁, then

Thus we may assume that p₀ ≠ p₁, and in particular that pt < ∞ for 0 < t < 1.

Let ΣX (resp. ΣY) be the space of all simple functions on X (resp. Y) that vanish outside sets of finite measure. Then ΣX ∞ Lp(μ) for all p and ΣX is dense in Lp(μ) for p < ∞, by Proposition 6.7; similarly for ΣY. The main part of the proof consists of showing that ||T f||,qt ≤ M₀1−t M₁t||f||pt, for all f ΣX. However, by Theorem 6.14,

where q′t is the conjugate exponent to qt. (Note that T f Lq0 Lq1, so {y : T f(y) ≠ 0} must be σ-finite unless q₀ = q₁ = ∞; hence the hypotheses of Theorem 6.14 are satisfied.) Moreover, we may assume that f ≠ 0 and rescale f so that ||f||pt = 1. We therefore wish to establish the following claim:

If f ΣX and ||f||pt = 1, then | ∫(T f)g dv ≤ M₀1−t M₁t for all g ε ΣY such that ||g||q′t = 1.

Let f = Σ₁m cj χ Ej and g = Σ₁n dk χ Fk where the Ej’s and the Fk’s are disjoint in X and Y and the cj’s and dk’s are nonzero. Write cj and dk in polar form: . Also, let

thus α(t) = pt⁻¹ and β(t) = qt⁻¹ for 0 < t < 1. Fix t (0,1); we have assumed that pt < ∞ and hence α(t) > 0, so we may define

If β(t) < 1, we define

while if β(t) = 1 we define gx = g for all z. (We henceforth assume that β(t) < 1 and leave the easy modification for β(t) = 1 to the reader.) Finally, we set

Thus,

where

so that ϕ is an entire holomorphic function of z that is bounded in the strip 0 ≤ Re z ≤ 1. Since ∫(T f)g dv = ϕ(t), by the three lines lemma it will suffice to show that |ϕ(z)| ≤ M₀ for Re z = 0 and |ϕ(z)| ≤ M₁ for Re z = 1. However, since

for s , we have

Therefore, by Hölder’s inequality,

A similar calculation shows that |ϕ(1 + is)| ≤ M₁, so the claim is proved.

We have now shown that for f Σx, so in view of Proposition 6.7, T|Σx has a unique extension to Lpt (μ) satisfying the same estimate there. It remains to show that this extension is T itself, that is, that T satisfies this estimate for all f Lpt (μ). Given such an f, choose a sequence {fn} in Σx such that |fn| ≤ |f| and fn → f pointwise. Also, let E = {x : |f(x)| > 1}, g = f χ E, gn = fn χ E, h = f − g, and hn = fn − gn. Then if p₀ < P₁ (which we may assume, by relabeling the p′s), we have g Lp₀ (μ), h Lpp₁ (μ), and by the dominated convergence theorem, ||fn − f||pt → 0, ||gn − g||p0 → 0, and ||hn − h||p1 → 0. Hence ||Tgn − Tg||q0 → 0 and ||Thn − Th||q1 → 0, so by passing to a suitable subsequence we may assume that Tgn → Tg a.e. and Thn → Th a.e. (Exercise 9). But then T fn → T f a.e., so by Fatou’s lemma,

and we are done.

The conclusion of the Riesz-Thorin theorem can be restated in a slightly stronger form. Let M(t) be the operator norm of T as a map from Lpt (μ) to Lqt (μ). We have shown that M(t) ≤ M₀1−t M₁t. It is possible for strict inequality to hold; however, if 0 < s < t < u < 1 and t = (1 − τ)s + τu, the theorem may be applied again to show that M(t) ≤ (s)1−r(u)τ. In short, the conclusion is that log (t) is a convex function of t.

We now turn to the Marcinkiewicz theorem, for which we need some more terminology. Let T be a map from some vector space of measurable functions on (X, M, μ) to the space of all measurable functions on (Y, N, v).

T is called sublinear if |T(f + g)| ≤ |T f| + |T g| and |T(cf)| = c|T f| for all f, g Δ and c > 0.

A sublinear map T is strong type (p, q) (1 ≤ p, q ≤ ∞) if Lp(μ) ⊂ , T maps Lp(μ) into Lq(v), and there exists C > 0 such that ||T f||q ≤ C||f||p for all f Lp(μ).

A sublinear map T is weak type (p, q) (1 ≤ p ≤ ∞, 1 ≤ q < ∞) if Lp(μ) ⊂ D, T maps Lp(μ) into weak Lq(v), and there exists C > 0 such that [T f]q ≤ C||f||p for all f Lp(μ). Also, we shall say that T is weak type (p, ∞) iff T is strong type (p, ∞).

6.28 The Marcinkiewicz Interpolation Theorem. Suppose that (X, M, μ) and (Y, N, v) are measure spaces; p₀, p₁, q₀, q₁ are elements of [1, ∞] such that p₀ ≤ q₀, p₁ ≤ q₁ and q₀ ≠ q₁; and

If T is a sublinear map from Lp₀ (μ) + Lp₁ (μ) to the space of measurable functions on Y that is weak types (p₀, q₀) and (p₁, q₁), then T is strong type (p, q). More precisely, if [T f]qj ≤ Cj||f||pj for j = 0,1, then ||T f||q ≤ Bp||f||p where Bp depends only on pj, qj, Cj in addition to p; and for j = 0,1, Bp|p − pj| (resp. Bp) remains bounded as p → pj if pj < ∞ (resp. pj = ∞).

Proof. The case p₀ = p₁ is easy and is left to the reader (Exercise 42). Without loss of generality we may therefore assume that p₀ < p₁, and for the time being we also assume that q₀ < ∞ and q₁ < ∞ (whence also p₀ < p₁ < ∞). Given f Lp(μ) and A > 0, let gA and hA be as in Proposition 6.25. Then by Propositions 6.24 and 6.25,

(6.29) equation

Likewise,

(6.30) equation

Since T is sublinear, by Proposition 6.22d we have

(6.31) equation

This is true for all α > 0 and A > 0, so we may take A to depend on α. We now make a specific choice of A. Namely, it follows from the equations defining p and g that

(6.32) equation

we denote the common value of these quantities by σ, and we take A = ασ. Then by (6.29), (6.30), (6.31), and the weak type estimates on T,

(6.33) equation

where, denoting by χ₀ and χ₁ the characteristic functions of {(α, β) : β > ασ) and {(α, β): β < ασ},

Since q₀/p₀ ≥ 1 and q₁/p₁ ≥ 1, we may apply Minkowski’s inequality for integrals to obtain

(6.34) equation

Let τ = 1/σ. If q₁ > q₀ then q − q₀ and σ are positive and the inequality β > ασ is equivalent to α < βτ, so

where we have used (6.32) to simplify the exponent of β. On the other hand, if q₁ < q₀, then q − q₀ and σ are negative and the inequality β > ασ is equivalent to α > βτ, so as above,

A similar calculation shows that

Combining these results with (6.33) and (6.34), we see that

But since |T(cf)| = c|T f) for c > 0, this implies that ||T f||q ≤ Bp||f||p for all f Lp(μ), and we are done. (The verification of the asserted properties of Bp is left as an easy exercise.)

It remains to show how to modify this argument to deal with the exceptional cases q₀ = ∞ or q₁ = ∞. We distinguish three cases.

Case I: p₁ = qi = ∞ (so p₀ < q₀ < ∞). Instead of taking A = ασ in the decomposition of f, we take A = α/C₁. Then ||ThA||∞ ≤ C₁||hA||∞ ≤ α, so λThA (α) = 0, and we obtain (6.33) with ϕ₁ = 0 and ασ replaced by α/C₁ in the definition of ϕ₀. The same argument as above then gives

Case II: p₀ < p₁ < ∞, q₀ < q₁ = ∞. Again the idea is to choose A so that λThA (α) = 0, and the proper choice is A = {α/d)σ where d = C₁ [p₁ ||f||pp/p]1/p₁ and σ = p₁/(p₁ − p) (the limiting value of the σ defined by (6.32) as q₁ → ∞). Indeed, since p₁ > p, we have

As in Case I, then, we find that ϕ₁ = 0 in (6.33) and the integral involving ϕ₀ is majorized by a constant Bp when ||f||p = 1, which yields the desired result.

Case III: p₀ < p₁ < ∞, q₁ < q₀ = ∞. The argument is essentially the same as in Case II, except that we take A = (α/d)σ with d chosen so that λTgA (α) = 0.

The lengthy formulas in this proof may seem daunting, but the ideas are reasonably simple. To elucidate them, we recommend the exercise of writing out the proof for two special (but important) cases: (i) p₀ = q₀ = 1, p₁ = q₁ = 2, and (ii) p₀ = q₀ = 1. p₁ = q₁ = ∞.

Let us compare our two interpolation theorems. The Marcinkiewicz theorem requires some restrictions on pj and qj that are not present in the Riesz-Thorin theorem; these restrictions, however, are satisfied in all the interesting applications. Apart from this, the hypotheses of the Marcinkiewicz theorem are weaker: T is allowed to be sublinear rather than linear, and it needs only to satisfy weak-type estimates at the endpoints. The conclusion in both cases is that T is bounded from Lp(μ) to Lq(v), but the Riesz-Thorin theorem produces a much sharper estimate for the operator norm of T. Thus neither theorem includes the other.

We conclude with two applications of the Marcinkiewicz theorem. The first one concerns the Hardy-Littlewood maximal operator H discussed in §3.4,

H is obviously sublinear and satisfies ||H f||∞ ≤ ||f||∞ for all f L∞. Moreover, Theorem 3.17 says precisely that H is weak type (1,1). We conclude:

6.35 Corollary. There is a constant C > 0 such that if 1 < p < ∞ and f Lp(n), then

Our second application is a theorem on integral operators related to Theorem 6.18.

6.36 Theorem. Suppose (X, M, μ) and (Y, M, v) are σ-finite measure spaces, and 1 < q < ∞. Let K be a measurable function on X x Y such that, for some C > 0, we have [K(x, ·)]q ≤ C for a.e. x X and [K(·, y)]q ≤ C for a.e. y Y. If 1 ≤ p <∞ and f Lp(v), the integral

converges absolutely for a.e. x X, and the operator T thus defined is weak type (1, q) and strong type (p, r) for all p, r such that 1 < p < r < ∞ and p⁻¹ + q⁻¹ = r⁻¹ + 1. More precisely, there exist constants Bp independent of K such that

Proof. Let p′, q′ be the conjugate exponents to p, q; then

so p < q′ and q < p′. Suppose 0 ≠ f Lp (1 ≤ p < q′); by multiplying f and K by constants, we may assume that ||f||p = C = 1. Given a positive number A whose value will be fixed later, define

and let T₁, T₂ be the operators corresponding to K₁, K₂. Then by Propositions 6.24 and 6.25, since q > 1 we have

and likewise

Hence, by Theorem 6.18, the integral defining T₁f(x) converges for a.e. x and

(6.37) equation

Similarly, since q < p′,

Therefore, by Hölder’s inequality, the integral defining T₂f(x) converges for every x, and

(6.38) equation

We have thus established that T f = T₁f + T₂f is well defined a.e.

Next, given α > 0, we wish to estimate λT f (α). But by Proposition 6.22d,

we will have ||T₂f||∞ ≤ , so that λT₂f (α) = 0. With this choice of A, then, by (6.37) and Chebyshev’s inequality we obtain

because ||f||p = 1 and

A simple homogeneity argument now yields the estimate ≤T f(α) ≤ Cp(||f||p/α)r with no restriction on ||f||p, so we have shown that T is weak type (p, r), and in particular (for p = 1) weak type (1, q).

Finally, given p (1, q′), choose (p, q′) and define by . Then T is weak types (1, q) and , so it follows from the Marcinkiewicz theorem that T is strong type (p, r).

Exercises

41. Suppose 1 < p ≤ ∞ and p⁻¹ + q⁻¹ = 1. If T is a bounded operator on Lp such that ∫(T f)g = ∫ f(Tg) for all f, g Lp Lq, then T extends uniquely to a bounded operator on Lr for all r in [p, q] (if p < q) or [q, p] (if q < p).

42. Prove the Marcinkiewicz theorem in the case p₀ = p₁. (Setting p = p₀ = p₁. we have λT f(α) ≤ (C₀||f||p/α)q₀ and λT f(α) ≤ (C₁||f||p/α)q₁. Use whichever estimate is better, depending on a, to majorize q ∫₀∞ αq−1 λT f(α) dα.)

43. Let H be the Hardy-Littlewood maximal operator on . Compute Hχ(0,1) explicitly. Show that it is in Lp for all p > 1 and in weak L¹ but not in L¹, and that its Lp norm tends to ∞ like (p − 1)⁻¹ as p → 1, although ||χ(0,1)||p = 1 for all p.

44. Let Iα be the fractional integration operator of Exercise 61 in §2.6. If 0 < α < 1, 1 < p < α⁻¹, and r⁻¹ = p⁻¹ − α, then Iα is weak type (1, (1 − α)⁻¹) and strong type (p, r) with respect to Lebesgue measure on (0,∞).

45. If 0 < α < n, define an operator Tα on functions on n by

Then Tα is weak type (1, (n-α)⁻¹) and strong type (p, r) with respect to Lebesgue measure on n, where 1 < p < nα⁻¹ and r⁻¹ = p⁻¹ − αn⁻¹. (The case n = 3, α = 1 is of particular interest in physics: If f represents the density of a mass or charge distribution, − (4π)⁻¹ T₁f represents the induced gravitational or electrostatic potential.)

6.6 NOTES AND REFERENCES

The importance of the space L² ([a, b]) was recognized soon after the invention of the Lebesgue integral because of its connection with Fourier series and other orthogonal expansions; and one of the early triumphs of the Lebesgue theory was the discovery in 1907 by Fischer [44] and F. Riesz [114] that L²([a, b]) is isomorphic to l², or what amounts to the same thing, that L²([a, b]) is complete. The spaces Lp([a, b]) for 1 < p < ∞ were first investigated by F. Riesz [117], who proved all of the major results in §§6.1–2 for them as well as the weak sequential compactness of the closed unit ball in Lp. The fact that (L¹)* = L∞ was first proved by Steinhaus [143].

In some respects it is unfortunate that Lp spaces were not named L1/p spaces, for — as one sees in the conjugacy relation p⁻¹ + q⁻¹ = 1 and in the results of §6.5 — relationships among different Lp spaces usually involve linear equations in p⁻¹.

A discussion of some of the deeper aspects of Lp spaces and their applications in other areas of analysis can be found in Lieb and Loss [93].

§6.1: Hölder’s inequality, in the case p = 2, is commonly associated with the names of Cauchy (who proved it for finite sums) and Buniakovsky and Schwarz (who proved it, independently, for integrals). For general p it was discovered independently by Hölder and Rogers. Minkowski’s original inequality was for finite sums. (See Hardy, Littlewood, and Pólya [65] for references.) A neat proof of Hölder’s inequality using complex function theory can be found in Rubel [122].

The relations among the spaces Lp + Lq, defined in Exercise 4, are studied in Alvarez [5]. See Romero [120] for a discussion of Exercise 5, including some other conditions for the inclusion Lp ⊂ L⁹ to hold, and Miamee [100] for a discussion of the more general relation Lp(μ) ⊂ Lq(v).

§6.2: A quite different approach to the Lp duality theory for 1 < p < ∞ can be found in Hewitt and Stromberg [76, §15]. J. Schwartz [130] has found a characterization of (L¹)* that is valid on arbitrary measure spaces.

The proof of Theorem 6.15 breaks down for p = ∞ because the set function v(E) = ϕ(χE) need not be countably additive. It is, however, a bounded, finitely additive complex measure on (X, M) that is absolutely continuous with respect to μ in the sense that v(E) = 0 whenever μ(E) = 0. Conversely, given a bounded, finitely additive complex measure v on (X, M), one can define the integral of a bounded measurable function with respect to v. (One defines ∫ f dv in the obvious way when f is simple and then shows that | ∫ f dv| ≤ C||f||u, so that the integral extends to all uniform limits of simple functions.) In this way one obtains a representation of (L∞)* as a space of finitely additive complex measures. See Hewitt and Stromberg [76, §20], and for a more general treatment of finitely additive integrals, Dunford and Schwartz [35, Chapter 3]. (The example of a ϕ (L∞)*\ L¹ that we presented at the end of §6.2 shows how horrible finitely additive measures can be: If v(E) = ϕ(χE), then v m, but v behaves like the point mass at zero when integrated against any continuous function.)

§6.3: Theorem 6.18 generalizes results of Schur [129] (for the case p = 2) and W. H. Young [164] (for the case K(x, y) = k(x − y); see §8.2). Theorem 6.20 is also essentially due to Schur [129].

The reader whose appetite for inequalities is not satisfied by this section can find a feast in Hardy, Littlewood, and Pólya [65].

§6.4: The weak Lp spaces first appeared implicitly in weak-type estimates, instances of which go back to the 1920s; see also the notes for §6.5 below. Decreasing rearrangements (Exercise 40) were introduced by Hardy and Littlewood [64], who give an entertaining motivation of their principal theorem on rearrangements in terms of cricket averages.

§6.5: The Riesz-Thorin theorem was first proved by M. Riesz (F. Riesz’s younger brother) [118] under the assumption that pj ≤ qj for j = 0, 1; the proof in the general case and the idea of using the three lines lemma are due to Thorin [149]. E. M. Stein has proved a very powerful generalization of the Riesz-Thorin theorem. It deals with a family {Tz : 0 ≤ Re z ≤ 1} of operators that (roughly speaking) depend holomorphically on z and satisfy some mild growth conditions as |Im z| → ∞, and it asserts that if Tz is bounded from Lpj to Lqj for Re z = j (J = 0, 1), then Tz is bounded from LPt to Lqt for Re z = t (0 < t < 1), where pt, qt are defined as in the Riesz-Thorin theorem. The precise statement and proof can be found in Bennett and Sharpley [15, §4.3], Stein and Weiss [142, §V.4], or Zygmund [167, §XII.1]. For a further extension of these ideas, see Coifman et al. [28].

The Marcinkiewicz interpolation theorem was announced by Marcinkiewicz [97] for the case pj = qj (j = 0, 1); after his untimely death in World War II, the work was completed by Zygmund [166], The theorem can be proved under still weaker hypotheses on T; an extra twist to the argument we have given yields the same result under the sole assumption that |T(f + g)| ≤ C(|T f| + |T g|) for some constant C. See Zygmund [166], [167, §XII.4]. The spaces Lp and weak Lp form part of a two-parameter family {L(p, q) : 1 ≤ p, q ≤ ∞} of function spaces, the so-called Lorentz spaces, such that Lp = L(p, p) and weak Lp = L(p, ∞), and the Marcinkiewicz theorem can be extended to a result about interpolation of operators on the L(p, q) spaces. See Bennett and Sharpley [15, §4.4], or Stein and Weiss [142, §5.3]

There are many other examples of “continuous families” of Banach spaces for which interpolation theorems can be proved — for example, the spaces λα discussed in Exercise 11 in §5.1 and the Sobolev spaces discussed in §9.3. There are also two general techniques for constructing “intermediate spaces” between pairs of Banach spaces, known as the “complex method” and the “real method,” which may be regarded as abstract forms of the Riesz-Thorin and Marcinkiewicz theorems. An account of these theories and their applications can be found in Bergh and Löfström [16]; see also Bennett and Sharpley [15] for the real method and its applications.

Corollary 6.35 is due to Hardy and Littlewood [64], Theorem 6.36 appears first in Folland and Stein [51], but the essential idea of the proof was discovered by Stein several years earlier (see, e.g., Stein [140, §5.1]), and the special case discussed in Exercise 44 goes back to Hardy and Littlewood [63].

Chapter 7

Radon Measures

The subject of this chapter is measure and integration theory on locally compact Hausdorff (LCH) spaces. We have seen in §2.6 that Lebesgue measure on n interacts nicely with the topology on n — measurable sets can be approximated by open or compact sets, and integrable functions can be approximated by continuous functions — and it is of interest to study measures having similar properties on more general spaces. Moreover, it turns out that certain linear functionals on spaces of continuous functions are given by integration against such measures. This fact constitutes an important link between measure theory and functional analysis, and it also provides a powerful tool for constructing measures.

Throughout this chapter, X will denote an LCH space. We continue to employ the terminology developed in Chapter 1 in the context of metric spaces: BX will denote the Borel σ-algebra on X, that is, the σ-algebra generated by the open sets; measures on BX will be called Borel measures; countable unions (intersections) of closed (open) sets will be called Fσ(Gδ) sets, and so forth.

7.1 POSITIVE LINEAR FUNCTIONALS ON CC(X)

We recall that Cc(X) is the space of continuous functions on X with compact support. A linear functional I on Cc(X) will be called positive if I(f) ≥ 0 whenever f ≥ 0. In this definition there is no mention of continuity, but it is worth noting that positivity itself implies a rather strong continuity property.

7.1 Proposition. If I is a positive linear functional on Cc(X), for each compact K ⊂ X there is a constant CK such that |I(f)| ≤ CK||f||u for all f Cc(X) such that supp(f) ⊂ K.

Proof. It suffices to consider real-valued f. Given a compact K, choose ϕ Cc(X, [0, 1]) such that ϕ = 1 on K (Urysohn’s lemma). Then if supp(f) ⊂ K, we have |f| ≤ ||f||uϕ, that is, ||f||uϕ − f ≥ 0 and ||f||uϕ + f ≥ 0. Thus ||f||u I(ϕ) − I(f) ≥ 0 and ||f||uI(ϕ) + I(f) ≥ 0, so that |I(f)| ≤ I(ϕ)||f||u.

If μ is a Borel measure on X such that μ(K) < ∞ for every compact K ⊂ X, then clearly Cc(X) ⊂ L¹(μ), so the map f is a positive linear functional on Cc(X). The principal result of this section is that every positive linear functional on Cc(X) arises in this fashion; moreover, one can impose some additional regularity conditions on μ, subject to which μ is unique. These conditions are as follows.

Let μ be a Borel measure on X and E a Borel subset of X. The measure μ is called outer regular on E if

and inner regular on E if

If μ is outer and inner regular on all Borel sets, μ is called regular. It turns out that regularity is a bit too much to ask for when X is not σ-compact, so we adopt the following definition. A Radon measure on X is a Borel measure that is finite on all compact sets, outer regular on all Borel sets, and inner regular on all open sets. We shall show in §7.2 that Radon measures are also inner regular on all of their σ-finite sets.

One further bit of notation: If U is open in X and f Cc(X), we shall write

to mean that 0 ≤ f ≤ 1 and supp(f) ⊂ U. (This is slightly stronger than the condition 0 ≤ f ≤ χU, which implies only that supp(f) .)

7.2 The Riesz Representation Theorem. If I is a positive linear functional on Cc(X), there is a unique Radon measure μ on X such that I(f) = ∫ f dμ for all f Cc(X). Moreover, μ satisfies

(7.3) equation

and

(7.4) equation

Proof. Let us begin by establishing uniqueness. If μ is a Radon measure such that I(f) = ∫ f dμ for all f Cc(X), and U ⊂ X is open, then clearly I(f) ≤ μ(U) whenever f U. On the other hand, if K ⊂ U is compact, by Urysohn’s lemma there is an f Cc(X) such that f U and f = 1 on K, whence μ(K) ≤ ∫ f dμ = I(f). Since μ is inner regular on U, it follows that (7.3) is satisfied. Thus μ is determined by I on open sets, and hence on all Borel sets because of outer regularity.

This argument proves the uniqueness of μ and also suggests how to go about proving existence. Namely, we begin by defining

for U open, and we then define μ*(E) for an arbitrary E ⊂ X by

Clearly μ(U) ≤ μ(V) if U ⊂ V, and hence μ* (U) = μ(U) if U is open.

The outline of the proof is now as follows. We shall establish that

i. μ* is an outer measure.

ii. Every open set is μ*-measurable.

At this point it follows from Carathéodory’s theorem that every Borel set is μ*-measurable and that μ = μ*|BX is a Borel measure. (The notation is consistent because μ*(U) = μ(U) for U open.) The measure μ is outer regular and satisfies (7.3) by definition. We next show that

iii. μ satisfies (7.4).

This clearly implies that n is finite on compact sets, and inner regularity on open sets also follows easily. Indeed, if U is open and α < μ(U), choose f Cc(X) such that f U and I(f) > α, and let K = supp(f). If g Cc(X) and g ≥ χK, then g − f ≥ 0 and hence I(g) ≥ I(f) > α. But then μ(K) > α by (7.4), so μ is inner regular on U. Finally, we prove that

With this, the proof of the theorem will be complete.

Proof of (i): It suffices to show that if {Uj} is a sequence of open sets and then μ(U) ≤ ∑₁∞ μ(Uj). Indeed, from this it follows that for any E ⊂ X,

and the expression on the right defines an outer measure by Proposition 1.10. If , and f U, let K = supp(f). Since K is compact, we have for some finite n, so by Proposition 4.41 there exist with and ∑₁n gj = 1 on K. But then f = ∑₁n fgj and fgj Uj, so

Since this is true for any f U, we conclude that μ(U) ≤ ∑₁∞ μ(Uj) as desired.

Proof of (ii): We must show that if U is open and E is any subset of X such that μ*(E) < ∞, then μ*(E) ≥ μ*(E U) + μ*(E \ U). First suppose that E is open. Then E U is open, so given > 0 we can find f Cc(X) such that f E U and I(f) > μ(E U) − . Also, E\ (supp(f)) is open, so we can find g Cc(X) such that supp(f) and I(g) > μ(E \ supp(f)) − . But then f + g E, so

Letting → 0, we obtain the desired inequality. For the general case, if μ* (E) < ∞ we can find an open V ⊂ E such that μ(V) < μ*(E) + , and hence

Letting → 0, we are done.

Proof of (iii): If K is compact, f Cc(X), and f ≥ χK, let U = {x : f(x) > 1 − }. Then U is open, and if g U, we have (1 − )⁻¹ f − g ≥ 0 and so I(g) ≤ (1 − )⁻¹ I(f). Thus μ(K) ≤ μ(U) ≤ (1 − )⁻¹ I(f), and letting → 0 we see that μ(K) ≤ I(f). On the other hand, for any open U ⊂ K, by Urysohn’s lemma there exists f Cc(X) such that f ≥ χK and f U, whence I(f) ≤ μ(U). Since μ is outer regular on K, (7.4) follows.

Proof of (iv): If suffices to show that I(f) = ∫ f dμ if f Cc(X, [0, 1]), as Cc(X) is the linear span of the latter set. Given N , for 1 ≤ j ≤ N let Kj = {x : f(x) ≥ jN⁻¹} and let K₀ = supp(f). Also, define f₁, …, fN Cc(X) by fj(x) = 0 if x Kj−1 fj(x) = f(x) − (j − 1)N⁻¹ if x Kj−1\ Kj, and fj(x) = N⁻¹ if x Kj. In other words,

Then hence

Also, if U is an open set containing Kj−1 we have N fj U and so I(fj) ≤ N⁻¹μ(U). Hence, by (7.4) and outer regularity,

Moreover, so that

It follows that

Since μ(supp(f)) < ∞ and N is arbitrary, we conclude that I(f) = ∫ f dμ.

The proof of this theorem yields something stronger than the statement: We obtain not just a Borel measure μ but an extension of μ to the σ-algebra of μ*-measurable sets. However, it follows from outer regularity that for any E ⊂ X,

so μ* is the outer measure induced by μ in the sense of §1.4. According to Exercise 22 in §1.4, therefore, is the completion of μ if μ is σ-finite and is the saturation of the completion of μ in general.

On the other hand, some authors prefer to restrict attention to a smaller σ-algebra than , namely, the σ-algebra generated by Cc(X) (that is, the smallest σ-algebra with respect to which every f Cc(X) is measurable). The elements of are called Baire sets. For more about Baire sets, see Exercises 4–6.

Exercises

1. Let X be an LCH space, Y a closed subset of X (which is an LCH space in the relative topology), and μ a Radon measure on Y. Then I(f) = ∫(f|Y) dμ is a positive linear functional on Cc(X), and the induced Radon measure v on X is given by v(E) = μ(E Y).

2. Let μ be a Radon measure on X.

a. Let N be the union of all open U ⊂ X such that μ(U) = 0. Then N is open and μ(N) = 0. The complement of N is called the support of μ.

b. x supp(μ) iff ∫ f dμ > 0 for every f Cc(X, [0, 1]) such that f(x) > 0.

3. Let μ be the one-point compactification of a set with the discrete topology. If μ is a Radon measure on X, then supp(μ) (see Exercise 2) is countable.

4. Let X be an LCH space.

a. If f Cc(X, [0, ∞)), then f⁻¹([a, ∞)) is a compact Gδ set for all a > 0.

b. If K ∞ X is a compact Gδ set, there exists f Cc(X, [0, 1]) such that K = f⁻¹({1}).

c. The σ-algebra BX⁰ of Baire sets is the σ-algebra generated by the compact Gδ sets.

5. Let X be a second countable LCH space.

a. Every compact subset of X is a Gδ set.

b. BX = BX⁰.

6. Let X be an uncountable set with the discrete topology, or the one-point compactification of such a set. Then BX ≠ BX⁰.

7.2 REGULARITY AND APPROXIMATION THEOREMS

In this section we explore the properties of Radon measures in more detail.

7.5 Proposition. Every Radon measure is inner regular on all of its σ-finite sets.

Proof. Suppose that μ is Radon and E is σ-finite. If μ(E) < ∞, for any > 0 we can choose an open U ⊃ E such that μ(U) < μ(E) + and a compact F ⊃ U such that μ(F) > μ(U) − . Since μ(U\E) < , we can also choose an open V ⊃ U\E such that μ(V) < . Let K = F\V. Then K is compact, K ⊂ E, and

Thus μ is inner regular on E. On the other hand, if μ(E) = ∞, E is an increasing union of sets Ej with μ(Ej) < ∞ and μ(Ej) → ∞. Thus for any N there exists j such that μ(Ej) > N and hence, by the preceding argument, a compact K ⊂ Ej with μ(K) > N. Hence μ is inner regular on E.?

7.6 Corollary. Every σ-finite Radon measure is regular. If X is σ-compact, every Radon measure on X is regular.

For an example of a nonregular Radon measure, see Exercise 12.

7.7 Proposition. Suppose that μ, is a σ-finite Radon measure on X and E is a Borel set in X.

a. For every > 0 there exist an open U and a closed F with F ⊂ E ⊂ U and μ(U\F) < .

b. There exist an Fσ set A and a Gδ set B such that A ⊂ E ⊂ B and μ(B\A) = 0.

Proof. Write E = ∪₁∞ Ej where the Ej’s are disjoint and have finite measure. For each j, choose an open Uj ⊃ Ej with μ(Uj) < μ(Ej) + 2−j−1 and let U = ∪₁∞ Uj. Then U is open, U ⊃ E, and μ(U\E) ≤ ∑₁∞ μ(Uj\Ej) < /2-Applying the same reasoning to Ec, we obtain an open V ⊃ Ec with μ(V\Ec) < /2. Let F = Vc. Then F is closed, F ⊂ E, and

This proves (a), and (b) follows easily; details are left to the reader.

7.8 Theorem. Let X be an LCH space in which every open set is σ-compact (which is the case, for example, if X is second countable). Then every Borel measure on X that is finite on compact sets is regular and hence Radon.

Proof. If μ is a Borel measure that is finite on compact sets, then Cc (X) ⊂ L¹(μ), so the map I(f) = ∫ f dμ is a positive linear functional on Cc(X). Let v be the associated Radon measure according to Theorem 7.2. If U ⊂ X is open, let U = ∪₁∞ Kj where each Kj is compact. Choose f₁ Cc(X) so that f U and f = 1 on K₁. Proceeding inductively, for n > 1 choose fn Cc(X) so that fn U and fn = 1 on ∪₁n Kj and on ∪₁n−1 supp(fj). Then fn increases pointwise to χU as n → ∞, so

by the monotone convergence theorem. Next, if E is any Borel set and > 0, by Proposition 7.7 there exist an open V ⊃ E and a closed F ⊂ E with v(V \ F) < . But V \ F is open, so μ(V \ F) = v(V \ F) < . In particular, μ(V) ≤ μ(E) + , so μ is outer regular. Also, μ(F) ≥ μ(E) − and F is σ-compact (since X is), so there exist compact Kj ⊂ F with μ(Kj) → μ(F), whence μ is inner regular. Thus μ is regular (and equal to v, by the uniqueness part of Theorem 7.2.)

Examples of non-Radon measures are considered in Exercises 13–15. In particular, Exercise 15 exhibits an example of a finite, non-Radon Borel measure on a compact Hausdorff space.

We now turn to some approximation theorems for measurable functions.

7.9 Proposition. If p is a Radon measure on X, Cc(X) is dense in Lp(μ) for 1 ≤ p < ∞.

Proof. Since the Lp simple functions are dense in Lp (Proposition 6.7), it suffices to show that for any Borel set E with μ(E) < ∞, χE can be approximated in the Lp norm by elements of Cc(X). Given > 0, by Proposition 7.5 we can choose a compact K ⊂ E and an open U ⊃ E such that μ(U \ K) < , and by Urysohn’s lemma we can choose f Cc(X) such that χK ≤ f ≤ χU. Then ||χE − f||p ≤ μ(U \ K)1/p < 1/p, so we are done.

7.10 Lusin’s Theorem. Suppose that p is a Radon measure on X and f : X → is a measurable function that vanishes outside a set of finite measure. Then for any > 0 there exists ϕ Cc(X) such that ϕ = f except on a set of measure < . If f is bounded, ϕ can be taken to satisfy ||ϕ||u ≤ ||f||u.

Proof. Let E = {x : f(x) ≠ 0}, and suppose to begin with that f is bounded. Then f L¹(μ), so by Proposition 7.9 there is a sequence {gn} in Cc(X) that converges to f in L¹, and hence by Corollary 2.32 a subsequence (still denoted by {gn}) that converges to f a.e. By Egoroff’s theorem there is a set A ⊂ E such that μ(E \ A) < /3 and gn → f uniformly on A, and there exist a compact B ⊂ A and an open U ⊃ E such that μ(A \ B) < /3 and μ(U \ E) < /3. Since gn → f uniformly on B, f|B is continuous, so by Theorem 4.34 there exists h Cc(X) such that h = f on B and supp(h) ⊂ U. But then {x : f(x) ≠ h(x)} is contained in U \ B, which has measure < .

To complete the proof for f bounded, define β : → by β(z) = z if |z| ≤ ||f||u and β(z) = ||f||u sgn z if |z| > ||f||u, and set ϕ = β o h. Then ϕ Cc(X) since β is continuous and β(0) = 0. Moreover, ||ϕ||u ≤ ||f||u, and ϕ = f on the set where h = f, so we are done.

If f is unbounded, let An = {x : 0 < |f(x)| ≤ n}. Then An increases to E as n → ∞, so μ(E \ An) < /2 for sufficiently large n. By the preceding argument there exists ϕ Cc(X) such that ϕ = fχAn except on a set of measure < /2, and hence ϕ = f except on a set of measure < .

Our final group of results in this section concerns semicontinuous functions. If X is a topological space, a function f : is called lower semicontinuous (LSC) if {x : f(x) > a} is open for all a , and f : X → [−∞,∞) is called upper semicontinuous (USC) if {x : f(x) < a} is open for all a .

7.11 Proposition. Let X be a topological space.

a. If U is open in X, then χU is LSC.

b. If f is LSC and c [0, ∞), then cf is LSC.

c. If g is a family of LSC functions and f(x) = sup{g(x) : g g}, then f is LSC.

d. If f₁ and f₂ are LSC, so is f₁ + f₂.

e. If X is an LCH space and f is LSC and nonnegative, then

Proof. (a) and (b) are obvious, and (c) follows from the observation that

As for (d), if f₁(x₀) + f₂(x₀) > a, choose > 0 so that f₁(x₀) > a − f₂(x₀) + Then

which is a neighborhood of x₀. Thus f₁ + f₂ is LSC. Finally, if X is LCH, f(x) > 0, and 0 < a < f(x), then U = {y : f(y) > a} is an open set containing x, so by Urysohn’s lemma there exists g Cc(X) such that g(x) = a and 0 ≤ g ≤ aχu ≤ f. This establishes (e) when f(x) > 0, and (e) is trivial when f(x) = 0.

There is, of course, a corresponding set of results for USC functions, whose formulation is left to the reader. The following result is a monotone convergence theorem for nets of LSC functions.

7.12 Proposition. Let g be a family of nonnegative LSC functions on an LCH space X that is directed by ≤ (that is, for every g₁, g₂ g there exists g g such that g₁ ≤ g and g₂ ≤ g). Let f = sup{g : g g}. lf μ is any Radon measure on X, then ∫ f dμ = g}.

Proof. By Proposition 7.11c, f is LSC and hence Borel measurable, and clearly . To prove the reverse inequality, consider the sequence ϕn of simple functions increasing to f that was constructed in Theorem 2.10:

By the monotone convergence theorem, given a < ∫ f dμ, we can fix n large enough so that 2−n ∑j μ(Unj) = ∫ ϕn dμ > a. Since Unj is open, there exist compact Kj ⊂ Unj (1 ≤ j ≤ 22n) such that 2−n ∑j μ(Kj) > a. Let = 2−n ∑j χKj. For each x Uj Kj we have f(x) > ϕn(x) ≥ (x), so we can pick gx g such that gx(x) > (x). But −χKj is LSC, so gx- is LSC by Proposition 7.11d, and hence the set Vx = {y : (y) < gx(y)} is open. Thus {Vx : x ∪j Kj} is an open cover of ∪j Kj, so there is a finite subcover Vx1, …, Vxm. Pick g g such that gxk ≤ g for k = 1, …, m; then ≤ g, so ∫ g dμ > a. Since a was any number less than ∫ f dμ, we are done.

7.13 Corollary. If μ is Radon and f is nonnegative and LSC, then

Proof. Combine Propositions 7.11e and 7.12.

7.14 Proposition. If μ is a Radon measure and f is a nonnegative Borel measurable function, then

If{x : f(x) > 0} is σ-finite, then

Proof. Let {ϕn} be a sequence of nonnegative simple functions that increase pointwise to f. Then f = ϕ₁ + ∑₂∞(ϕn − ϕn−1), and each term in this series is a nonnegative simple function, so we can write f = ∑₁∞ ajχEj where aj > 0. Given > 0, for each j choose an open Uj ⊃ Ej such that μ(Uj) ≤ μ(Ej)+/(2jaj). Then g = ∑₁∞ ajχUj is LSC by Proposition 7.11, g ≤ f’, and . This establishes the first assertion. For the second, if a < ∫ f dμ, let N be large enough so that ∑₁N ajμ(Ej) > a. Since the Ej’s are σ-finite, by Proposition 7.5 there are compact sets Kj ⊂ Ej such that ∑₁N ajμ(Kj) > a. Thus if g = ∑₁n ajχKj, then g is USC, g ≤ f, and ∫ g dμ > a.

Exercises

7. If μ is a σ-finite Radon measure on X and A Bx me Borel measure μA defined by μA(E) = μ(E A) is a Radon measure. (See also Exercise 13.)

8. Suppose that μ is a Radon measure on X. If ϕ L¹(μ) and ϕ ≥ 0, then v(E) = ∫E ϕ dμ is a Radon measure. (Use Corollary 3.6.)

9. Suppose that μ is a Radon measure on X and ϕ C(X, (0, ∫)). Let v(E) = ∫E ϕ dμ, and let v’ be the Radon measure associated to the functional f f ϕ dμ on Cc(X).

a. If U is open, v(U) = v′(U). (Apply Corollary 7.13 to ϕχU.)

b. v is outer regular on all Borel sets. (Hint: The open sets Vk = {x : 2k < ϕ(x) < 2k+2}, k , cover X.)

c. v = v′, and hence v is a Radon measure. (See also Exercise 13.)

10. If μ is a Radon measure and f L¹(μ) is real-valued, for every > 0 there exist an LSC function g and a USC function h such that h ≤ f ≤ g and ∫(g − h)dμ < .

11. Suppose that μ is a Radon measure on X such that μ({x}) = 0 for all x X, and A Bx satisfies 0 < μ(A) < ∞. Then for any α such that 0 < α < μ(A) there is a Borel set B ⊂ A such that μ(B) = α.

12. Let X = x d, where d denotes with the discrete topology. If f is a function on X, let fy(x) = f(x, y); and if E ⊂ X, let Ey = {x : (x, y) E).

a. f Cc(X) iff fy Cc() for all y and fy = 0 for all but finitely many y.

b. Define a positive linear functional on Cc(X) by f(x, y) dx, and let μ be the associated Radon measure on X. Then μ(E) = ∞ for any E such that for uncountably many y.

c. Let E = {0} x d. Then μ(E) = ∞ but μ(K) = 0 for all compact K ⊂ E.

13. In the setting of Exercise 12, let A = ( \ {0}) x d and ϕ(x, y) = |x|. Then the measures μA(E) = μ(A E) and v (E) = vEϕ dμ are not Radon. (Thus, the hypotheses that μ be σ-finite in Exercise 7, that ϕ L¹ (μ) in Exercise 8, and that ϕ > 0 in Exercise 9, cannot be dropped.)

14. Let μ be a Radon measure on X, and let μ₀ be the semifinite part of μ₀ (see Exercise 15 in §1.3).

a. μ₀ is inner regular on all Borel sets.

b. μ₀ is outer regular on all Borel sets E such that μ(E) < ∞.

c. ∫ f dμ = ∫ f dμ₀ for all f Cc(X).

d. If μ is the measure of Exercise 12 and m is Lebesgue measure on , then μ₀(E) = ∑y m(Ey) for any Borel set E.

15. Let Ω be the set of countable ordinals, ω₁ the first uncountable ordinal, and Ω* = Ω∪{ω₁}. Let Ω* be endowed with the order topology (see Exercise 9 in §4.1).

a. Ω* is a compact Hausdorff space. (Hint: Ω* contains no infinite strictly decreasing sequences.)

b. Ω is an open set in Ω* that is not σ-compact.

c. A subset E of Ω is uncountable iff for each x Ω there exists y E such that x < y.

d. If {En} is a sequence of uncountable closed sets in Ω*, then ₁∞ En is uncountable. (If {xj} is an increasing sequence in Ω such that each En contains infinitely many xj’s, then exists and is in .)

e. If E BΩ*, then either E ∪{ω₁} or Ec U {ω₁} contains an uncountable closed set. (Hint: The set of all E satisfying the latter condition is a σ-algebra.)

f. Define μ on BΩ* by μ(E) = 1 if E ∪ {ω₁} contains an uncountable closed set, (μ(E) = 0 otherwise. Then μ is a measure, μ({ω₁}) = 0. but μ(U) = 1 for every open U containing ω₁.

g. If f C(Ω*), there exists x Ω such that f(y) = f(ω₁) for y ≥ x. (If En = {x : |f(x) − f(ω₁)| < n⁻¹}, then Enc is countable.)

h. With μ as in (f), the Radon measure on Ω* associated to the functional f f dμ is the point mass at ω₁.

7.3 THE DUAL OF C₀(X)

We recall that for any LCH space X, C₀(X) is the uniform closure of Cc(X) (Proposition 4.35), and hence if μ is a Radon measure on X, the functional I(f) = ∫ f dμ extends continuously to C₀(X) iff it is bounded with respect to the uniform norm. In view of the equality

(a special case of (7.3)) together with the fact that |∫ f dμ| ≤ ∫ |f| dμ, this happens precisely when μ(X) < ∞, in which case μ(X) is the operator norm of I.

We have therefore identified the positive bounded linear functionals on C₀(X): they are given by integration against finite Radon measures. Our object in this section is to extend this result to give a complete description of Co (X)*. The key fact is that real linear functionals on C₀(X, ) have a “Jordan decomposition.”

7.15 Lemma. If I C₀(X, )*, there exist positive functionals I± C₀(X, )* such that I = I⁺ − I⁻.

Proof. If f C₀(X, [0, ∞)), we define

Since |I(g)| ≤ ||I|| ||g||u ≤ ||I|| ||f||u for 0 ≤ g ≤ f, and I(0) = 0, we have 0 ≤ I⁺(f) ≤ ||I|| ||f||u. We claim that I⁺ is the restriction to C₀(X,[0,∞)) of a linear functional; the proof is much the same as the proof of the linearity of the integral in §2.3.

Obviously I⁺(cf) = cI⁺(f) if c ≥ 0. Also, whenever 0 ≤ g₁ ≤ f₁ and 0 ≤ g₂ ≤ f₂ we have 0 ≤ g₁ + g₂ ≤ + f₁ + f₂, so that I⁺ (f₁ + f₂) ≥ I(g₁) + I(g₂), and it follows that I⁺(f₁ + f₂) ≥ I⁺(f₁) + I⁺ (f₂). On the other hand, if 0 ≤ g ≤ f₁ + f₂, let g₁ = min(g, f₁) and g₂ = g − g₁. Then 0 ≤ g₁ ≤ f₁ and therefore . In short .

Now, if f C₀(X, ), then its positive and negative parts f⁺ and f⁻ are in C₀(X, [0, ∞)), and we define I⁺(f) = I⁺(f⁺) − I⁺(f⁻). If also f = g − h where g, h ≥ 0, then g + f⁻ = h + f⁺, whence I⁺(g) + I⁺ (f⁻) = I⁺(h) + I⁺(f⁺). Thus I⁺(f) = I⁺(g) − I⁺(h), and it follows easily as in the proof of Proposition 2.21 that I⁺ is linear on C₀(X, ). Moreover,

so that ||I⁺|| ≤ ||I||.

Finally, let I⁻ = I⁺ − I. Then I⁻ C₀(X, )*, and it is immediate from the definition of I⁺ that I⁺ and I⁻ are positive.

Any I C₀(X)* is uniquely determined by its restriction J to C₀(X, ), and we have J = J₁ + iJ₂ where J₁, J₂ are real linear functional. We therefore conclude from Lemma 7.15 and the discussion preceding it that for any I ε C₀ (X)* there are finite Radon measures μ₁, …, μ₄ such that I(f) = ∫ f dμ where μ = μ₁ − μ₂ + i(₃ − ₄).

At this point we need some more definitions. A signed Radon measure is a signed Borel measure whose positive and negative variations are Radon, and a complex Radon measure is a complex Borel measure whose real and imaginary parts are signed Radon measures. (It is worth noting that on a second countable LCH space, every complex Borel measure is Radon. This follows from Theorem 7.8 since complex measures are bounded.) We denote that space of complex Radon measures on X by M(X), and for μ M(X) we define

where, of course, |μ| is the total variation of μ.

7.16 Proposition. If μ is a complex Borel measure, then μ is Radon iff|μ| is Radon. Moreover, M(X) is a vector space and μ ||μ|| is a norm on it.

Proof. We observe that a finite positive Borel measure v is Radon iff for every Borel set E and every > 0 there exist a compact K and an open U such that and v(U \ K) < , by Propositions 7.5 and 7.7. The first assertion follows easily from this. Indeed, if μ = μ₁ − μ₂ + i(μ₃ − μ₄) and |μ|(U \ K) < , then μj(U \ K) < for all j; conversely, if μj(Uj \ Kj) < /4 for all j, then μ(U | K) < where K = ∪₁⁴ Kj and U = ₁⁴ Uj. The same argument shows that M(X) is closed under addition and scalar multiplication. Finally, that || · || is a norm on M(X) follows from Proposition 3.14.

7.17 The Riesz Representation Theorem. Let X be an LCH space, and for μ M(X) and f C₀(X) let Iμ(f) = f f dμ. Then the map p Iμ is an isometric isomorphism from M(X) to C₀(X)*.

Proof. We have already shown that every I C₀(X)* is of the form Iμ. On the other hand, if μ M(X), by Proposition 3.13c we have

so Iμ C₀(X)* and ||Iμ|| ≤ ||μ||. Moreover, if h = dμ, then |h| = 1 by Proposition 3.13b, so by Lusin’s theorem, for any > 0 there exists f Cc(X) such that ||f||u = 1 and except on a set E with |μ|(E) < /2. Then

It follows that ||μ|| ≤ ||Iμ||, so the proof is complete.

7.18 Corollary. If X is a compact Hausdorff space, then C(X)* is isometrically isomorphic to M(X).

Let μ be a fixed positive Radon measure on X. lf f L¹ (μ), the complex measure dvf = f dμ is easily seen to be Radon (Exercise 8), and ||vf|| = ∫ |f| dμ. = ||f||₁. Thus f vf is an isometric embedding of L¹(μ) into M(X) whose range consists precisely of those v M(X) such that v μ. (The last statement follows from the Radon-Nikodym theorem, which applies even if μ is not σ-finite; see §7.5.) The most important example of this situation is μ = m = Lebesgue measure on n, and we shall identify L¹(m) with a subspace of M(n).

The weak* topology on M(X) = C₀(X)*, in which μα → μ iff ∫ f dμα → ∫ f dμ for all f C₀(X), is of considerable importance in applications; we shall call it the vague topology on M(X), (The term “vague” is common in probability theory and has the advantage of forming an adverb more gracefully than “weak*.”) The vague topology is sometimes called the weak topology, but this terminology conflicts with ours, since C₀(X) is rarely reflexive (see Exercise 20). Weak convergence arguments for Lp(μ) generally fail for p = 1 because L¹(μ) is not the dual of L∞(μ), but good substitute results can often be obtained by regarding L¹(μ) as a subspace of M(X) as in the preceding paragraph and using the vague topology there.

We conclude by presenting a useful criterion for vague convergence in M().

7.19 Proposition. Suppose μ, μ₁, μ₂, … M(), and let Fn(x) = μn((−∞, x]) and F(x) − μ((−∞o, x]).

a. If supn ||μn|| < ∞ and Fn(x) → F(x) for every x at which F is continuous, then μn → μ vaguely.

b. If μn ≥ 0, lim:x→-∞[supn Fn(x)] = 0, and μn → μ vaguely, then Fn(x) → F(x) at every x at which F is continuous.

Proof. (a) Since F is continuous except at countably many points (Theorems 3.27 and 3.29), Fn → F a.e. with respect to Lebesgue measure. Also, ||Fn||u ≤ ||μn||, so the Fn’s are uniformly bounded. If f is continuously differentiate and has compact support, then, integration by parts (Theorem 3.36) and the dominated convergence theorem yield

But by Theorem 4.52, the set of all such f’s is dense in C₀(), so ∞ f dμ → ∞ f dμ for all f C₀() by Proposition 5.17. Thus μn → μ vaguely.

(b) Given a and δ, > 0, choose N large enough so that supn Fn(−N) < δ, and let f Cc() be the function that is 1 on [−N, a], 0 on (−∞, − N − ] and [a + , ∞), and linear in between. If n is large enough so that ∫ f dμn < ∫ f dμ + δ, we have

Since δ is arbitrary, it follows that

Similarly, by considering the function that is 1 on [−N + , a − ], 0 on (−∞, N] and [a, ∞), and linear in between, we see that

Since is arbitrary, if F is continuous at a we have Fn(a) → F(a) as desired.

Exercises

16. Suppose that I C₀(X, )* and I⁺, I⁻ are the functionals constructed in the proof of Lemma 7.15. If μ is the signed Radon measure associated to I, then the positive and negative variations of μ are the Radon measures associated to I⁺ and I⁻.

17. If μ is a positive Radon measure on X with μ(X) = ∞, there exists f C₀(X) such that ∫ f dμ = ∫. Consequently, every positive linear functional on C₀(X) is bounded.

18. If μ is a σ-finite Radon measure on X and v M(X), let v = v₁ + v₂ be the Lebesgue decomposition of v with respect to μ. Then v₁ and v₂ are Radon. (Use Exercise 8.)

19. Let X be a completely regular space and A a completely regular subalgebra of BC(X) (see Exercise 73 in §4.8). Find a description of A* as a space of measures.

20. Some examples of nonreflexivity of C₀ (X):

a. If μ, M(X), let Φ(μ) = . This sum is well defined, and Φ M(X)*. If there exists a nonzero μ M(X) such that μ({x}) = 0 for all x X, then Φ is not in the image of C₀(X) in M(X)* C₀(X)**.

b. At the other extreme, let X = with the discrete topology; then (Note: C₀() is usually denoted by c₀.)

21. Let {fα}αA be a subset of C₀(X) and {cα}αA a family of complex numbers. If for each finite set B ⊂ A there exists μB M(X) such that ||μB|| ≤ 1 and ∫ fα dμB = cα for α B, then there exists μ M(X) such that ||μ|| ≤ 1 and ∫ fαdμ = cα for all α A.

22. A sequence {fn} in C₀(X) converges weakly to f C₀(X) iff sup||fn||u < ∞ and fn → f pointwise.

23. The hypotheses μn ≥ 0 and limx→-∞ [supn Fn(x)] = 0 in Proposition 7.19b are necessary. (Take μn to be the difference of the point masses at n⁻¹ and −n⁻¹, or the point mass at −n.)

24. Find examples of sequences {μn} in M() such that

a. μn → 0 vaguely, but ||μn|| 0.

b. μn → 0 vaguely, but ∫ f dμn → f dμ for some bounded measurable f with compact support.

c. μn ≥ 0 and μn → 0 vaguely, but there exists x such that Fn(x) ∫ F(x) (notation as in Proposition 7.19).

25. Let μ be a Radon measure on X such that every nonempty open set has positive measure (e.g., Lebesgue measure). For each x X there is a net {fα} in L¹(μ) that converges vaguely in M(X) to the point mass at x. If X is first countable, the net can be taken to be a sequence. (Consider functions of the form μ(U)⁻¹χU.)

26. If {μn} ⊂ M(X), μn → μ vaguely, and ||μn|| → ||μ||, then ∫ f dμn → ∫ f dμ for every f BC(X). (If μ = 0 the result is trivial. Otherwise, there exists g Cc(X) with ||g||u ≤ 1 such that ∫ g dμ > ||μ|| − , and ∫ gfdμn → ∫ gfdμ for f BC(X).) Moreover, the hypothesis ||μn|| → ||μ|| cannot be omitted.

27. Let Ck ([0, 1]) be as in Exercise 9 in §5.1. If I Ck([0,1])*, there exist μ M([0,1]) and constants c₀, …, ck−1, all unique, such that

(The functional f f(j) (0) could be replaced by any set of k functionals that separate points in the space of polynomials of degree < k.)

7.4 PRODUCTS OF RADON MEASURES

In this section we study Radon measures on product spaces. X and Y will denote LCH spaces, and πX and πY will denote the projections of X x Y onto X and Y, respectively.

7.20 Theorem.

a. .

b. If X and Y are second countable, then .

c. If X and Y are second countable and μ, and v are Radon measures on X and Y, then μ, x u is a Radon measure on X x Y.

Proof. Parts (a) and (b) are direct generalizations of Proposition 1.5, and the proof is essentially the same. The main tool is Proposition 1.4. It implies, first, that is generated by the sets U x V where U is open in X and V is open in Y. Since these sets are open in X x Y, we have . If the topologies on X and Y have countable bases ε and F, then every open set in X, Y, or X x Y is a countable union of sets in ε, F, or {U x V : U ε, V F}. It follows that BX, BY, and BXxY are generated by these families and hence that As for (c), μ x v is a Borel measure by (b), so by Theorem 7.8 we need only show that (μ x v)(K) is finite for every compact . But this is easy: πX(K) and πY(K) are compact, and π₁(K) x π₂(K), so (μ x v)(K) = μ(πX(K))v(πY(K)) < ∞.

When X or Y is not second countable it can happen that ; see Exercises 28 and 29. In this case the product of Radon measures is certainly not a Radon measure. However, there is a natural way of manufacturing a Radon measure from it. To see this, we need a couple of facts about continuous functions. If g and h are functions on X and Y, we define h on X x Y by

7.21 Proposition. Let P be the vector space spanned by the functions g h with g Cc(X), h Cc(Y). Then P is dense in Cc(X Y) in the uniform norm. More precisely, given f Cc(X x Y), > 0, and precompact open sets U ⊂ X and V ⊂ Y containing πX(supp(f)) and πY(supp(f)), there exists F P such that ||F − f||u and supp(F) ⊂ U x V.

Proof. is a compact Hausdorff space. It follows easily from the Stone-Weierstrass theorem that the linear span of is dense in . In particular, there is an element G of this linear span such that . Also, by Urysohn’s lemma there exist ϕ Cc(U, [0, 1]) and Cc(V, [0, 1]) such that ϕ = 1 on πX(supp(f)) and = 1 on πY(supp(f)). Thus if we define on ∞ and F = 0 elsewhere, we have , supp(F) , and ||F − f||u < .

7.22 Proposition. Every f is -measurable. Moreover, if μ, and v are Radon measures on X and Y, then Cc(X x Y) ⊂ L¹(μ x v), and

Proof. If g Cc(X) and h Cc(Y), we have . Since πX and πY are measurable from to BX and BY (by definition of and g and h are continuous, and are -measurable. Since products, sums, and pointwise limits of measurable functions are measurable, the first assertion follows from Proposition 7.21. Also, every f Cc(X x Y) is bounded and supported in a set of finite (μ x v)-measure, hence is in L¹(μ x v). Fubini’s theorem holds for such f even if μ, and v are not σ-finite because one can replace μ and v by the finite measures μ|πX(supp(f)) and v|πY(supp(f)).

It is now clear how to obtain a Radon measure on X x Y from Radon measures μ and v on X and Y. Namely, by Proposition 7.22 the formula I(f) = ∫ f d(μ x v) defines a positive linear functional on Cc(X x Y), so it determines a Radon measure on X x Y by the Riesz representation theorem. We call this measure the Radon product of μ and v and denote it by . The obvious question is: Does agree with μ x v on ? In general, the answer is no. Indeed, a counterexample may be obtained by taking X = , Y = d ( with the discrete topology), μ = Lebesgue measure, and v = counting measure. It is not hard to see that in this case , but Exercises 12 and 14 show that is not semifinite and that μ x v is the semifinite part of . However, some results are still available, and in the σ-finite case everything works out beautifully. In what follows, we employ the notation of x-sections and y-sections introduced in §2.5.

7.23 Lemma.

a. If E BXxY, then Ex BY for all x X and Ey BX for all y Y.

b. If f : X x Y → is BXxY-measurable, then fx is BY-measurable for all x X and fy is BX-measurable for all y Y.

Proof. The collection of all E ⊂ X x Y such that Ex BY and Ey BX for all x, y is easily seen to be a σ-algebra. It contains all open sets — if E is open, so are Ex and Ey, being inverse images of E under the maps y’ (x, y’) and x’ (x’, y) — and hence it contains BXxY. This proves (a), and (b) follows since (fx)⁻¹(A) = (f⁻¹(A))x and (fy)⁻¹(A) = (f⁻¹(A))y.

7.24 Lemma. If f Cc(X x Y) and μ, and v are Radon measures on X and Y, then the functions x fxdv and y fy dμ are continuous.

Proof. We write out the proof only for fx. It suffices to show that for any x₀ X and > 0 there is a neighborhood U of x₀ such that ||fx − fx₀||u < for x U, since then

However, for each y πY(supp(f)) there exist neighborhoods Uy, Vy of x₀ and y such that if (x, z) Uy x Vy, then |f(x₀, y) − f(x, z)| < . We may choose a finite subcover Vy₁, …, Vyn of πY (supp(f)) and then take U = ₁m Uyj; details are left to the reader.

7.25 Proposition. Let μ and v be Radon measures on X and Y. If U is open in X x Y, then the functions x v(Ux) and y μ(Uy) are Borel measurable on X and Y, and

Proof. Let . By Proposition 7.11 we have χU = sup{f : g F} and hence χUx = sup(fx : f F} and χUv = sup{fy : f F}. Thus by Proposition 7.12,

From Lemma 7.24 and Proposition 7.11 it follows that x v(Ux) and y μ(Uy) are LSC and hence Borel measurable. Another application of Proposition 7.12, together with Proposition 7.22, yields

and likewise .

7.26 Theorem. Suppose that μ, and v are σ-finite Radon measures on X and Y. If E BXxY, then the functions x v(Ex) and y μ(Ey) (which make sense by Lemma 7.23) are Borel measurable on X and Y, and

Moreover, the restriction of μ v to is μ x v.

Proof. For the moment, let us fix open sets and with μ(U) and v(V) finite, and let W = U x V. Let M be the collection of all sets E BXxY such that E W satisfies the conclusions of the theorem. We then have

i. contains all open sets, by Proposition 7.25.

ii. If E, F M and , then ; in particular, if , then . Indeed, we have

and likewise for v((E W)x) and μ((E W)y). Since the conclusions are true for E W and F W and all the sets involved have finite measure (this is why we introduced W), we can subtract to obtain the conclusions for (E \ F) W.

iii. M is closed under finite disjoint unions. (This is simply the additivity of the measures.)

iv. M is closed under countable increasing unions, and hence (by (ii)) under countable decreasing intersections. (This follows from the monotone convergence theorem.)

Now, let ε = {A| B : A, B open in X x Y}, and let A be the collection of finite disjoint unions of sets in ε. Since

ε is an elementary family, so by Proposition 1.7, A is an algebra. By Lemma 2.35, the monotone class generated by A coincides with the σ-algebra generated by A, which is clearly . But by (i)–(iv) (since A \ B = A \ (A B)), M contains this monotone class, so .

Next, since μ and v are σ-finite and outer regular, we have X = ∪₁∞ Un and Y = ∪₁∞ Vn where Un and Vn are open and have finite measure, and we may assume that the sequences {Un} and {Vn} are increasing. If E BXxY, the preceding argument shows that E (Un x Vn) satisfies the conclusions of the theorem for all n, and the monotone convergence theorem then implies that E satisfies the conclusions too.

Finally, if ∞ by Tonelli’s theorem we have

and the proof is complete.

7.27 The Fubini-Tonelli Theorem for Radon Products. Let μ and v be σ-finite Radon measures on X and Y, and let f be a Borel measurable function on X x Y. Then fx and fy are Borel measurable for every x and y. If f ≥ 0, then x fx dv and y fy dμ are Borel measurable on X and Y. If f L¹(μ v), then fx L¹(v) for a.e. x, fy L¹(μ) for a.e. y, and x fxdv and y fydμ are in L¹(μ) and L¹(v). In both cases, we have

Proof. The measurability of fx and fy was established in Lemma 7.23. The rest of the proof is identical to the proof of the ordinary Fubini-Tonelli theorem, except that Theorem 7.26 is used in place of Theorem 2.36.

The extension of the notion of Radon products to any finite number of factors is straightforward. More interestingly, the theory can be extended to infinitely many factors provided that the spaces in question are compact and the measures on them are normalized to have total mass 1.

To be precise, suppose that {Xα}αA is a family of compact Hausdorff spaces and, for each α, μα is a Radon measure on Xα such that μα(Xα) = 1. Let X = ΠαA Xα, a compact Hausdorff space by Tychonoff’s theorem. We would like to define a Radon measure μ on X such that if Eα is a Borel set in Xα for each α and Eα = Xα for all but finitely many α, then . (The product on the right is well defined since all but finitely many factors are equal to 1.) A bit of notation will be helpful: Given α₁, …, αn A, let π(α₁, …, αn) be the natural projection from X onto Π₁n Xαj,

Thus where Eα = Xα for α ≠ α₁, …, αn.

7.28 Theorem. Suppose that, for each α A, μα is a Radon measure on the compact Hausdorff space Xα such that μα(Xα) = 1. Then there is a unique Radon measure μ on X = ΠαA Xα such that for any α₁, …, αn A and any Borel set E in π₁n Xαj,

Proof. Let CF (X) be the set of all f C(X) that depend on only finitely many coordinates, that is, all f of the form f = g π(α₁, …, αn) for some α₁, …, αn A and g C(Π₁n Xαj). If f is such a function, we define

Adding on some extra coordinates to the set α₁, …, αn has no effect on this formula since μα(Xα) = 1 for all α. Thus I(f) is a well-defined positive linear functional on CF(X), and |I(f)| ≤ ||f||u with equality when f is constant.

Now, CF(X) is clearly an algebra that separates points, contains constant functions, and is closed under complex conjugation, so by the Stone-Weierstrass theorem it is dense in C(X). Hence the functional I extends uniquely to a positive linear functional of norm 1 on C(X), and the Riesz representation theorem therefore yields a unique Radon measure μ on X such that for all f CF(X).

Given α₁, …, αn A, let μ(α₁, …, αn) = . Then μ(α₁, …, αn) is a Borel measure on Π₁n Xαj that satisfies

when g is the characteristic function of a Borel set, and hence (by the usual linearity and approximation arguments) when g is any bounded Borel function. In particular, from the definition of μ, for all g C(Π₁n Xαj) we have

If we can show that μ(α₁, …, αn) is Radon, the uniqueness of the Riesz representation will imply that , which will complete the proof.

Let E be a Borel set in , and write π = π(α₁, …, αn) for short. Since μ, is regular, for any > 0 there is a compact K ⊂ π⁻¹(E) such that μ(K) > μ(π⁻¹ (E)) − . Then K’ = π(K) is a compact subset of E, and since K ⊂ π₋₁(K’), so μ(α₁, …, αn)(K) > μ(π⁻¹ (E)) − = μ(α₁, …, αn) (E) − . Thus μ(α₁, …, αn) is inner regular, and the same argument applied to Ec shows that it is outer regular. Thus μ(α₁, …, αn) is Radon, and we are done.

Exercises

28. If X is the set of ordinals less than or equal to the first uncountable ordinal ω₁, with the order topology, then . In fact, {(x, y) : x < y < ω₁} is open but not in . (Reexamine Exercise 47 in §2.5 in the light of Exercise 15 in §7.2.)

29. If X is a set of cardinality > c with the discrete topology, then . In fact, D = {(x, y) : x = y} is closed but not in . (Use Exercise 5 in §1.2 and Proposition 1.23. If , then D M where M is a sub-σ-algebra of generated by a countable family of rectangles, hence where N is a countably generated sub-σ-algebra of BX. Then {x} = Dx N for all x, but card(N) ≤ c.) The same reasoning applies if X is replaced by its one-point compactification.

30. Let μ and v be Radon measures on X and Y, not necessarily σ-finite. If f is a nonnegative LSC function on X x Y, then and are Borel measurable and .

31. Some results concerning Baire sets on product spaces:

a. . (Hint: Proposition 7.22 remains true if is replaced by ⁰.)

b. If X and Y are either compact or second countable, then .

c. If X is an uncountable set with the discrete topology, then .

7.5 NOTES AND REFERENCES

§7.1: The Riesz representation theorem is actually the work of many hands. F. Riesz [116] first proved it for the case ; he formulated the result in terms of Riemann-Stieltjes integrals and used no measure theory. It was extended to compact subsets of n by Radon [111], to compact metric spaces by Banach (see Saks [128]), and to compact Hausdorff spaces by Kakutani [80]. For the noncompact case, the first general results were obtained by Markov [98], who characterized positive linear functionals on BC(X) for a normal space X in terms of certain finitely additive set functions. A theorem essentially equivalent to Theorem 7.2 was apparently known to Bourbaki about 1940 (see Weil [158, §6]), but his treatment of integration was not published until 1952, by which time several others had obtained similar results. For more detailed references, see Dunford and Schwartz [35, §IV.16] and Hewitt and Ross [75, §11]. See also König [86] for a generalization of the Riesz representation theorem to spaces that are not locally compact.

Our use of the term “Radon measure,” which derives from Radon’s seminal paper [111], is common but not entirely standard. Some authors refer to such measures as “regular Borel measures”; others use the term “Radon measure” to mean a positive linear functional on Cc(X), and still others define Radon measures to be inner regular rather than outer regular on all Borel sets. It should also be noted that some older texts define the Borel σ-algebra to be the σ-algebra generated by the compact sets, which is in general smaller than our BX.

If μ is a Radon measure, let denote the complete, saturated extension of μ discussed at the end of §7.1. It is a significant fact that is always decomposable in the sense of Exercise 15 in §3.2; see Hewitt and Stromberg [76, Theorem 19.30]. Consequently, the extension of the Radon-Nikodym theorem in that exercise and the fact that (Exercise 25 in §6.2) are available. In this connection one should note that Lp() is essentially identical to Lp(μ) for p < ∞, by Propositions 2.12 and 2.20.

§7.2 See Cohn [27, Proposition 7.2.3] for a proof of Theorem 7.8 that does not use the Riesz representation theorem.

Propositions 7.12 and 7.14 suggest an alternative way of constructing the Radon measure μ associated to a positive linear functional I on Cc(X) in the spirit of the Daniell integral (see §2.8). Namely, one first extends I to nonnegative LSC functions g by setting

and then extends I to arbitrary nonnegative functions h by setting

It is then not difficult to verify that if E ⊂ X, I(χE) = μ*(E) where μ* is the outer measure in the proof of Theorem 7.2. For details, see Hewitt and Ross [75] or Hewitt and Stromberg [76].

Kupka and Prikry [88] contains a readable discussion of some of the more advanced topics in the theory of measures on LCH spaces.

§7.3: Theorem 7.17 is frequently stated only for the case where X is compact; however, the more general formulation follows easily from the compact case by considering the one-point compactification of X. An interesting proof of the Baire measure version of this result, quite different from ours, can be found in Hartig [66].

§7.4: The Fubini-Tonelli theorem for Radon products, as presented here, is essentially due to deLeeuw [31]; see also Cohn [27, §7.6]. Another variant of this theorem, which includes some further results for the non-σ-finite case, can be found in Hewitt and Ross [75, § 13].

Theorem 7.28 is essentially due to Nelson [103]. There is also a purely measure-theoretic version of this result: If is a family of measure spaces with μα(Xα) = 1 for all α, one can define a product measure Πα μα on (). See Saeki [127], Halmos [61, §38], or Hewitt and Stromberg [76, §22]. The hypotheses of Theorem 7.28 are more restrictive, but the conclusion is stronger; in particular, the domain of the measure μ in Theorem 7.28 is the Borel σ-algebra on Πα Xα, which is much larger than when the index set A is uncountable.

Chapter 8

Elements of Fourier Analysis

It is easy to say that Fourier analysis, or harmonic analysis, originated in the work of Euler, Fourier, and others on trigonometric series; it is much harder to describe succinctly what the subject comprises today, for it is a meeting ground for ideas from many parts of analysis and has applications in such diverse areas as partial differential equations and algebraic number theory. Two of the central ingredients of harmonic analysis, however, are convolution operators and the Fourier transform, which we study in this chapter.

8.1 PRELIMINARIES

We begin by making some notational conventions. Throughout this chapter we shall be working on n, and n will always refer to the dimension. In any measure-theoretic considerations we always have Lebesgue measure in mind unless we specify otherwise. Thus, if E is a measurable set in n, we shall denote Lp(E, m) by Lp(E). If U is open in n and k , we denote by Ck(U) the space of all functions on U whose partial derivatives of order ≤ k all exist and are continuous, and we set . Furthermore, for any E ⊂ n we denote by the space of all C∞ functions on n whose support is compact and contained in E. If E = n or U = n, we shall usually omit it in naming function spaces: thus, . If x, y n, we set

It will be convenient to have a compact notation for partial derivatives. We shall write

and for higher-order derivatives we use multi-index notation. A multi-index is an ordered n-tuple of nonnegative integers. If α = {α₁, …, αn) is a multi-index, we set

and if x = (x₁, …, xn) n,

(The notation is inconsistent with the notation , but the meaning will always be clear from the context.) Thus, for example, Taylor’s formula for functions f Ck reads

and the product rule for derivatives becomes

(Exercise 1).

We shall often avail ourselves of the sloppy but handy device of using the same notation for a function and its value at a point. Thus, “xα” may be used to denote the function whose value at any point x is xα.

Two spaces of C∞ functions on n will be of particular importance for us. The first is the space Cc∞ of C∞ functions with compact support. The existence of nonzero functions in Cc∞ is not quite obvious; the standard construction is based on the fact that the function even at me origin (Exercise 3). If we set

(8.1) equation

it follows that , and supp() is the closed unit ball. In the next section we shall use this single function to manufacture elements of in great profusion; see Propositions 8.17 and 8.18.

The other space of C∞ functions we shall need is the Schwartz space S consisting of those C∞ functions which, together with all their derivatives, vanish at infinity faster than any power of |x|. More precisely, for any nonnegative integer N and any multi-index α we define

then

Examples of functions in § are easy to find: for instance, where α is any multi-index. Also, clearly .

It is an important observation that if , then for all α and all p [1, ∞]. Indeed, for all N, and (1 + |x|)−N Lp for N > n/p by Corollary 2.52.

8.2 Proposition. S is a Fréchet space with the topology defined by the norms ||·||(N, α).

Proof. The only nontrivial point is completeness. If {fk} is a Cauchy sequence in S, then ||fj − fk||(N, α) → 0 for all N, α. In particular, for each α the sequence converges uniformly to a function gα. Denoting by ej the vector (0, …, 1, …, 0) with the 1 in the jth position, we have

Letting k → ∞, we obtain

The fundamental theorem of calculus implies that , and an induction on |α| then yields for all α. It is then easy to check that for all α.

Another useful characterization of S is the following.

8.3 Proposition. If f C∞, then f S iff is bounded for all multi-indices α, β iff is bounded for all multi-indices α, β.

Proof. Obviously . On the other hand, is strictly positive on the unit sphere |x| = 1, so it has a positive minimum δ there. It follows that for all x since both sides are homogeneous of degree N, and hence

This establishes the first equivalence. The second one follows from the fact that each is a linear combination of terms of the form and vice versa, by the product rule (Exercise 1).

We next investigate the continuity of translations on various function spaces. The following notation for translations will be used throughout this chapter and the next one: If f is a function on n and y n,

We observe that and that . A function f is called uniformly continuous if 0. (The reader should pause to check that this is equivalent to the usual -δ definition of uniform continuity.)

8.4 Lemma. If f Cc(n), then f is uniformly continuous.

Proof. Given > 0, for each x supp(f) there exists δx > 0 such that . Since supp(f) is compact, there exist x₁, …, xN such that the balls of radius about xj cover supp(f). If δ = 1/2 min{δxj}, then, one easily sees that ||τyf − f||u < whenever |y| < δ.

8.5 Proposition. If 1 ≤ p < ∞, translation is continuous in the Lp norm; that is, if f Lp and z n, then lim.

Proof. Since τy+z = τyτz, by replacing f by τzf it suffices to assume that z = 0. First, if g Cc, for |y| ≤ 1 the functions τyg are all supported in a common compact set K, so by Lemma 8.4,

Now suppose f Lp. If > 0, by Proposition 7.9 there exists g Cc with ||g − f||p < /3, so

and ||τyg − g||p < /3 if y is sufficiently small.

Proposition 8.5 is false for p = ∞, as one should expect since the L∞ norm is closely related to the uniform norm; see Exercise 4.

Some of our results will concern multiply periodic functions in n, and for simplicity we shall take the fundamental period in each variable to be 1. That is, we define a function f on n to be periodic if f(x + k) = f(x) for all x n and k n. Every periodic function is thus completely determined by its values on the unit cube

Periodic functions may be regarded as functions on the space of cosets of n, which we call the n-dimensional torus and denote by n. (When n = 1 we write rather than ¹.) n is a compact Hausdorff space; it may be identified with the set of all z = {z₁, …, zn) n such that |zj| = 1 for all j, via the map

On the other hand, for measure-theoretic purposes we identify n with the unit cube Q, and when we speak of Lebesgue measure on n we mean the measure induced on n by Lebesgue measure on Q. In particular, m(n) = 1. Functions on n may be considered as periodic functions on n or as functions on Q; the point of view will be clear from the context when it matters.

Exercises

1. Prove the product rule for partial derivatives as stated in the text. Deduce that

for some constants cγδ and c’γδ with cγδ = c’γδ = 0 unless |γ| < |α| and |δ| < |β|.

2. Observe that the binomial theorem can be written as follows:

Prove the following generalizations:

a. The multinomial theorem: If x n,

b. The n-dimensional binomial theorem: If x, y n,

3. Let η(t) = e−1/t for t > 0, η(t) = 0 for t < 0.

a. For k and t > 0, η(k)(t) = Pk(1/t)e−1/t where Pk is a polynomial of degree 2k.

b. η(k) (0) exists and equals zero for all k .

4. If f L∞ and ||τyf − f||∞ → 0 as y → 0, then f agrees a.e. with a uniformly continuous function. (Let Arf be as in Theorem 3.18. Then Arf is uniformly continuous for r > 0 and uniformly Cauchy as r → 0.)

8.2 CONVOLUTIONS

Let f and g be measurable functions on n. The convolution of f and g is the function f * g defined by

for all x such that the integral exists. Various conditions can be imposed on f and g to guarantee that f * g is defined at least almost everywhere. For example, if f is bounded and compactly supported, g can be any locally integrable function; see also Propositions 8.7–8.9 below.

In what follows, we shall need the fact that if f is a measurable function on n, then the function K(x, y) = f(x − y) is measurable on n x n. We have K = f o s where s(x, y) = x − y; since s is continuous, K is Borel measurable if f is Borel measurable. This can always be assumed without affecting the definition of f * g, by Proposition 2.12. However, the Lebesgue measurability of K also follows from the Lebesgue measurability of f; see Exercise 5.

The elementary properties of convolutions are summarized in the following proposition.

8.6 Proposition. Assuming that all integrals in question exist, we have

a. f * g = g * f,

b. (f * g) * h = f * (g * h).

c. For z n, τz(f * g) = (τz f) * g = f * (τz g).

d. If A is the closure of {x + y : x supp(f), y supp(g)}, then supp (f * g) ⊂ A.

Proof. (a) is proved by the substitution z = x − y:

(b) follows from (a) and Fubini’s theorem:

As for (c),

and by (a),

For(d), we observe that if x A, then for any y supp(g) we have x-y supp(f); hence f(x − y)g(y) = 0 for all y, so f * g(x) = 0.

The following two propositions contain the basic facts about convolutions of Lp functions.

8.7 Young’s Inequality. If f L¹ and g Lp (1 ≤ p ≤ ∞), then f * g(x) exists for almost every x, f * g Lp, and .

Proof. This is a special case of Theorem 6.18, with K(x, y) = f(x − y). Alternatively, one can use Minkowski’s inequality for integrals:

8.8 Proposition. If p and q are conjugate exponents, f Lp, and g Lq, then f * g(x) exists for every x, f * g is bounded and uniformly continuous, and . If 1 < p < ∞ (so that 1 < q < ∞ also), then f * g C₀(n).

Proof. The existence of f * g and the estimate for ||f * g||u follow immediately from Hölder’s inequality. In view of Propositions 8.5 and 8.6, so does the uniform continuity of f * g: If 1 ≤ p < ∞,

(If p = ∞, interchange the roles of f and g.) Finally, if 1 < p, q < ∞, choose sequences {fn} and {gn} of functions with compact support such that ||fn − f||p → 0 and ||gn − g||q → 0. By Proposition 8.6d and what we have just proved, fn * gn Cc. But

so f * g C₀ by Proposition 4.35.

The preceding results are all we shall use, but for the sake of completeness we state also the following generalization.

8.9 Proposition. Suppose 1 ≤ p, q, r ≤ ∞ and p⁻¹ + q⁻¹ x = r⁻¹ + 1.

a. (Young’s Inequality, General Form) If f Lp and g Lq, then f * g and ||f * g||r ≤ ||f||p||g||q.

b. Suppose also that p > 1, q > 1, and r < ∞. If f Lp and g weak Lq, then f * g Lr and ||f * g||r ≤ Cpq||f||p[g]q where Cpq is independent of f and g.

c. Suppose that p = 1 and r = q > 1. If f L¹ and g weak Lq, then f * g weak Lq and [f * g]q ≤ Cq||f||₁, where Cq is independent of f and g.

Proof. To prove (a), let q be fixed. The special cases p = 1, r = q and p = q/(q − 1), r = ∞ are Propositions 8.7 and 8.8. The general case then follows from the Riesz-Thorin interpolation theorem. (See also Exercise 6 for a direct proof.) (b) and (c) are special cases of Theorem 6.36.

One of the most important properties of convolution is that, roughly speaking, f * g is at least as smooth as either f or g, because formally we have

and similarly . To make this precise, one needs only to impose conditions on f and g so that differentiation under the integral sign is legitimate. One such result is the following; see also Exercises 7 and 8.

8.10 Proposition. If f L¹, g Ck, and is bounded for |α| ≤ k, then f * g Ck and for |α| ≤ k.

Proof. This is clear from Theorem 2.27.

8.11 Proposition. If f, g , then f * g .

Proof. First, f * g C∞ by Proposition 8.10. Since

(8.12) equation

we have

which is finite by Corollary 2.52.

Convolutions of functions on the torus n are defined just as for functions on n. (If one regards functions on n as periodic functions on n, of course, the integration is to be extended over the unit cube rather than n.) All of the preceding results remain valid, with the same proofs.

The following theorem underlies many of the important applications of convolutions on n. We introduce a bit of notation that will be used frequently hereafter: If ϕ is any function on n and t > 0, we set

(8.13) equation

We observe that if ϕ L¹, then ϕt is independent of t, by Theorem 2.44:

Moreover, the “mass” of ϕt becomes concentrated at the origin as t → 0. (Draw a picture if this isn’t clear.)

8.14 Theorem. Suppose ϕ L¹ and .

a. If f Lp (1 ≤ p < ∞), then f * ϕt → a f in the Lp norm as t → 0.

b. If f is bounded and uniformly continuous, then f * ϕt → a f uniformly as t → 0.

c. If f L∞ and f is continuous on an open set U, then f * ϕt → a f uniformly on compact subsets of U as t → 0.

Proof. Setting y = tz, we have

Apply Minkowski’s inequality for integrals:

Now, ||τtz f − f||p is bounded by 2||f||p and tends to 0 as t → 0 for each z, by Proposition 8.5. Assertion (a) therefore follows from the dominated convergence theorem.

The proof of (b) is exactly the same, with ||·||p replaced by ||·||u. The estimate for ||f * ϕt − a f||u is obvious, and ||τtz f − f||u → 0 as t → 0 by the uniform continuity of f.

As for (c), given ε > 0 let us choose a compact such that . Also, let K be a compact subset of U. If t is sufficiently small, then, we will have x − tz U for all x K and z E, so from the compactness of K it follows as in Lemma 8.4 that

for small t. But then

from which (c) follows.

If we impose slightly stronger conditions on ϕ, we can also show that f * ϕt → a f almost everywhere for f Lp. The device in the following proof of breaking up an integral into pieces corresponding to the dyadic intervals [2k, 2k+1] and estimating each piece separately is a standard trick of the trade in Fourier analysis.

8.15 Theorem. Suppose |ϕ(x)| ≤ C(1 + |x|)−n-ε for some C, ε > 0 (which implies that ϕ L¹ by Corollary 2.52), and , then f * ϕt(x) → a f(x) as t → 0 for every x in the Lebesgue set of f — in particular, for almost every x, and for every x at which f is continuous.

Proof. If x is in the Lebesgue set of f, for any δ > 0 there exists η > 0 such that

(8.16) equation

Let us set

We claim that I₁ is bounded by Aδ where A is independent of t, whereas I₂ → 0 as t → 0. Since

we will have

and since δ is arbitrary, this will complete the proof.

To estimate I₁, let K be the integer such that 2K ≤ η/t < 2K+1 if η/t ≥ 1, and K = 0 if η/t < 1. We view the ball |y| < η as the union of the annuli 2−kη ≤ |y| < 21-k η (1 ≤ k ≤ K) and the ball |y| < 2−K η. On the kth annulus we use the estimate

and on the ball |y| < 2−K η we use the estimate |ϕt(y)| ≤ Ct−n. Thus

Therefore, by (8.16) and the fact that 2K ≤ η/t < 2K+1,

As for I₂, if p′ is the conjugate exponent to p and χ is the characteristic function of , by Hölder’s inequality we have

so it suffices to show that for 1 ≤ q ≤ ∞, and in particular for q = 1 and q = p′, ||χϕt||q → 0 as t → 0. If q = ∞, this is obvious:

If q < ∞, by Corollary 2.51 we have

In either case, is dominated by tε, so we are done.

In most of the applications of the preceding two theorems one has a = 1, although the case a = 0 is also useful. If a = 1, {ϕt}t0 is called an approximate identity, as it furnishes an approximation to the identity operator on Lp by convolution operators. This construction is useful for approximating Lp functions by functions having specified regularity properties. For example, we have the following two important results:

8.17 Proposition. Cc∞ (and hence also ∞) is dense in Lp (1 ≤ p < ∞) and in C₀.

Proof. Given f Lp and ε > 0, there exists g Cc with ||f − g||p < ε/2, by Proposition 7.9. Let ϕ be a function in such that for example, take where is as in (8.1). Then g * ϕt Cc∞ by Propositions 8.6d and 8.10, and ||g * ϕt − g||p < ε/2 for sufficiently small t by Theorem 8.14. The same argument applies if Lp is replaced by C₀, ||·||p by ||·||u, and Proposition 7.9 by Proposition 4.35.

8.18 The C∞ Urysohn Lemma. Of K ∞ n is compact and U is an open set containing K, there exists f Cc∞ such that 0 ≤ f ≤ 1, f = 1 on K, and supp(f) ⊂ U.

Proof. Let δ = ρ(K, Uc) (the distance from K to Uc, which is positive since K is compact), and let V = {x : ρ(x, K) < δ/3}. Choose a nonnegative such that ϕ = 1 and ϕ(x) = 0 for |x| ≥ δ/3 (for example, ( )⁻¹ δ/3 with as in (8.1)), and set f = χV * ϕ. Then f Cc∞ by Propositions 8.6d and 8.10, and it is easily checked that 0 ≤ f ≤ 1, f = 1 on K, and supp.

Exercises

5. If s : n x n → n is defined by s(x, y) = x − y, then s⁻¹(E) is Lebesgue measurable whenever E is Lebesgue measurable. (For n = 1, draw a picture of s⁻¹(E) ⊂ ². It should be clear that after rotation through an angle π/4, s⁻¹(E) becomes F x where F = {x : , and Theorem 2.44 can be applied. The same idea works in higher dimensions.)

6. Prove Theorem 8.9a by using Exercise 31 in §6.3 to show that

7. If f is locally integrable on n and g Ck has compact support, then f * g Ck.

8. Suppose that f Lp(). If there exists h Lp() such that

we call h the (strong) Lp derivative of f. If f Lp(n), Lp partial derivatives of f are defined similarly. Suppose that p and q are conjugate exponents, f Lp, g Lq, and the Lp derivative ∂jf exists. Then ∂j(f * g) exists (in the ordinary sense) and equals (∂jf) * g.

9. If f Lp(), the Lp derivative of f (call it h; see Exercise 8) exists iff f is absolutely continuous on every bounded interval (perhaps after modification on a null set) and its pointwise derivative f′ is in Lp, in which case h = f′ a.e. (For “only if,” use Exercise 8: If g Cc with g = 1, then f * gt → f and (f * gt)’ → h as t → 0. For “if,” write

and use Minkowski’s inequality for integrals.)

10. Let ϕ satisfy the hypotheses of Theorem 8.15. If f Lp (1 ≤ p ≤ ∞), define the ϕ-maximal function of f to be Mϕf(x) = supt0|f * ϕt(x)|. (Observe that the Hardy-Littlewood maximal function H f is Mϕ|f| where ϕ is the characteristic function of the unit ball divided by the volume of the ball.) Show that there is a constant C, independent of f, such that Mϕf ≤ C · H f. (Break up the integral as the sum of the integrals over |y| ≤ t and over (k = 0, 1, 2, …), and estimate ϕt on each region.) It follows from Theorem 3.17 that Mϕ is weak type (1,1), and the proof of Theorem 3.18 can then be adapted to give an alternate demonstration that .

11. Young’s inequality shows that L¹ is a Banach algebra, the product being convolution.

a. If J is an ideal in the algebra L¹, so is its closure in L¹.

b. If f L¹, the smallest closed ideal in L¹ containing f is the smallest closed subspace of L¹ containing all translates of f. (If g Cc, f * g(x) can be approximated by sums Σ f(x − yj)g(yj)Δyj. On the other hand, if {ϕt} is an approximate identity, f * τy(ϕt) → τy f as t → 0.)

8.3 THE FOURIER TRANSFORM

One of the fundamental principles of harmonic analysis is the exploitation of symmetry. To be more specific, if one is doing analysis on a space on which a group acts, it is a good idea to study functions (or other analytic objects) that transform in simple ways under the group action, and then try to decompose arbitrary functions as sums or integrals of these basic functions.

The spaces we are studying are n and n, which are Abelian groups under addition and act on themselves by translation. The building blocks of harmonic analysis on these spaces are the functions that transform under translation by multiplication by a factor of absolute value one, that is, functions f such that for each x there is a number ϕ(x) with |ϕ(x)| = 1 such that f(y + x) = ϕ(x)f(y). If f and ϕ have this property, then f(x) = ϕ(x)f(0), so f is completely determined by ϕ once f(0) is given; moreover,

so that (unless f = 0) ϕ(x + y) = ϕ(x)ϕ(y). In short, to find all f’s that transform as described above, it suffices to find all ϕ’s of absolute value one that satisfy the functional equation ϕ(x + y) = ϕ(x)ϕ(y). Upon imposing the natural requirement that ϕ should be measurable, we have a complete solution to this problem.

8.19 Theorem. If ϕ is a measurable function on n (resp. n) such that ϕ(x + y) = ϕ(x)ϕ(y) and |ϕ| = 1, there exists ξ n (resp. ξ n) such that ϕ>(x) = e2πiξ·x.

Proof. We first prove this assertion on . Let a be such that such an a surely exists, for otherwise the Lebesgue differentiation theorem would imply that ϕ = 0 a.e. Setting A = , then, we have

Thus ϕ, being the indefinite integral of a locally integrable function, is continuous; and then, being the integral of a continuous function, it is C¹. Moreover,

It follows that (d/dx)(e−Bxϕ(x)) = 0, so that e−Bxϕ(x) is constant. Since ϕ(0) = 1, we have ϕ(x) = eBx, and since |ϕ| = 1, B is purely imaginary, so B = 2πiξ for some ξ . This completes the proof for ; as for , the ϕ we have been considering will be periodic (with period 1) iff e2πiξ = 1 iff ξ .

The n-dimensional case follows easily, for if e₁, …, en is the standard basis for n, the functions j(t) = ϕ(tej) satisfy j(t + s) = j(t)j(s) on , so that j(t) = e2πiξjt, and hence

The idea now is to decompose more or less arbitrary functions on n or n in terms of the exponentials ε2πiξ·x. In the case of n this works out very simply for L² functions:

8.20 Theorem. Let Eκ(x) = e2πiκ·x, Then {Eκ : κ n} is an orthonormal basis of L²(n).

Proof. Verification of orthonormality is an easy exercise in calculus; by Fubini’s theorem it boils down to the fact that equals 1 if k = 0 and equals 0 otherwise. Next, since EκEλ = Eκ+λ, the set of finite linear combinations of the Eκ’s is an algebra. It clearly separates points on n; also, E₀ = 1 and κ = E-κ. Since n is compact, the Stone-Weierstrass theorem implies that this algebra is dense in C(n) in the uniform norm and hence in the L² norm, and C(n) is itself dense in L²(n) by Proposition 7.9. It follows that {Eκ} is a basis.

To restate this result: If f L²(n), we define its Fourier transform , a function on n, by

and we call the series

the Fourier series of f. The term “Fourier transform” is also used to mean the map . Theorem 8.20 then says that the Fourier transform maps L²(n) onto l²(n), that (Parseval’s identity), and that the Fourier series of f converges to f in the L² norm. We shall consider the question of pointwise convergence in the next two sections.

Actually, the definition of makes sense if f is merely in L¹ (n), and , so the Fourier transform extends to a norm-decreasing map from L¹(n) to . (The Fourier series of an L¹ function may be quite badly behaved, but there are still methods for recovering f from when f L¹, as we shall see in the next section.) Interpolating between L¹ and L², we have the following result.

8.21 The Hausdorff-Young Inequality. Suppose that 1 ≤ p ≤ 2 and q is the conjugate exponent to p. If .

Proof. Since the assertion follows from the Riesz-Thorin interpolation theorem.

The situation on n is more delicate. The formal analogue of Theorem 8.20 should be

These relations turn out to be valid when suitably interpreted, but some care is needed. In the first place, the integral defining is likely to diverge if f L². However, it certainly converges if f L¹. We therefore begin by defining the Fourier transform of f L(n) by

(We use the notation for the Fourier transform only in certain situations where it is needed for clarity.) Clearly , and is continuous by Theorem 2.27; thus

We summarize the elementary properties of in a theorem.

8.22 Theorem. Suppose f, g L¹(n).

a. and where .

b. If T is an invertible linear transformation of n and S = (T*)⁻¹ is its inverse transpose, then . In particular, if T is a rotation, then and if T x = t⁻¹x (t > 0), then , so that in the notation of (8.13).

c.

d. If xαf L¹ for |α| ≤ k, then .

e. If for |α| ≤ k, and , then .

f. (The Riemann-Lebesgue Lemma) .

Proof. a. We have

and similarly for the other formula.

b. By Theorem 2.44,

c. By Fubini’s theorem,

d. By Theorem 2.27 and induction on |α|,

e. First assume n = |α| = 1. Since f C₀, we can integrate by parts:

The argument for n > 1, |α| = 1 is the same — to compute , integrate by parts in the jth variable — and the general case follows by induction on |α|.

f. By (e), if f C¹ Cc, then is bounded and hence . But the set of all such f’s is dense in L¹ by Proposition 8.17, and uniformly whenever fn → f in L¹. Since C₀ is closed in the uniform norm, the result follows.

Parts (d) and (e) of Theorem 8.22 point to a fundamental property of the Fourier transform: Smoothness properties of f are reflected in the rate of decay of at infinity, and vice versa. Parts (a), (c), (e), and (f) of this theorem are valid also on n, as is (b) provided that T leaves the lattice n invariant (Exercise 12).

8.23 Corollary. maps the Schwartz class continuously into itself.

Proof. If f S, then for all α, β, so by Theorem 8.22d,e, is C∞ and

Thus is bounded for all α, β, whence S by Proposition 8.3. Moreover, since

It then follows that by the proof of Proposition 8.3, so the Fourier transform is continuous on .

At this point we need to compute an important specific Fourier transform.

8.24 Proposition. If f(x) = where a > 0, then .

Proof. First consider the case n = 1. Since the derivative of e−πax² is −2πae−πax², by Theorem 8.22d,e we have

It follows that = 0, so that is constant. To evaluate the constant, set ξ = 0 and use Proposition 2.53:

The n-dimensional case follows by Fubini’s theorem, since

We are now ready to invert the Fourier transform. If f L¹, we define

and we claim that if f L¹ and then . A simple appeal to Fubini’s theorem fails because the integrand in

is not in L¹(n x n). The trick is to introduce a convergence factor and then pass to the limit, using Fubini’s theorem via the following lemma:

8.25 Lemma. If f, g L¹ then .

Proof. Both integrals are equal to .

8.26 The Fourier Inversion Theorem. If f L¹ and , then f agrees almost everywhere with a continuous function f₀, and .

Proof. Given t > 0 and x n, set

By Theorem 8.22a and Proposition 8.24,

where g(x) = e−π|x|² and the subscript t has the meaning in (8.13). By Lemma 8.25, then,

Since , by Theorem 8.14 we have f * gt → f in the L¹ norm as t → 0. On the other hand, since L¹ the dominated convergence theorem yields

It follows that a.e., and similarly a.e. Since and are continuous, being Fourier transforms of L¹ functions, the proof is complete.

8.27 Corollary. If f L¹ and = 0, then f = 0 a.e.

8.28 Corollary. is an isomorphism of onto itself.

Proof. By Corollary 8.23, maps continuously into itself, and hence so does , since . By the Fourier inversion theorem, these maps are inverse to each other.

At last we are in a position to derive the analogue of Theorem 8.20 on n.

8.29 The Plancherel Theorem. If f L¹ L², then L²; and extends uniquely to a unitary isomorphism on L².

Proof. Let . Since implies f L∞, we have by Proposition 6.10, and is dense in L² because and is dense in L² by Proposition 8.17. Given . By the inversion theorem,

Hence, by Lemma 8.25,

Thus preserves the L² inner product; in particular, by taking g = f, we obtain . Since by the inversion theorem, extends by continuity to a unitary isomorphism on L².

It remains only to show that this extension agrees with on all of L¹ L². But if f L¹ L² and g(x) = e−π|x|² as in the proof of the inversion theorem, we have f * gt L¹ by Young’s inequality and because and is bounded. Hence f * gt X; moreover, by Theorem 8.14, f * gt → f in both the L¹ and L² norms. Therefore both uniformly and in the L² norm, and we are done.

We have thus extended the domain of the Fourier transform from L¹ to L¹ + L². Just as on n, the Riesz-Thorin theorem yields the following result for the intermediate Lp spaces:

8.30 The Hausdorff-Young Inequality. Suppose that 1 ≤ p < 2 and q is the conjugate exponent to p. If f , then and .

If f L¹ and L¹, the inversion formula

exhibits f as a superposition of the basic functions ; it is often called the Fourier integral representation of f. This formula remains valid in spirit for all f L², although the integral (as well as the integral defining ) may not converge pointwise. The interpretation of the inversion formula will be studied further in the next section.

We conclude this section with a beautiful theorem that involves an interplay of Fourier series and Fourier integrals. To motivate it, consider the following problem: Given a function f L¹ (n), how can one manufacture a periodic function (that is, a function on n) from it? Two possible answers suggest themselves. One way is to “average” f over all periods, producing the series . This series, if it converges, will surely define a periodic function. The other way is to restrict to the lattice n and use it to form a Fourier series . The content of the following theorem is that these methods both work and both give the same answer.

8.31 Theorem. If f L¹(n), the series f converges pointwise a.e. and in L¹(n) to a function P f such that . Moreover, for (Fourier transform on n) equals (κ) (Fourier transform on n).

Proof. Let . Then n is the disjoint union of the cubes Q + k = , so

Now apply Theorem 2.25. First, it shows that the series Στkf converges a.e. and in L¹(n) to a function P f L¹(n) such that , since n is measure-theoretically identical to Q. Second, it yields

If we impose conditions on f to guarantee that the series in question converge absolutely, we obtain a more refined result.

8.32 The Poisson Summation Formula. Suppose f C(n) satisfies |f(x)| ≤ C(1 + |x|)−n- and for some C, > 0. Then

where both series converge absolutely and uniformly on n. In particular, taking x = 0,

Proof. The absolute and uniform convergence of the series follows from the fact that which can be seen by comparing the latter series to the convergent integral . Thus the function is in C(n) and hence in L²(n), so Theorem 8.35 implies that the series converges in L²(n) to P f. Since it also converges uniformly, its sum equals P f pointwise. (The replacement of k by -k in the formula for P f is immaterial since the sum is over all k n.)

Exercises

12. Work out the analogue of Theorem 8.22 for the Fourier transform on n.

13. Let f(x) = 1/2 − x on the interval [0, 1), and extend f to be periodic on .

a. (0) = 0, and (κ) = (2πiκ)⁻¹ if κ ≠ 0.

b. . (Use the Parseval identity.)

14. (Wirtinger’s Inequality) If f C¹([a, b]) and f(a) = f(b) = 0, then

(By a change of variable it suffices to assume a = 0, b = 1/2. Extend by setting f(−x) = −f(x), and then extend f to be periodic on . Check that f, thus extended, is in C¹ () and apply the Parseval identity.)

15. Let sine x = (sin πx)/πx (sinc 0 = 1).

a. If sinc 2ax.

b. Let . Then a is a Hilbert space and sinc(2ax − k) : k } is an orthonormal basis for a.

c. (The Sampling Theorem) If f Ha, then f C₀ (after modification on a null set), and sinc(2ax − k), where the series converges both uniformly and in L². (In the terminology of signal analysis, a signal of bandwidth 2a is completely determined by sampling its values at a sequence of points {k/2a} whose spacing is the reciprocal of the bandwidth.)

16. Let

a. Compute fk(x) explicitly and show that ||f||u = 2.

b. sin 2πkx sin 2πx, and as k → ∞. (Use Exercise 15a, and substitute y = 2πkx in the integral defining

c. is a proper subset of C₀. (Consider and use the open mapping theorem.)

17. Given a > 0, let for x > 0 and f(x) − 0 for x ≤ 0.

a. f L¹, and f L² if a > .

b. . (Here we are using the branch of za in the right half plane that is positive when z is positive. Cauchy’s theorem may be used to justify the complex substitution in the integral defining .)

c. If a, b > then

.

18. Suppose f L²().

a. The L² derivative f’ (in the sense of Exercises 8 and 9) exists iff , in which case .

b. If the L² derivative f’ exists, then

(If the integrals on the right are finite, one can integrate by parts to obtain

c. (Heisenberg’s Inequality) For any b, β ,

(The inequality is trivial if either integral on the right is infinite; if not, reduce to the case b = β = 0 by considering This inequality, a form of the quantum uncertainty principle, says that f and cannot both be sharply localized about single points b and β.

19. (A variation on the theme of Exercise 18) If f L²(n) and the set = {x : f(x) ≠ 0} has finite measure, then for any measurable .

20. If f L¹ (n+m), define . (Here x n and y m.) Then , and .

21. State and prove a result that encompasses both Theorem 8.31 and Exercise 20, in the setting of Fourier transforms on closed subgroups and quotient groups of n.

22. Since commutes with rotations, the Fourier transform of a radial function is radial; that is, if F L¹(n) and F(x) = f(|x|), then , where f and g are related as follows.

a. Let where σ is surface measure on the unit sphere S in n (Theorem 2.49). Then J is radial — say, and .

b. J satisfies .

c. j satisfies . (This equation is a variant of Bessel’s equation. The function j is completely determined by the fact that it is a solution of this equation, is smooth at ρ = 0, and satisfies j(0) = σ(S) = 2πn/2/Γ(n/2). In fact, where Jα is the Bessel function of the first kind of order α.)

d. If n = 3, j(ρ) = 4πρ⁻¹ sin ρ. (Set f(ρ) = ρj(ρ) and use (c) to show that f” + f = 0. Alternatively, use spherical coordinates to compute the integral defining J(0, 0, ρ) directly.)

23. In this exercise we develop the theory of Hermite functions.

a. Define operators T, T* on S() by and . Then and .

b. Let , and for k ≥ 1 let is the kth normalized Hermite function.) We have and , and hence TT*hk = khk.

c. Let S = 2TT* + I. Then S f(x) = x²f(x) − f“(x) and Shk = (2k + 1)hk. (S is called the Hermite operator.)

d. is an orthonormal set in L²(). (Check directly that ||h₀||₂ = 1. then observe that for and use (a) and (b).)

e. We have

(use induction on k), and in particular,

f. Let . Then Hk is a polynomial of degree k, called the kth normalized Hermite polynomial. The linear span of H₀, …, Hm is the set of all polynomials of degree ≤ m. (The kth Hermite polynomial as usually defined is

g. is an orthonormal basis for L²(). (Suppose f ⊥ hk for all k, and let g(x) = f(x)e−x²/2. Show that = 0 by expanding in its Maclaurin series and using (f).)

h. Define , and define = A⁻¹ FA f for f L². Then A is unitary and . Moreover, , and hence . Therefore, if is an orthonormal basis for L² consisting of eigenfunctions for ; namely, .

8.4 SUMMATION OF FOURIER INTEGRALS AND SERIES

The Fourier inversion theorem shows how to express a function f on n in terms of provided that f and are in L¹. The same result holds for periodic functions. Namely, if f L¹(n) and l¹(n), then the Fourier series converges absolutely and uniformly to a function g. Since , it follows from Theorem 8.20 that f L² and that the series converges to f in the L² norm. Hence f = g a.e., and f = g everywhere if f is assumed continuous at the outset.

Two questions therefore arise. What conditions on f will guarantee that is integrable? And how can f be recovered from if is not integrable?

As for the first question, since is bounded for f L¹, the issue is the decay of at infinity, and this is related to the smoothness properties of f. For example, by Theorem 8.22e, if f Cn+1(n) and , then and hence by Corollary 2.52. The same result holds for periodic functions, for the same reason: If f Cn+1(n), then and hence .

To obtain sharper results when n > 1 requires a generalized notion of partial derivatives, so we shall postpone this task until §9.3. (See Theorem 9.17.) However, for n = 1 we can easily obtain a better theorem that covers the useful case of functions that are continuous and piecewise C¹. We state it for periodic functions and leave the nonperiodic case to the reader (Exercise 24).

8.33 Theorem. Suppose that f is periodic and absolutely continuous on , and that f′ LP() for some p > 1. Then .

Proof. Since p > 1, we have and since for p > 2, we may assume that p ≤ 2. Integration by parts (Theorem 3.36) shows that . Hence, by the inequalities of Holder and Hausdorff-Young, if q is the conjugate exponent to p,

Adding to both sides, we see that .

We now turn to the problem of recovering f from under minimal hypotheses on f, and we consider first the case of n. The proof of the Fourier inversion theorem contains the essential idea: Replace the divergent integral by where Φ is a continuous function that vanishes rapidly enough at infinity to make the integral converge. If we choose Φ to satisfy Φ(0) = 1, then Φ(tξ) → 1 as t → 0, and with any luck the corresponding integral will converge to f in some sense. One Φ that works is the function Φ (ξ) = e−π|ξ|² used in the proof of the inversion theorem, but we shall see below that there are others of independent interest. We therefore formulate a fairly general theorem, for which we need the following lemma that complements Theorem 8.22c.

8.34 Lemma. If f, g L²(n), then .

Proof. by Plancherel’s theorem and Hölder’s inequality, so makes sense. Given x n, let . It is easily verified that , so since F is unitary on L²,

8.35 Theorem. Suppose that , and . Given f L¹ + L², for t >0 set

a. If f Lp(1 ≤ p < ∞), then ft Lp and ||ft − f||p → 0 as t → 0.

b. If f is bounded and uniformly continuous, then so is ft, and ft → f uniformly as t → 0.

c. Suppose also that |ϕ(x)| ≤ C(1 + |x|)−n- for some C, > 0. Then ft(x) → f(x) for every x in the Lebesgue set of f.

Proof. We have f = f₁ + f₂ where f₁ L¹ and f₂ L². Since , and , the integral defining ft converges absolutely for every x. Moreover, if ϕt(x) = t−n ϕ(t⁻¹x), we have by the inversion theorem and Theorem 8.22b, and . Since we have , so by Theorem 8.22c and the inversion formula,

Also, ϕ L² by the Plancherel theorem, so by Lemma 8.34,

In short, ft = f * ϕt, so the assertions follow from Theorems 8.14 and 8.15.

By combining this theorem with the Poisson summation formula, we obtain a corresponding result for periodic functions.

8.36 Theorem. Suppose that Φ C(n) satisfies and Φ(0) = 1. Given f L¹(n), for t > 0 set

(which converges absolutely since .

a. If f Lp(n) (1 ≤ p < ∞), then ||ft − f||p → 0 as t → 0, and if f C(n), then ft → f uniformly as t → 0.

b. ft(x) → f(x) for every x in the Lebesgue set of f.

Proof. Let and . Then , and ϕt satisfies the hypotheses of the Poisson summation formula, so

Let us denote the common value of these sums by t(x). Then

so ft = f * t. Hence, by Young’s inequality and Theorem 8.31 we have

so the operators are uniformly bounded on Lp, 1 ≤ p ≤ ∞.

Now, since Φ is continuous and Φ(0) − 1, we clearly have ft → f uniformly (and hence in Lp(n)) if f is a trigonometric polynomial — that is, if (κ) = 0 for all but finitely many κ. But the trigonometric polynomials are dense in C(n) in the uniform norm by the Stone-Weierstrass theorem, and hence also dense in Lp(n) in the Lp norm for p < ∞. Assertion (a) therefore follows from Proposition 5.17.

To prove (b), suppose that x is in the Lebesgue set of f; by translating f we may assume that x = 0, which simplifies the notation. With , we have

Since

for x Q and k ≠ 0 we have , and hence

which vanishes as t → 0. On the other hand, if we define g = fχQ L¹ (n), then 0 is in the Lebesgue set of g (because 0 is in the interior of Q, and the condition that 0 be in the Lebesgue set of g depends only on the behavior of g near 0), so by Theorem 8.15,

Let us examine some specific examples of functions Φ that can be used in Theorems 8.35 and 8.36. The first is the one already used in the proof of the inversion theorem,

This ϕ is called the Gauss kernel or Weierstrass kernel. It is important for a number of reasons, including its connection with the heat equation that we shall explain in §8.7. When n = 1, its periodized version

in terms of which the ft in Theorem 8.36 is given by ft = f * ϕt is essentially one of the Jacobi theta functions, which are connected with elliptic functions and have applications in number theory.

The second example is Φ(ξ) = e−2π|ξ|, whose inverse Fourier transform ϕ is called the Poisson kernel on n. When n = 1, we have

(8.37) equation

The formula for ϕ in higher dimensions is worked out in Exercise 26; it turns out that ϕ(x) is a constant multiple of (1 + |x|²)−(n+1)/2. Like the Gauss kernel, the Poisson kernel has an interpretation in terms of partial differential equations that we shall explain in §8.7.

If we take n = 1 and in Theorem 8.36, make the substitution , and write Arf in place of ft, we obtain

(8.38) equation

This formula is a special case of one of the classical methods for summing a (possibly) divergent series. Namely, if is a series of complex numbers, for 0 < r < 1 its rth Abel mean is the series . If the latter series converges for r < 1 to the sum S(r) and the limit exists, the series is said to be Abel summable to S. If converges to the sum S, then it is also Abel summable to S (Exercise 27), but the Abel sum may exist even when the series diverges.

In (8.38), Arf(x) is the rth Abel mean of the Fourier series of f, in which the kth and (−k)th terms are grouped together to make a series indexed by the nonnegative integers. It has the following complex-variable interpretation: If we set z − re2πix, then

The two series on the right define, respectively, a holomorphic and an antiholomorphic function on the unit disc |z| < 1. In particular, Arf(x) is a harmonic function on die unit disc, and the fact that Arf → f as r → 1 means that f is the boundary value of this function on the unit circle. See also Exercise 28.

Our final example is the function Φ(ξ) = max(1 − |ξ|, 0) with n = 1. Its inverse Fourier transform is

If we use this Φ in Theorem 8.36, take t = (m + 1)⁻¹ (m = 0, 1, 2, …), and write σmf(x) for f1/(m+1)(x), we obtain

(8.39) equation

This is an instance of another classical method for summing divergent series. Namely, if is a series of complex numbers, its mth Cesàro mean is the average of its first m + 1 partial sums, , where . If the sequence of Cesàro means converges as m → ∞ to a limit S, the series is said to be Cesàro summable to S. It is easily verified that if converges to S, then it is Cesàro summable to S (but perhaps not conversely), and that σmf(x) is the mth Cesàro mean of the Fourier series of f with the kth and (−k)th terms grouped together. See Exercise 29, and also Exercise 33 in the next section.

Exercises

24. State and prove an analogue of Theorem 8.33 for functions on . (In addition to the hypotheses that f be locally absolutely continuous and that f′ Lp for some p > 1, you will need some further conditions f and/or f’ at infinity to make the argument work. Make them as mild as possible.)

25. For 0 < α ≤ 1, let λα() be the space of Hölder continuous functions on of exponent α as in Exercise 11 in §5.1. Suppose 1 < p < ∞ and p⁻¹ + q⁻¹ = 1.

a. If f satisfies the hypotheses of Theorem 8.33, then f λ1/q(), but f need not lie in λα() for any α > 1/q. (Hint: f(b) −

b. If α < 1, λα() contains functions that are not of bounded variation and hence are not absolutely continuous. (But cf. Exercise 37 in §3.5.)

26. The aim of this exercise is to show that the inverse Fourier transform of e−2π|ξ| on n is

a. If (Use (8.37).)

b. If (Use(a), Proposition 8.24, and the formula (

c. Let then the formula in (b) expresses as a superposition of dilated Gauss kernels. Use Proposition 8.24 again to derive the asserted formula for ϕ.

27. Suppose that the numerical series is convergent.

a. Let Smn = Σmn ak. Then (“summation by parts”).

b.

c. The series is uniformly convergent for 0 ≤ r ≤ 1, and hence its sum S(r) is continuous there. In particular, .

28. Suppose that f L¹(), and let Arf be given by (8.38).

a. Arf = f * Pr where is the Poisson kernel for .

b.

29. Given , let and .

a.

b. If exists, then so does limm→∞ σm, and the two limits are equal.

c. The series diverges but is Abel and Cesàro summable to 1/2.

30. If f L¹(n), f is continuous at 0, and ≥ 0, then L¹. (Use Theorem 8.35c and Fatou’s lemma.)

31. Suppose a > 0. Use (8.37) to show that

Then subtract a⁻² from both sides and let a → 0 to show that Σ₁∞ k⁻² = π²/6.

32. A C∞ function f on is real-analytic if for every x , f is the sum of its Taylor series based at x in some neighborhood of x. If f is periodic and we regard f as a function on S = {z : |z| = 1}, this condition is equivalent to the condition that f be the restriction to S of a holomorphic function on some neighborhood of S. Show that f C∞() is real-analytic iff |(κ)| ≤ Ce−|κ for some C, > 0. (See the discussion of the Abel means Arf in the text, and note that when |z| = 1.)

8.5 POINTWISE CONVERGENCE OF FOURIER SERIES

The techniques and results of the previous two sections, involving such things as Lp norms and summability methods, are relatively modern; they were preceded historically by the study of pointwise convergence of one-dimensional Fourier series. Although the latter is one of the oldest parts of Fourier analysis, it is also one of the most difficult — unfortunately for the mathematicians who developed it, but fortunately for us who are the beneficiaries of the ideas and techniques they invented in doing so. A thorough study of this issue is beyond the scope of this book, but we would be remiss not to present a few of the classic results.

To set the stage, suppose f L¹(). We denote by Smf the mth symmetric partial sum of the Fourier series of f:

From the definition of (k), we have

where Dm is the mth Dirichlet kernel:

The terms in this sum form a geometric progression, so

Multiplying top and bottom by e−πix yields the standard closed formula for Dm:

(8.40) equation

The difficulty with the partial sums Smf, as opposed to (for example) the Abel or Cesàro means, can be summed up in a nutshell as follows. Smf can be regarded as a special case of the construction in Theorem 8.36; in fact, with the notation used there, Smf = f1/m if we take Φ = χ[−1,1]. But χ[−1,1] does not satisfy the hypotheses of Theorem 8.36, because its inverse Fourier transform (πx)⁻¹ sin 2πx (Exercise 15a) is not in L¹(). On the level of periodic functions, this is reflected in the fact that although Dm L¹() for all m, ||Dm||₁ → ∞ as m → ∞ (Exercise 34).

Among the consequences of this is that the Fourier series of a continuous function f need not converge pointwise, much less uniformly, to f; see Exercise 35. (This does not contradict the fact that trigonometric polynomials are dense in C()! It just means that if one wants to approximate a function f C() uniformly by trigonometric polynomials, one should not count on the partial sums Smf to do the job; the Cesàro means defined by (8.39) work much better in general.) To obtain positive results for pointwise convergence, one must look in other directions.

The first really general theorem about pointwise convergence of Fourier series was obtained in 1829 by Dirichlet, who showed that for every x provided that f is piecewise continuous and piecewise monotone. Later refinements of the argument showed that what is really needed is for f to be of bounded variation. We now prove this theorem, for which we need two lemmas. The first one is a slight generalization of one of the more arcane theorems of elementary calculus, the “second mean value theorem for integrals.”

8.41 Lemma. Let ϕ and be real-valued functions on [a, b]. Suppose that is monotone and right continuous on [a, b] and is continuous on [a, b]. Then there exists η [a, b] such that

Proof. Adding a constant c to ϕ changes both sides of the equation by the amount , so we may assume that ϕ(a) = 0. We may also assume that ϕ is increasing; otherwise replace ϕ by -ϕ. Let (so that ’ = -) and apply Theorem 3.36:

The endpoint evaluations vanish since ϕ(a) = (b) = 0. Since ϕ is increasing and , if m and M are the minimum and maximum values of on [a, b] we have . By the intermediate value theorem, then, there exists η [a, b] such that , which is the desired result.

8.42 Lemma. There is a constant C < ∞ such that for every m ≥ 0 and every ,

Moreover, for all m.

Proof. By (8.40),

Since (sin πx)⁻¹ − (πx)⁻¹ is bounded on and |sin(2m + 1)πx| ≤ 1, the second integral on the right is bounded in absolute value by a constant. With the substitution y = (2m + 1)πx, the first one becomes

where sin y dy. But Si(x) is continuous and approaches the finite limits (see Exercise 59b in §2.6), so Si(x) is bounded. This proves the first assertion. As for the second one,

(only the term with k = 0 is nonzero), so since Dm is even,

8.43 Theorem. If f BV() — that is, if f is periodic on and of bounded variation on [−1/2, 1/2] — then

In particular, lim at every x at which f is continuous.

Proof. We begin by making some reductions. In examining the convergence of Smf(x), we may assume that x = 0 (by replacing f with the translated function τ-xf), that f is real-valued (by considering the real and imaginary parts separately), and that f is right continuous (since replacing f(t) by f(t+) affects neither Smf nor 1/2[f(0+) + f(0−)]). In this case, by Theorem 3.27b, on the interval we can write f as the difference of two right continuous increasing functions g and h. If these functions are extended to by periodicity, they are again of bounded variation, and it is enough to show that and likewise for h.

In short, it suffices to consider the case where x = 0 and f is increasing and right continuous on . Since Dm is even, we have Smf(0) = f * Dm(0) = , so by Lemma 8.42,

We shall show that the first integral on the right tends to zero as m → ∞; a similar argument shows that the second integral also tends to zero, thereby completing the proof.

Given > 0, choose δ > 0 small enough so that f(δ) − f(0+) < /C where C is as in Lemma 8.42. Then by Lemma 8.41, for some η [0, δ],

which is less than . On the other hand, by (8.40),

where g± is the periodic function given on the interval [−, ) by

But as m → ∞ by the Riemann-Lebesgue lemma (the periodic analogue of Theorem 8.22f). Therefore,

for every > 0, and we are done.

One of the less attractive features of Fourier series is that bad behavior of a function at one point affects the behavior of its Fourier series at all points. For example, if f has even one jump discontinuity, then cannot be in l¹() and so the series cannot converge absolutely at any point. However, to a limited extent the convergence of the series at a point x depends only on the behavior of f near x, as explained in the following localization theorem.

8.44 Theorem. If f and g are in L¹ () and f = g on an open interval I, then Smf − Smg → 0 uniformly on compact subsets of I.

Proof. It is enough to assume that g = 0 (consider f − g), and by translating f we may assume that I is centered at 0, say I = (−c, c) where c ≤ . Fix δ < c; we shall show that if f = 0 on I then Smf → 0 uniformly on [−δ, δ].

The first step is to show that Smf → 0 pointwise on [−δ, δ], and the argument is similar to the preceding proof. Namely, by (8.40) we have

where

Since f(x − y) = 0 on a neighborhood of the zeros of sin πy, the functions gx,± are in by the Riemann-Lebesgue lemma.

The next step is to show that if x₁, x₂ [−δ, δ], then Smf(x₁) − Smf(x₂) vanishes as x₁ − x₂ → 0, uniformly in m. By (8.40) again,

But f(x₁ − y) − f(x₂ − y) = 0 for |y| < c − δ, and for we have

where A is independent of m. Hence

which vanishes as x₁ − x₂ → 0 by (the periodic analogue of) Proposition 8.5.

Now, given > 0, we can choose η small enough so that if x₁, x₂ [−δ, δ] and |x₁ − x₂| < η, then Choose x₁, …, xk [−δ, δ] so that the intervals |x − xj| < η cover [−δ, δ]. Since Smf(xj) → 0 for each j, we can choose M large enough so that |Smf(xj)| < /2 for m > M and 1 ≤ j ≤ k. If |x| ≤ δ, then, we have |x − xj| < η for some j, so

for m > M, and we are done.

8.45 Corollary. Suppose that f L¹() and I is an open interval of length ≤ 1.

a. If f agrees on I with a function g such that , then Smf → f uniformly on compact subsets of I.

b. If f is absolutely continuous on I and f’ Lp(I) for some p > 1, then Smf → f uniformly on compact subsets of I.

Proof. If f = g on I, then on I, and if , then Smg → g uniformly on ; (a) follows. As for (b), given [a₀, b₀] ⊂ I, pick a < a₀ and b > b₀ so that [a, b] ⊂ I, and let g be the continuous periodic function that equals f on [a, b] and is linear on [b, a + 1] (which is unique since g(b) = f(b) and g(a + 1) = g(a) = f(a)). Under the hypotheses of (b), g is absolutely continuous on and by Theorem 8.33. Thus Smf → f uniformly on [a₀, b₀] by (a).

Finally, we discuss the behavior of Smf near a jump discontinuity of f. Let us first consider a simple example: Let

(8.46) equation

Then ϕ is periodic and is C∞ except for jump discontinuities at the integers, where ϕ(j+) − ϕ(j−) = 1. It is easy to check that and for k ≠ 0 (Exercise 13a), so that

From Corollary 8.45 it follows that Smϕ → ϕ uniformly on any compact set not containing an integer, and it is obvious that Smϕ(x) = 0 when x is an integer. But near the integers a peculiar thing happens: Smϕ contains a sequence of spikes that overshoot and undershoot ϕ, as shown in Figure 8.1, and as m → ∞ the spikes tend to zero in width but not in height. In fact, when m is large the value of Smϕ at its first maximum to the right of 0 is about 0.5895, about 18% greater than ϕ(0+) = . This is known as the Gibbs phenomenon; the precise statement and proof are given in Exercise 37.

Fig. 8.1 The Gibbs phenomenon: the graph y = .

Now suppose that f is any periodic function on having a jump discontinuity at x = a (that is, f(a+) and f(a−) exist and are unequal). Then the function

is continuous at every point where f is, and also at x = a provided that we (re)define g(a) to be [f(a+) + f(a−)], as the jumps in f and ϕ cancel out. If g satisfies one of the hypotheses of Corollary 8.45 on an interval I containing a, the Fourier series of g will converge uniformly near a, and hence the Fourier series of f will exhibit the same Gibbs phenomenon as that of ϕ.

Finally, suppose that f is periodic and continuous except at finitely many points a₁, …, ak, , where f has jump discontinuities. We can then subtract off all the jumps to form a continuous function g:

If f satisfies some mild smoothness conditions — for example, if f is absolutely continuous on any interval not containing any aj and f’ Lp for some p > 1 — then will be in l¹(). Conclusion: Smf → f uniformly on any interval not containing any , and Smf exhibits the Gibbs phenomenon near every aj.

Exercises

33. Let σm be the Cesàro means of the Fourier series of f given by (8.39).

a. σmf = f * Fm where Fm = (m + 1)⁻¹ Σ₀m Dk and Dk is the kth Dirichlet kernel. (See Exercise 29a.) Fm is called the mth Fejér kernel.

b. Fm(x) = sin²(m + 1)πx/(m + 1)sin² πx. (Use (8.40) and the fact that sin(2k + 1)πx = Im e(2k+1)πix.)

34. If Dm is the mth Dirichlet kernel, ||Dm||₁ → ∞ as m ∞. (Make the substitution y = (2m + 1)πx and use Exercise 59a in §2.6.)

35. The purpose of this exercise is to show that the Fourier series of “most” continuous functions on do not converge pointwise.

a. Define ϕm(f) = Smf(0). Then ϕ C()* and ||ϕ|| = ||Dm||₁.

b. The set of all f C() such that the sequence {Smf(0)} converges is meager in C(). (Use Exercise 34 and the uniform boundedness principle.)

c. There exist f C() (in fact, a residual set of such f’s) such that {Smf(x)} diverges for every x in a dense subset of . (The result of (b) holds if the point 0 is replaced by any other point in . Apply Exercise 40 in §5.3.)

36. The Fourier transform is not surjective from L¹() to C₀(). (Use Exercise 34, and cf. Exercise 16c.)

37. Let ϕ be given by (8.46) and let Δm = Smϕ − ϕ.

a. (d/dx)Δm(x) = Dm(x) for x .

b. The first maximum of Δm to the right of 0 occurs at x = (2m + 1)⁻¹, and

(Use (8.40) and the fact that

c. More generally, the jth critical point of Δm to the right of 0 occurs at x = j/(2m + 1) (j = 1, …, 2m), and

These numbers are positive for j odd and negative for j even. (See Exercise 59b in §2.6.)

8.6 FOURIER ANALYSIS OF MEASURES

We recall that M(n) is the space of complex Borel measures on n (which are automatically Radon measures by Theorem 7.8), and we embed L¹(n) into M(n) by identifying f L¹ with the measure dμ = f dm. We shall need to define products of complex measures on Cartesian product spaces, which can easily be done in terms of products of positive measures by using Radon-Nikodym derivatives. Namely, if μ, v M(n), we define μ x v M(n x n) by

If μ, v M(n), we define their convolution μ * v M(n) by μ * v(E) = μ x v(α⁻¹(E)) where α : n x n → n is addition, α(x, y) = x + y. In other words,

(8.47) equation

8.48 Proposition.

a. Convolution of measures is commutative and associative.

b. For any bounded Borel measurable function h,

c. .

d. If dμ = f dm and dv = g dm, then d(μ * v) = (f * g) dm; that is, on L¹ the new and old definitions of convolution coincide.

Proof. Commutativity is obvious from Fubini’s theorem, as is associativity, for λ * μ * v is unambiguously defined by the formula

Assertion (b) follows from (8.47) by the usual linearity and approximation arguments. In particular, taking h = d|μ * v|(d(μ * v), since |h| = 1 we obtain

which proves (c). Finally, if dμ = f dm and dv = g dm, for any bounded measurable h we have

whence d(μ * v) = (f * g) dm.

We can also define convolutions of measures with functions in Lp(n, m), which we implicitly assume to be Borel measurable. (By Proposition 2.12, this is no restriction.)

8.49 Proposition. If f Lp(n) (1 ≤ p ≤ ∞) and μ M(n), then the integral exists for a.e. x, f * μ Lp, and . (Here “Lp” and “a.e.” refer to Lebesgue measure.)

Proof. If f and μ are nonnegative, then f * μ(x) exists (possibly being equal to ∞) for every x, and by Minkowski’s inequality for integrals,

In particular, f * μ(x) < ∞ for a.e. x. In the general case this argument applies to |f| and |μ|, and the result follows easily.

In the case p = 1, the definition of f * μ in Proposition 8.49 coincides with the definition given earlier in which f is identified with f dm, for

for any Borel set E. Thus L¹(n) is not merely a subalgebra of M(n) with respect to convolution but an ideal.

We extend the Fourier transform from L¹(n) to M(n) in the obvious way: If μ is the function defined by

(The Fourier transform on measures is sometimes called the Fourier-Stieltjes transform.) Since e−2πiξ·x is uniformly continuous in x, it is clear that is a bounded continuous function and that . Moreover, by taking h(x) = e−2πiξ·x in Proposition 8.48b,w.

We conclude by giving a useful criterion for vague convergence of measures in terms of Fourier transforms.

8.50 Proposition. Suppose that μ₁, μ₂, …, and μ are in M for all k and pointwise, then μk → μ vaguely.

Proof. If f S, then (Corollary 8.23), so by the Fourier inversion theorem,

Since and , the dominated convergence theorem implies that . But S is dense in C₀(n) (Proposition 8.17), so by Proposition 5.17, for all , that is, μk → μ vaguely.

This result has a partial converse: If μk → μ vaguely and ||μk|| → ||μ||, then pointwise. This follows from Exercise 26 in §7.3.

Exercises

38. Work out the analogues of the results in this section for measures on the torus n.

39. If μ is a positive Borel measure on with μ() = 1, then for all k ≠ 0 unless μ is a linear combination, with positive coefficients, of the point masses at for some m , in which case for all j .

40. L¹(n) is vaguely dense in M(n). (If μ M(n), consider ϕt * μ where {ϕt}t0 is an approximate identity.)

41. Let Δ be the set of finite linear combinations of the point masses δx, x n. Then Δ is vaguely dense in M(n). (If f is in the dense subset Cc(n) of L¹(n) and g C₀(n), approximate by Riemann sums. Then use Exercise 40.)

42. A function ϕ on n that satisfies ≥ 0 for all and all x₁, …, xm n, for any m , is called positive definite. If μ, M(n) is positive, then is positive definite.

8.7 APPLICATIONS TO PARTIAL DIFFERENTIAL EQUATIONS

In this section we present a few of the many applications of Fourier analysis to the theory of partial differential equations; others will be found in Chapter 9. We shall use the term differential operator to mean a linear partial differential operator with smooth coefficients, that is, an operator L of the form

If the aα’s are constants, we call L a constant-coefficient operator. In this case, if for all sufficiently well-behaved functions f (for example, we have

It is therefore convenient to write L in a slightly different form: We set bα = (2πi)|α|aα and introduce the operators

so that

Thus, if P is any polynomial in n complex variables, say , we can form the constant-coefficient operator , and we then have . The polynomial P is called the symbol of the operator P(D).

Clearly, one potential application of the Fourier transform is in finding solutions of the differential equation P(D)u = f. Indeed, application of the Fourier transform to both sides yields , whence . Moreover, if P⁻¹ is the Fourier transform of a function ϕ, we can express u directly in terms of f as u = f * ϕ. For these calculations to make sense, however, the functions f and must be ones to which the Fourier transform can be applied, which is a serious limitation within the theory we have developed so far. The full power of this method becomes available only when the the domain of the Fourier transform is substantially extended. We shall do this in §9.2; for the time being, we invite the reader to work out a fairly simple example in Exercise 43. (It must also be pointed out that even when this method works, is far from being the only solution of P(D)u = f; there are others that grow too fast at infinity to be within the scope even of the extended Fourier transform.)

Let us turn to some more concrete problems. The most important of all partial differential operators is the Laplacian

The reason for this is that Δ is essentially the only (scalar) differential operator that is invariant under translations and rotations. (If one considers operators on vector-valued functions, there are others, such as the familiar grad, curl, and div of 3-dimensional vector analysis.) More precisely, we have:

8.51 Theorem. A differential operator L satisfies L(f o T) = (Lf)o T for all translations and rotations T iff there is a polynomial P in one variable such that L = P(Δ).

Proof. Clearly L is translation-invariant iff L has constant coefficients, in which case L = Q(D) for some polynomial Q in n variables. Moreover, since and the Fourier transform commutes with rotations, L commutes with rotations iff Q is rotation-invariant. Let Q = where Qj is homogeneous of degree j; then it is easy to see that Q is rotation-invariant iff each Qj is rotation-invariant. (Use induction on j and the fact that . But this means that Qj(ξ) depends only on by homogeneity. Moreover, is a polynomial precisely when j is even, so cj = 0 for j odd. Setting , then, we have , that is, L = .

One of the basic boundary value problems for the Laplacian is the Dirichlet problem: Given an open set and a function f on its boundary , find a function u on such that Δu = 0 on Ω and . (This statement of the problem is deliberately a bit imprecise.) We shall solve the Dirichlet problem when Ω is a half-space.

For this purpose it will be convenient to replace n by n + 1 and to denote the coordinates on n+1 by x₁, …, xn, t. We continue to use the symbol A to denote the Laplacian on n, and we set

so the Laplacian on . We take the half-space Ω to be . Thus, given a function f on n, satisfying conditions to be made more precise below, we wish to find a function u on such that and .

The idea is to apply the Fourier transform on n, thus converting the partial differential equation into the simple ordinary differential equation . The general solution of this equation is

(8.52) equation

and we require that . We therefore obtain a solution to our problem by taking (more about the reasons for this choice below); this gives where is the Poisson kernel introduced in §8.4. As we calculated in Exercise 26,

So far this is all formal, since we have not specified conditions on f to ensure that these manipulations are justified. We now give a precise result.

8.53 Theorem. Suppose . Then the function u(x, t) = (f * Pt)(x) satisfies , and limt→0 u(x, t) = f(x) for a.e. x and for every x at which f is continuous. Moreover, limt→0 ||u(·, t) − f||p = 0 provided p < ∞.

Proof. Pt and all of its derivatives are in , since a rough calculation shows that and for large x. Also, = 0, as can be verified by direct calculation or (more easily) by taking the Fourier transform. Hence f * Pt is well defined and

Since Pt(x) = t−n P₁(t⁻¹ x) and , the remaining assertions follow from Theorems 8.14 and 8.15.

The function u(x, t) = (f * Pt)(x) is not the only one satisfying the conclusions of Theorem 8.53; for example, v(x, t) = u(x, t) + ct also works, for any c . For f L¹, we could also obtain a large family of solutions by taking c² in (8.52) to be an arbitrary function in Cc∞ and c₁ = − c₂. (But there is no nice convolution formula for the resulting function u, because e2πt|ξ| is not the Fourier transform of a function or even a distribution.) The solution u(x, t) = (f * Pt)(x) is distinguished, however, by its regularity at infinity; for example, it can be shown that if f BC(n), then u is the unique solution in BC(n x [0, ∞)).

The same idea can be used to solve the heat equation

on n × (0, ∞) subject to the initial condition u(x, 0) = f(x). (Physical interpretation: u(x, t) represents the temperature at position x and time t in a homogeneous isotropic medium, given that the temperature at time 0 is f(x).) Indeed, Fourier transformation leads to the ordinary differential equation with initial condition . The unique solution of the latter problem is . In view of Proposition 8.24, this yields

Here we have Gt(x) = t−n/2G₁(t−x), so after the change of variable s = . Theorems 8.14 and 8.15 apply again, and we obtain an exact analogue of Theorem 8.53 for the initial value problem . Actually, in the present case the hypotheses on f can be relaxed considerably because see Exercise 44.

Another fundamental equation of mathematical physics is the wave equation

(Physical interpretation: u(x, t) is the amplitude at position x and time t of a wave traveling in a homogeneous isotropic medium, with units chosen so that the speed of propagation is 1.) Here it is appropriate to specify both u(x, 0) and :

(8.54) equation

After applying the Fourier transform, we obtain

the solution to which is

(8.55) equation

Since

it follows that

But here there is a problem: is the Fourier transform of a function only when n ≤ 2 and the Fourier transform of a measure only when n ≤ 3; for these cases the resulting solution of the wave equation is worked out in Exercises 45–47. To carry out this analysis in higher dimensions requires the theory of distributions, which we shall examine in Chapter 9. (We shall not, however, derive the explicit formula for Wt, which becomes increasingly complicated as n increases.)

Exercises

43. Let ϕ(x) = e−|x|/2 on . Use the Fourier transform to derive the solution u = f * ϕ of the differential equation u − u” = f, and then check directly that it works. What hypotheses are needed on f?

44. Let Gt(x) = (4πt)−n/2e−|x|²/4t, and suppose that f Lloc¹ (n) satisfies for every ε > 0. Then u(x, t) = f * Gt(x) is well defined for all x n and on and limt→0 u(x, t) = f(x) for a.e. x and for every x at which f is continuous. (To show u(x, t) → f(x) a.e. on a bounded open set V, write f = ϕf + (1 − ϕ)f where ϕ Cc and ϕ = 1 on V, and show that [(1 − ϕ)f) * Gt → 0 on V.)

45. Let n = 1. Use (8.55) and Exercise 15a to derive d’Alembert’s solution to the initial value problem (8.54):

Under what conditions on f and g does this formula actually give a solution?

46. Let n = 3, and let σt denote surface measure on the sphere |x| = t. Then

(See Exercise 22d.) What is the resulting solution of the initial value problem (8.54), expressed in terms of convolutions? What conditions on f and g ensure its validity?

47. Let n = 2. If ξ ², let . Rewrite the result of Exercise 46,

in terms of an integral over the disc Dt = {y : |y| ≤ t} in ² by projecting the upper and lower hemispheres of the sphere |x| = t in ³ onto the equatorial plane. Conclude that (2π|ξ|)⁻¹ sin 2πt|ξ| is the Fourier transform of

and write out the resulting solution of the initial value problem (8.54).

48. Solve the following initial value problems in terms of Fourier series, where f, g, and u(·, t) are periodic functions on . Make some precise hypotheses on f and g, and state precise conclusions.

a. .

b.

49. In this exercise we discuss heat flow on an interval.

a. Solve with boundary conditions u(x, 0) = f(x) for x (a, b), u(a, t) = u(b, t) = 0 for t > 0, in terms of Fourier series. (This describes heat flow on (a, b) when the endpoints are held at a constant temperature. It suffices to assume a = 0, b = ; extend f to by requiring f to be odd and periodic, and use Exercise 48a.)

b. Solve the same problem with the condition u(a, t) = u(b, t) = 0 replaced by . (This describes heat flow on (a, b) when the endpoints are insulated. This time, extend f to be even and periodic.)

50. Solve with boundary conditions u(x, 0) = f(x) and for x (a, b), u(a, t) = u(b, t) = 0 for t > 0, in terms of Fourier series by the method of Exercise 49a. (This problem describes the motion of a vibrating string that is fixed at the endpoints. It can also be solved by extending f to be odd and periodic and using Exercise 45. That form of the solution tells you what you see when you look at a vibrating string; this one tells you what you hear when you listen to it.)

8.8 NOTES AND REFERENCES

The scope of Fourier analysis is much wider than we have been able to indicate in this chapter. Dym and McKean [36] gives a more comprehensive treatment with many interesting applications. Also recommended are Körner’s delightful book [87], which discusses various aspects of classical Fourier analysis and their role in science, and the excellent collection of expository articles edited by Ash [7], which gives a broader view of the mathematical ramifications of the subject. On the more advanced level, the reader should consult Zygmund [167] for the classical theory and Stein [140], [141] and Stein and Weiss [142] for some of the more recent developments.

§8.1: The formulas given in most calculus books for the remainder term Rk(x) = f(x) − Pk(x) in Taylor’s formula (where Pk is the Taylor polynomial of degree k) require f to possess derivatives of order k + 1, but this is not really necessary. The version of Taylor’s theorem stated in the text is derived in Folland [45].

§8.3: Trigonometric series and integrals have a very long history, but modern Fourier analysis only became possible after the invention of the Lebesgue integral. When that tool became available, the L² theory was quickly established: the Riesz-Fischer theorem [44], [114] for Fourier series (essentially Theorem 8.20), and the Plancherel theorem [110] for Fourier integrals. Since then the subject has developed in many directions.

There is no universal agreement on where to put the factors of 2π in the definition of the Fourier transform. Other common conventions are

whose inverse transforms are

₁ has the disadvantage of not being unitary , whereas F₂ is unitary but does not convert convolution into multiplication (F₂(f * g) = (2π)n/2(F₂f)(F₂g)). To make both L² norms and convolutions come out right, one can either put the 2π’s in the exponent, as we have done, or omit them from the exponent but replace Lebesgue measure dx by (2π)−n/2 dx in defining both the Fourier transform (as in ₂) and convolutions.

The Hausdorff-Young inequality is sharp on n, since equality holds when f is a constant function; but on n the optimal result, a deep theorem of Beckner [14], is that .

One of the fundamental qualitative features of the Fourier transform is the fact that, roughly speaking, a nonzero function and its Fourier transform cannot both be sharply localized, that is, they cannot both be negligibly small outside of small sets. This general principle has a number of different precise formulations, two of which are derived in Exercises 18 and 19; see Folland and Sitaram [50] for a comprehensive discussion.

A nice complex-variable proof of the fact that the Fourier transform is injective on L¹ can be found in Newman [106].

§§8.4–5: The theory of convergence of one-dimensional Fourier series really began (as mentioned in the text) with Dirichlet’s theorem in 1829. The first construction of a continuous function whose Fourier series does not converge pointwise was obtained by du Bois Reymond in 1876, and the fact that the Fourier series of a continuous function f is uniformly Cesàro summable to f was proved by Fejér in 1904. In 1926 Kolmogorov produced an f Lp() such that {Smf(x)} diverges at every x; on the other hand, in 1927 M. Riesz proved that for 1 for every f Lp(). The culmination of this subject is the theorem of Carleson (1966, for p = 2) and Hunt (1967, for general p) that if f Lp() where p > 1, then Smf → f almost everywhere.

For more information, see Zygmund [167], the articles by Zygmund and Hunt in Ash [7], and Fefferman [42]. Also see Hewitt and Hewitt [74] for an interesting historical discussion of the Gibbs phenomenon.

Convergence of Fourier series in n variables is an even trickier subject. In the first place, one must decide what one means by a partial sum of a series indexed by n. It is a straightforward consequence of the Riesz and Carleson-Hunt theorems that if f Lp(n) with p > 1, the “cubical partial sums”

converge to f a.e. and (if p < ∞) in the Lp norm. On the other hand, C. Fefferman proved the rather shocking result that for the “spherical partial sums”

the convergence holds for all f Lp only when p = 2, if n > 1. Of course, one can consider modifications of the spherical partial sums in the hope of obtaining positive results; the most intensively studied of these are the Bochner-Riesz means

obtained by taking in Theorem 8.36. (When n = 1, σm+1¹f is essentially equivalent to the Cesàro mean σmf.) These Φ’s satisfy the hypotheses of Theorem 8.36 when α > (n − 1), and some positive results are also known for smaller values of α.

Davis and Chang [30] is a good source for all of this material; see also Stein and Weiss [142] and Ash [7].

§8.7: The solution of the initial value problem for the wave equation in arbitrary dimensions can be found in Folland [48]; see also Folland [49]. Further applications of Fourier analysis to differential equations can be found in Folland [46], [48], Körner [87], and Taylor [147].

Chapter 9

Elements of Distribution Theory

At least as far back as Heaviside in the 1890s, engineers and physicists have found it convenient to consider mathematical objects which, roughly speaking, resemble functions but are more singular than functions. Despite their evident efficacy, such objects were at first received with disdain and perplexity by the pure mathematicians, and one of the most important conceptual advances in modern analysis is the development of methods for dealing with them in a rigorous and systematic way. The method that has proved to be most generally useful is Laurent Schwartz’s theory of distributions, based on the idea of linear functionals on test functions. For some purposes, however, it is preferable to use a theory more closely tied to L² on which the power of Hilbert space methods and the Plancherel theorem can be brought to bear, namely, the (L²) Sobolev spaces. In this chapter we present the fundamentals of these theories and some of their applications.

9.1 DISTRIBUTIONS

In order to find a fruitful generalization of the notion of function on n, it is necessary to get away from the classical definition of function as a map that assigns to each point of n a numerical value. We have already done this to some extent in the theory of Lp spaces: If f Lp, the pointwise values f(x) are of little significance for the behavior of f as an element of Lp, as f can be modified on any set of measure zero without affecting the latter. What is more to the point is the family of integrals ∫ fϕ as ϕ ranges over the dual space Lq. Indeed, we know that f is completely determined by its action as a linear functional on Lq; on the other hand, if we take ϕ = ϕr = m(Br)⁻¹χBr where Br is the ball of radius r about x, by the Lebesgue differentiation theorem we can recover the pointwise value f(x), for almost every x, as limr→0 ∫ fϕr. Thus, we lose nothing by thinking of f as a linear map from Lq(n) to rather than as a map from n to .

Let us modify this idea by allowing f to be merely locally integrable on n but requiring ϕ to lie in C∞c. Again the map is a well-defined linear functional on , and again the pointwise values of f can be recovered a.e. from it, by an easy extension of Theorem 8.15. But there are many linear functionals on that are not of the form , and these — subject to a mild continuity condition to be specified below — will be our “generalized functions.”

Recall that for we have defined to be the set of all functions whose support is compact and contained in E. If is open, is the union of the spaces as K ranges over all compact subsets of U. Each of the latter is a Fréchet space with the topology defined by the norms

in which a sequence {ϕj} converges to ϕ iff ∂α ϕj → ∂αϕ uniformly for all α. (The completeness of C∞c(K) is easily proved by the argument in Exercise 9 in §5.1.) With this in mind, we make the following definitions, in which U is an open subset of n:

i. A sequence {ϕj} in Cc∞(U) converges in Cc∞ to ϕ if {ϕj} ⊂ Cc∞(K) for some compact set K ∞ U and ϕj → ϕ in the topology of Cc∞(K), that is, ∂α ϕj → ∂αϕ uniformly for all α.

ii. If X is a locally convex topological vector space and T : Cc∞(U) → X is a linear map, T is continuous if T|Cc∞(K) is continuous for each compact K ⊂ U, that is, if Tϕj → Tϕ whenever ϕj → ϕ in Cc∞(K) and K ⊂ U is compact.

iii. A linear map T : Cc∞(U) → Cc∞(U’) is continuous if for each compact K ⊂ U there is a compact K’ ∞ U’ such that T(Cc∞(K)) ⊂ Cc∞(K’), and T is continuous from Cc∞(K) to Cc∞(K’).

iv. A distribution on U is a continuous linear functional on Cc∞(U). The space of all distributions on U is denoted by D’(U), and we set D’ = D’(n). We impose the weak* topology on D’(U), that is, the topology of pointwise convergence on Cc∞(U).

Two remarks: First, the standard notation D’ for the space of distributions comes from Schwartz’s notation D for Cc∞, which is also quite common. Second, there is a locally convex topology on Cc∞ with respect to which sequential convergence in Cc∞ is given by (i) and continuity of linear maps T : Cc∞ → × and T : Cc∞ → Cc∞ is given by (ii) and (iii). However, its definition is rather complicated and of little importance for the elementary theory of distributions, so we shall omit it.

Here are some examples of distributions; more will be presented below.

Every f Lloc¹(U) — that is, every function f on U such that ∞K |f| < ∞ for every compact K ⊂ U — defines a distribution on U, namely, the functional ϕ → ∫ fϕ, and two functions define the same distribution precisely when they are equal a.e.

Every Radon measure μ on U defines a distribution by ϕ ϕ dμ.

If x₀ U and α is a multi-index, the map ϕ ∂αϕ(x₀) is a distribution that does not arise from a function; it arises from a measure μ precisely when α = 0, in which case μ is the point mass at x₀.

If f Lloc¹ (U), we denote the distribution also by f, thereby identifying Lloc¹(U) with a subspace of D’(U). In order to avoid notational confusion between f(x) and f(ϕ) = ∫ fϕ, we adopt a different notation for the pairing between Cc∞(U) and D’(U). Namely, if F D’(U) and ϕ Cc∞ (U), the value of F at ϕ will be denoted by . Observe that the pairing between D’(U) and Cc(U) is linear in each variable; this conflicts with our earlier notation for inner products but will cause no serious confusion. If μ is a measure, we shall also identify μ with the distribution dμ.

Sometimes it is convenient to pretend that a distribution F is a function even when it really is not, and to write ∫ F(x)ϕ(x) dx instead of . This is the case especially when the explicit presence of the variable x is notationally helpful.

At this point we set forth two pieces of notation that will be used consistently throughout this chapter. First, we shall use a tilde to denote the reflection of a function in the origin:

Second, we denote the point mass at the origin, which plays a central role in distribution theory, by δ:

As an illustration of the role of δ and the notion of convergence in D’, we record the following important corollary of Theorem 8.14:

9.1 Proposition. Suppose that f L¹(n) and ∫ f = a, and for t > 0 let ft(x) = t−n f(t⁻¹ x). Then ft → aδ in D’ as t → 0.

Proof. If ϕ Cc∞, by Theorem 8.14 we have

Although it does not make sense to say that two distributions F and G in D’(U) agree at a single point, it does make sense to say that they agree on an open set V ⊂ U; namely, F = G on V iff for all ϕ Cc∞ (V). (Clearly, if F and G are continuous functions, this condition is equivalent to the pointwise equality of F and G on V; if F and G are merely locally integrable, it means that F = G a.e. on V) Since a function in Cc∞(V₁ ∪ V₂) need not be supported in either V₁ or V₂, it is not immediately obvious that if F = G on V₁ and on V₂ then F = G on V₁ V₂. However, it is true:

9.2 Proposition. Let {Vα} be a collection of open subsets of U and let V = ∪α Vα. If F, G D’(U) and F = G on each Vα, then F = G on V.

Proof. If ϕ Cc∞(V), there exist α₁, … αm such that supp ϕ ₁m Vαj. Pick ₁, …, m Cc∞ such that supp(j) Vαj and ∑₁m j = 1 on supp(ϕ). (That this can be done is the analogue of Proposition 4.41, proved in the same way as that result by using the Urysohn lemma.) Then = .

According to Proposition 9.2, if F D’(U), there is a maximal open subset of U on which F = 0, namely the union of all the open subsets on which F = 0. Its complement in U is called the support of F.

There is a general procedure for extending various linear operations from functions to distributions. Suppose that U and V are open sets in n, and T is a linear map from some subspace X of Lloc¹(U) into Lloc¹(V). Suppose that there is another linear map such that

Suppose also that T’ is continuous in the sense denned above. Then T can be extended to a map from D’(U) to D’(V), still denoted by T, by

The intervention of the continuous map T” guarantees that the original T, as well as its extension to distributions, is continuous with respect to the weak* topology on distributions: If Fα → F D’(U), then TFα → TF in D’(V).

Here are the most important instances of this procedure. In each of them, U is an open set in n, and the continuity of T’ is an easy exercise that we leave to the reader.

i. (Differentiation) Let T f = ∂α f, denned on C|α|(U). If ϕ Cc∞(U), integration by parts gives there are no boundary terms since ϕ has compact support. Hence T’ = (−1)|α|T|Cc∞(U), and we can define the derivative ∂αF D’(U) of any F D’(U) by

Notice, in particular, that by this procedure we can define derivatives of arbitrary locally integrable functions even when they are not differentiable in the classical sense; this is one of the main reasons for the power of distribution theory. We shall discuss this matter in more detail below.

ii. (Multiplication by Smooth Functions) Given C∞(U), define T f = f. Then T’ = T|Cc∞(U), so we can define the product F D’(U) for F D’(U) by

Moreover, if Cc∞(U), this formula makes sense for any ϕ Cc∞(n) and defines F as a distribution on n.

iii. (Translation) Given y n, let V = U + y = {x + y : x U} and let T = τy. (Recall that we have defined τyf(x) = f(x − y).) Since ∫ f(x − y)ϕ(x)dx = ∫ f(x)ϕ(x + y)dx, we have T’ = τ-y|Cc∞(U + y). For F D’(U), then, we define the translated distribution τyF D’(U + y) by

For example, the point mass at y is τyδ.

iv. (Composition with Linear Maps) Given an invertible linear transformation S of n, let V = S⁻¹(U) and let T f = f S. Then T’ϕ = |det S|⁻¹ ϕ S⁻¹ by Theorem 2.44, so for F D’(U) we define by

In particular, for Sx = −x we have , and | det S| = 1, so we define die reflection of a distribution in the origin by

v. (Convolution, First Method) Given Cc∞, let

(V is open but may be empty.) If f Lloc¹(U), the integral

is well defined for all x V. The same definition works for F D’(U): the convolution F * is the function defined on V by

Since in Cc∞ as x → x₀, F * is a continuous function (actually C∞, as we shall soon see) on V. As an example, for any Cc∞ we have

so δ is the multiplicative identity for convolution.

vi. (Convolution, Second Method) Let , and V be as in (v). If f Lloc¹(U) and ϕ Cc∞(V), we have

That is, if T f = f * , then T maps Lloc¹(U) into Lloc¹(V) and T’ϕ = ϕ * . For F D’(U), we can therefore define F * as a distribution on V by

Again, we have δ * = , for

The definitions of convolution in (v) and (vi) are actually equivalent, as we shall now show.

9.3 Proposition. Suppose that U is open in n and Cc∞. Let V = {x : x − y U for y supp()}. For F D’(U) and x: V let F * (x) = . Then

a. F * C∞(V).

b.

c. For any ϕ Cc∞(V), .

Proof. Let e₁…, en be the standard basis for n. If x V, there exists t₀ > 0 such that x + tej U for |t| < t₀, and it is easily verified that

It follows that ∂j(F * )(x) exists and equals , so by induction, F * C∞(V) and ∂α(F * ) = F * ∂α. Moreover, since and , we have

Next, if ϕ Cc(V), we have

The integrand here is continuous and supported in a compact subset of U, so the integral can be approximated by Riemann sums. That is, for each (large) m we can approximate supp(ϕ) by a union of cubes of side length 2−m (and volume 2−nm) centered at points y₁m, …, yk(m)mm supp(ϕ); then the corresponding Riemann sums Sm = 2−nm are supported in a common compact subset of U and converge uniformly to ϕ * as m → . Likewise, converges uniformly to . Hence,

Next we show that although distributions may be highly singular objects, they can all be approximated in the (weak*) topology of distributions by smooth functions, even by compactly supported ones.

9.4 Lemma. Suppose that ϕ Cc∞, Cc∞, and , and let t(x) = t−n(t⁻¹x).

a. Given any neighborhood U of supp(ϕ), we have supp(ϕ * t) U for t sufficiently small.

b. ϕ * t → ϕ in Cc∞ as t → 0.

Proof. If supp() ⊂ {x : |x| ≤ R} then supp(ϕ * t) is contained in the set of points whose distance from supp(ϕ) is at most tR; this is included in a fixed compact set if t ≤ 1 and is included in U if t is small. Moreover, uniformly as t → 0, by Theorem 8.14. The result follows.

9.5 Proposition. For any open U ⊂ n, Cc∞(U) is dense in D’(U) in the topology of D’(U).

Proof. Suppose F D’(U). We shall first approximate F by distributions supported in compact subsets of U, then approximate the latter by functions in C∞c(U).

Let {Vj} be an increasing sequence of precompact open subsets of U whose union is U, as in Proposition 4.39. For each j, by the C∞ Urysohn lemma we can pick ζj Cc∞(U) such that ζj = 1 on j. Given ϕ Cc∞(U), for j sufficiently large we have supp(ϕ) ⊂ Vj and hence . Therefore ζjF → F as j → .

Now, as we noted in defining products of smooth functions and distributions, since supp(ζj) is compact, ζjF can be regarded as a distribution on n. Let , t be as in Lemma 9.4, and . Then also, so given ϕ Cc∞, we have ϕ * → ϕ in Cc∞ by Lemma 9.4. But then by Proposition 9.3, we have and . In short, every neighborhood of F in D’(U) contains the C∞ functions (ζj F) * t for j large and t small.

Finally, we observe that supp(ζj) ⊂ Vk for some k. If supp(ϕ) k = Ø, then for sufficiently small t we have supp( = Ø (Lemma 9.4 again) and hence . In other words, supp, so we are done.

We conclude this section with some further remarks and examples concerning differentiation of distributions. To restate the basic facts: Every F D’ (U) possesses derivatives ∂αF D’(U) of all orders; moreover, ∂α is a continuous linear map of D’(U) into itself. Let us examine a couple of one-dimensional examples to see what sort of things arise by taking distribution derivatives of functions that are not classically differentiable.

First, differentiating functions with jump discontinuities leads to “delta-functions,” that is, distributions given by measures that are point masses. The simplest example is the Heaviside step function . for which we have

so H’ = δ. See Exercises 5 and 7 for generalizations.

Second, distribution derivatives can be used to extract “finite parts” from divergent integrals. For example, let . f is locally integrable on | {0} and so defines a distribution there, but diverges whenever ϕ(0) ≠ 0. Nonetheless, there is a distribution on that agrees with f on \ {0}, namely, the distribution derivative of the locally integrable function L(x) = (log x)χ(0, ∞)(x). One way of seeing what is going on here is to consider the functions L(x) = (log x)χ(, ∞)(x). By the dominated convergence theorem we have Lϕ for any ϕ , that is, L → L in D’; it follows that L’ → L’ in D’. But

As , this last sum converges even though the two terms individually do not. Formally, passage to the limit gives ; that is, L’ is obtained from f by subtracting an infinite multiple of δ. (This process is akin to the “renormalizations” used by physicists to remove the divergences from quantum field theory.)

Another way to analyze this situation is to consider smooth approximations to L, such as L(x) = L(x)(x) where is a smooth function such that (x) = 0 for x ≤ 1 and (x) = 1 for x ≥ 2. The reader is invited to sketch the graphs of L and (L)’; the latter will look like the graph of f together with a large negative spike near the origin, which turns into “-∞ · δ” as → 0. See also Exercises 10 and 12.

Finally, we remark that one of the bugbears of advanced calculus, that equality of mixed partials need not hold for functions whose derivatives are not continuous, disappears in the setting of distributions: on Cc∞; therefore = on D’! In the standard counterexample, f(x, y) = xy(x² − y²)(x² + y²)⁻¹ (with f(0, 0) = 0), f and f are locally integrable functions that agree everywhere except at the origin; hence they are identical as distributions.

Exercises

1. Suppose that f₁, f₂, …, and f are in L¹loc(U). The conditions in (a) and (b) below imply that fn → f in D’(U), but the condition in (c) does not.

a. fn Lp(U) (1 ≤ p ≤ ∞) and fn → f in the Lp norm or weakly in Lp.

b. For all n, |fn| ≤ g for some g Lloc¹(U), and fn → f a.e.

c. fn → f pointwise.

2. The product rule for derivatives is valid for products of smooth functions and distributions.

3. On , if C∞ then δ(k) = , where the superscripts denote derivatives.

4. Suppose that U and V are open in n and Φ : V → U is a C∞ diffeomorphism. Explain how to define D′(U) for any F D′(V).

5. Suppose that f is continuously differentiable on except at x₁, …, xm, where f has jump discontinuities, and that its pointwise derivative df/dx (defined except at the xj’s) is in Lloc¹(). Then the distribution derivative f′ of f is given by f′ = (df/dx) + .

6. If f is absolutely continuous on compact subsets of an interval , the distribution derivative f′ D′(U) coincides with the pointwise (a.e.-defined) derivative of f.

7. Suppose f Lloc¹(). Then the distribution derivative f′ is a complex measure on iff f agrees a.e. with a function F N BV, in which case .

8. Suppose f Lp(n). If the strong Lp derivatives exist in the sense of Exercise 8 in §8.2, they coincide with the partial derivatives of f in the sense of distributions.

9. A distribution F on n is called homogeneous of degree λ if F Sr = rλF for all r > 0, where Sr(x) = rx.

a. δ is homogeneous of degree −n.

b. If F is homogeneous of degree λ, then ∂α F is homogeneous of degree λ − |α|.

c. The distribution (d/dx) [χ(0,∂) (x) log x] discussed in the text is not homogeneous, although it agrees on \{0} with a function that is homogeneous of degree −1.

10. Let f be a continuous function on n\{0} that is homogeneous of degree −n (i.e., f(rx) = r−nf(x)) and has mean zero on the unit sphere (i.e., f dσ = 0 where σ is surface measure on the sphere). Then f is not locally integrable near the origin (unless f = 0), but the formula

defines a distribution PV(f) — “PV” stands for “principal value” — that agrees with f on n \ {0} and is homogeneous of degree −n in the sense of Exercise 9. (Hint: For any a > 0, the indicated limit equals

and these integrals converge absolutely.)

11. Let F be a distribution on n such that supp(F) = {0}.

a. There exist N , C > 0 such that for all ϕ Cc∞,

b. Fix Cc∞ with (x) = 1 for |x| ≤ 1 and (x) = 0 for |x| ≥ 2. If ϕ Cc∞, let ϕk(x) = ϕ(x)[1 − (kx)]. If ∂αϕ(0) = 0 for |α| ≤ N, then ∂αϕk → ∂αϕ uniformly as k → for |α| ≤ N. (Hint: By Taylor’s theorem, |∂αϕ(x)| ≤ C|x|N+1-|α| for |α| ≤ N.)

c. If ϕ Cc∞ and ∂αϕ(0) = 0 for |α| ≤ N, then .

d. There exist constants cα (|α| ≤ N) such that F = .

12. Suppose λ > n; then the function x |x|−λ on n is not locally integrable near the origin. Here are some ways to make it into a distribution:

a. If ϕ Cc∞, let Pϕk be the Taylor polynomial of ϕ about x = 0 of degree k. Given k > λ − n − 1 and α > 0, define

Then Fak is a distribution on n that agrees with |x|−λ on n \ {0}.

b. If and we take k to be the greatest integer ≤ λ − n, we can let a → ∞ in (a) to obtain another distribution F that agrees with |x|−λ on n\| {0}:

c. Let n = 1 and let k be the greatest integer ≤ λ. Let

Then f Lloc¹(), and the distribution derivative f(k) agrees with |x|−λ on \{0}.

d. According to Exercise 11, the difference between any two of the distributions constructed in (a)–(c) is a linear combination of δ and its derivatives. Which one?

13. If F D′ and for j = 1, …, n, then F is a constant function. (Consider f * t where t is an approximate identity in Cc∞.)

14. For n ≥ 3, define F, Fε Lloc¹(n) by

where ωn = 2πn/2/Γ(n/2) is the volume of the unit sphere, and let Δ be the Laplacian.

a. ΔFε(x) = ε−ng(ε⁻¹x) where g(x) = nωn⁻¹(|x|² + 1)−(n+2)/2.

b. ∫ g = 1. (Use polar coordinates and set s = r²/(r² + 1).)

c. ΔF = δ. (Fε → F in D′; use Proposition 9.1.)

d. If ϕ Cc∞, the function f = F * ϕ satisfies Δf = 0.

e. The results of (c) and (d) hold also for n = 1 but can be proved more simply there. For n = 2, they hold provided F, Fε are defined by F(x) = (2π)⁻¹ log|x| and Fε = (4π)⁻¹ log(|x|² + ε²).

15. Define G on n × by G(x, t) = (4πt)−n/2e−|x|²/4t χ(0, )(t).

a. (∂t − Δ)G = δ, where Δ is the Laplacian on n. (Let ; then Gε → G in D’. Compute for ϕ Cc∞, recalling the discussion of the heat equation in §8.7.)

b. If ϕ Cc∞(n × ), the function f = G * ϕ satisfies (∂t − Δ)f = ϕ.

9.2 COMPACTLY SUPPORTED, TEMPERED, AND PERIODIC DISTRIBUTIONS

If U is an open set in n, the space of all distributions on U whose support is a compact subset of U is denoted by ’(U); as usual, we set ’ = ’(n). (U) turns out to be a dual space in its own right, as we shall now show.

The space C∞(U) of C∞ functions on U is a Fréchet space with the C∞ topology — that is, the topology of uniform convergence of functions, together with all their derivatives, on compact subsets of U. This topology can be defined by a countable family of seminorms as follows. Let {Vm}₁∞ be an increasing sequence of precompact open subsets of U whose union is U, as in Proposition 4.39; then for each m and each multi-index α we have the seminorm

(9.6) equation

Clearly uniformly on compact sets for all α iff ||fj − f||[m,α]→ 0 for all m, α; a different choice of sets Vm would yield an equivalent family of seminorms.

9.7 Proposition. Cc∞(U) is dense in C∞ (U).

Proof. Let {Vm}₁∞ be as in (9.6). For each m, by the C∞ Urysohn lemma we can pick m Cc∞(U) with m = 1 on m. If ϕ C∞(U), clearly ||mϕ–ϕ||[m₀,α] = 0 provided m ≥ m₀; thus mϕ → ϕ in the C∞ topology.

9.8 Theorem. ’(U) is the dual space of Cc∞(U). More precisely: If F ’(U), then F extends uniquely to a continuous linear functional on C∞(U); and if G is a continuous linear functional on C∞(U), then G|Cc∞(U) ε’(U).

Proof. If F ε’(U), choose Cc∞(U) with = 1 on supp(F), and define the linear functional G on C∞(U) by (G, ϕ) = . Since F is continuous on Cc∞(supp()), and the topology of the latter is defined by the norms ϕ ||∂αϕ||u, by Proposition 5.15 there exist N and C > 0 such that for ϕ C∞(U). By the product rule, if we choose m large enough so that supp() Vm, this implies that

so that G is continuous on C∞(U). That G is the unique continuous extension of F follows from Proposition 9.7.

On the other hand, if G is a continuous linear functional on C∞(U), by Proposition 5.15 there exist C, m, N such that for all ϕ C∞(U). Since , this implies that G is continuous on Cc∞(K) for each compact K U, so G|Cc∞(U) D′(U). Moreover, if [supp(ϕ)] , then ; hence supp(G) and .

The operations of differentiation, multiplication by functions, translation, and composition by linear maps discussed in §9.1 all preserve the class ε. As for convolution, there is more to be said.

First, if F ε’ and then F * ϕ Cc∞, as Proposition 8.6d remains valid in this setting. Second, if F ε’ and C∞, F * can be defined as a C∞ function or as a distribution just as before:

(see Exercise 16). Finally, a further dualization allows us to define convolutions of arbitrary distributions with compactly supported distributions. To wit, if F D′ and G ε’, we can define F * G D′ and G * F D′ as follows:

and likewise for . The proof that F * G and G * F are indeed distributions (i.e., that they are continuous on Cc∞) and that F * G = G * F requires a closer examination of the continuity of the maps involved. We shall not pursue this matter here; however, see Exercises 20 and 21.

A notable omission from our list of operations that can be extended from functions to distributions is the Fourier transform F. The trouble is that F does not map Cc∞ into itself; in fact, if ϕ Cc∞, then cannot vanish on any nonempty open set unless ϕ = 0. To see this, suppose on a neighborhood of ξ₀. Replacing ϕ by , we may assume that ξ₀ = 0. Since ϕ has compact support, we can expand in its Maclaurin series and integrate term by term to obtain

(see Exercise 2a in §8.1). But for all α by Theorem 8.22d. These derivatives all vanish by assumption, so and hence ϕ = 0.

However, we do have available a slightly larger space of smooth functions that is mapped into itself by , namely, the Schwartz class . We recall that S is a Fréchet space with the topology defined by the norms

9.9 Proposition. Suppose Cc∞ and (0) = 1, and let ε(x) = (εx). Then for any . In particular, Cc∞ is dense in .

Proof. Given N , for any we can choose a compact set K such that (1 + |x|)N < for x K. Since (εx) → 1 uniformly for x K as ε → 0, it follows easily that ||ε ϕ – ϕ||(N,0) → 0 for every N. For the norms involving derivatives, we observe that by the product rule,

where Eε is a sum of terms involving derivative of ε. Since

we have ||Eε||u → Cε → 0 as ε → 0. The preceding argument then shows that .

A tempered distribution is a continuous linear functional on . The space of tempered distributions is denoted by ; it comes equipped with the weak* topology, that is, the topology of pointwise convergence on . If F , then F|Cc∞ is clearly a distribution, since convergence in Cc∞ implies convergence in , and F|Cc∞ determines F uniquely by Proposition 9.9. Thus we may, and shall, identify ’ with the set of distributions that extend continuously from Cc∞ to . We say that a locally integrable function is tempered if it is tempered as a distribution.

The condition that a distribution be tempered means, roughly speaking, that it does not grow too fast at infinity. Here are a few examples:

Every compactly supported distribution is tempered.

If f Lloc¹(n) and for some N, then f is tempered, for .

The function f(x) = eax on is tempered iff a is purely imaginary. Indeed, suppose a = b + ic with b, c real. If b = 0, then f is bounded and hence tempered by (ii). If , choose a function Cc∞ such that ∫ = 1, and let j(x) = e−ax(x − j). It is easily verified that ϕj → 0 in as j → + (if b > 0) or j → (if b < 0), but for all j.

On the other hand, the function f(x) = ex cos ex on is tempered, because it is the derivative of the bounded function sin ex. Indeed, if , integration by parts yields

Intuitively, f(x) is not too large “on average” when x is large, because of its rapid oscillations.

We turn to the consideration of the basic linear operations on tempered distributions. The operations of differentiation, translation, and composition with linear transformations work just the same way for tempered distributions as for plain distributions; these operations all map and ’ into themselves. The same is not true of multiplication by arbitrary smooth functions, however. The proper requirement on C in order for the map F → F to preserve and is that and all its derivatives should have at most polynomial growth at infinity:

Such C∞ functions are called slowly increasing. For example, every polynomial is slowly increasing; so are the functions (1 + |x|²)s (s ), which will play an important role in the next section.

As for convolutions, for any and we can define the convolution , as before, and we have an analogue of Proposition 9.3:

9.10 Proposition. If F and then F * is a slowly increasing C∞ function, and for any ϕ we have .

Proof. That F * ) is established as in Proposition 9.3. By Proposition 5.15, the continuity of F implies that there exist m, N, C such that

and hence by (8.12),

The same reasoning applies with replaced by is slowly increasing. Next, by Proposition 9.3 we know that the equation holds when ϕ, . By Proposition 9.9, if we can find sequences {ϕj} and {j} in Cc∞ that converge to ϕ and in . Then in by (the proof of) Proposition 8.11, so . On the other hand, the preceding estimates show that |F * j(x)| ≤ C(1 + |x|)m with C and m independent of j, and likewise |ϕj(x)| ≤ C(1 + |x|)−m-n−1, so by the dominated convergence theorem.

Finally, we come to the principal raison tre of tempered distributions, the Fourier transform. We recall (Corollary 8.23) that the Fourier transform maps continuously into itself, and that for f, g L¹ (in particular, for f, g ) we have

We can therefore extend the Fourier transform to a continuous linear map from ’ to itself by defining

This definition clearly agrees with the one in Chapter 8 when F L¹ + L².

The basic properties of the Fourier transform in Theorem 8.22 continue to hold in this setting. To wit,

(The first four of these formulas involve products of slowly increasing C∞ functions, specified by their values at a general point x or ξ, and tempered distributions.) The easy verifications of these facts are left to the reader (Exercise 17).

Moreover, we can define the inverse transform in the same way:

The Fourier inversion theorem formula then extends to ’:

so that , and likewise . Thus the Fourier transform is an isomorphism on ’.

If F ε’, there is an alternative way to define . Indeed, makes sense for any ϕ C∞, and if we take ϕ(x) = e−2πiξ·x, we obtain a function of ξ that has a strong claim to be called (ξ). In fact, the two definitions are equivalent:

9.11 Proposition. If F ε’, then is a slowly increasing C∞ function, and it is given by (ξ) = where Eξ(x) = e2πiξ·x.

Proof. Let g(ξ) = . Consideration of difference quotients of g, as in the proof of Proposition 9.3, shows that g is a C∞ function with derivatives given by . Moreover, by Theorem 9.8 and Proposition 5.15, there exist m, N, C such that

so g is slowly increasing.

It remains to show that g = , and by Proposition 9.9 it suffices to show that ) for ϕ Cc∞. In this case gϕ Cc∞, so ∫ gϕ can be approximated by Riemann sums as in the proof of Proposition 9.3, say . The corresponding sums and their derivatives in × converge uniformly, for x in any compact set, to and its derivatives. Therefore, since F is a continuous functional on ,

It is time for some examples. First and foremost, the Fourier transform of the point mass at 0 is the constant function 1: . More generally, for point masses at other points and their derivatives, we have

In particular:

9.12 Proposition. The Fourier transforms of the linear combinations of δ and its derivatives are precisely the polynomials.

The Fourier inversion theorem then yields the formulas for the Fourier transforms of polynomials and imaginary exponentials:

(9.13) equation

As an illustration of the heuristics associated to these results, consider the formula

Although this is nonsensical as a pointwise equality, it is valid when viewed from the right angle. One the one hand, it expresses the fact that the Fourier transform of the constant function 1 is δ. More interestingly, it is a concise statement of the Fourier inversion theorem. Indeed, if we replace x by x − y, integrate both sides against ϕ S, and reverse the order of integration on the left, we obtain

The integral on the left is , and the integral on the right equals ϕ(x)!

It is an important fact that every distibution is, at least locally, a linear combination of derivatives of continuous functions. The Fourier transform yields an easy proof of this:

9.14 Proposition.

a. If F ε’, there exist N , constants cα (|α| ≤ N), and f C₀(n) such that F = ∑|α|≤Ncα∂αf.

b. If F D’(U) and V is a precompact open set with U, there exist N, cα, f as above such that F = ∑|α|≤N cα ∂α f on V.

Proof. By Proposition 9.11, if F ε’ then is slowly increasing, so the function g(ξ) = (1 + |ξ|²)−M (ξ) will be in L¹ if the integer M is chosen sufficiently large. Let ; then and = (1 + |ξ|²)M , so . This proves (a); for (b), choose such that = 1 on V, and apply (a) to F.

We conclude this section with a sketch of the theory of periodic distributions; some of the details are fleshed out in Exercises 22–24.

The space of smooth periodic functions is a Fréchet space with the topology defined by the seminorms , and a distribution on n is a continuous linear functional on this space; the space of distributions on n is denoted by D’(n). If F D’(n), its Fourier transform is the function on n defined by (κ) = (F, E-κ) where Eκ(x) = e2πiκ·x. Since F satisfies an estimate of the form there exist C, N such mat

(9.15) equation

and the Fourier transform is an isomorphism from D’ (n) to the space of all functions on n satisfying such an estimate. Moreover, if F D’(n), the Fourier series ∑κ (κ) Eκ converges in D’(n) to F.

Instead of defining periodic distributions as distributions on n (linear functionals on , one can define them as distributions on n (linear functionals on that are invariant under the translations τκ, κ n. Accordingly, let

The periodization map Pϕ = ∑κn τκϕ used in Theorem 8.31 is easily seen to map continuously into , so it induces a map P’ : D’(n) → D’(n) given by ). Since P o τκ = P for κ n, we have τκ o P’ = P’, that is, the range of P’ lies in D’(n)per. In fact, P’ : D’(n) → D’(n)per is a bijection. (The proof is nontrivial; see Exercise 24.) Moreover, if f L¹(n), then f and P’ f coincide as periodic functions on , for if ,

Thus the two descriptions of periodic distributions are equivalent.

If F D’(n), the Fourier series ∑ (κ)Eκ converges in D’(n) to F; on the other hand, it follows easily from (9.15) that it also converges in S′(n), and its sum there is P’ f. Thus , and by (9.13) we have

giving the relation between the n- and n-Fourier transforms for periodic distributions. In particular, if F = δn, the point mass at the origin in n, then (κ) = 1 for all κ; hence P’F and (P’F)^ are both equal to ∑ τκδ — a restatement of the Poisson summation formula.

Exercises

16. Suppose . Show that for any . (The result can be reduced to Proposition 9.3; given F and ϕ, the indicated expressions depend only on the values of in a compact set.)

17. Suppose that F S′. Show that

a.

b.

c.

d.

18. If n = l + m, let us write x n as (y, z) with y l and z m. Let F denote the Fourier transform on n and F₁, F₂ the partial Fourier transforms in the first and second sets of variables — i.e., dy and likewise for ₂. Then ₁ and ₂ are isomorphisms on S(n) and S′(n), and = ₁, ₂ = ₂₁.

19. On , let F₀ = PV(1/x) as defined in Exercise 10. Also, for > 0 let and S(x) = e−2π|x| sgn x.

a. lim→0 F = F₀ in the weak* topology of S’. (Theorem 8.14, with a = 0, may be useful.)

b. (Hint:

c. = (πi)⁻¹ F and hence .

d. From (c) it follows that ₀ = −πisgn. Prove this directly by showing that , where H,N(x) = x⁻¹ if < |x| < N and H,N(x) = 0 otherwise, and using Exercise 59b in §2.6.

e. Compute (0 by writing and using (c), (ii) by writing = lim (x) and using (b).

20. Suppose that F S’ and G ε’.

a. is well-defined element of S’.

b. If S, then G * S.

c. Let F * G (or G * F) be the tempered distribution such that (F * G)^ = . Then

21. Suppose that F, G, H S’.

a. If at most one of F, G, H has noncompact support, then (F * G) * H = F * (G * H), where the convolutions are defined as in Exercise 20.

b. On , let F be the constant function 1, G = dδ/dx, and . Then (F * G) * H and F * (G * H) are well defined in S’ but are unequal.

22. Let Eκ(x) = e2πiκ·x. If g : n → satisfies |g(κ)| ≤ C(1 + |κ|)N for some C, N > 0, then the series ∑κn (κ)Eκ converges in D’(n) to a distribution F that satisfies = g. It also converges in S’(n) to a tempered distribution G (= P’F) such that τκG = G for all κ.

23. Suppose that F, G D’(n).

a. There is a unique F * G D’(n) such that (F * G)^ = . (Use Exercise 22.)

b. If G , then and as on n.

24. Let P be the periodization map, Pϕ = ∑κn τκϕ.

a. P is a continuous linear map from to . (Note that for ϕ Cc∞ and x in a compact set, only finitely many terms of the series ∑ τκϕ(x) are nonzero.)

b. Choose γ Cc∞ with , and let ω = γ * χ[0,1)n. Then ω Cc∞ and Pω = 1.

c. If , then = P(ω) (where is regarded as a function on n on the left and as a function on n on the right). Consequently, P : is surjective and the dual map P′: D′(n) → D′(n)per is injective.

d. Given G D′(n)per, define F D’(n) by (with the same understanding as in part (c)). Then P’F = G, so P′ maps D′(n) onto D’(n)per.

25. Suppose that P is a polynomial in n variables such that only zero of P(ξ) in n is ξ = 0, and let P(D) be as in §8.7.

a. Every tempered distribution F that satifies P(D)F = 0 is a polynomial. (Use Proposition 9.12 and Exercise 11.)

b. Every bounded function f that satisfies P(D)f = 0 is a constant. (This result, for the special cases where P(D) is the Laplacian or the Cauchy-Riemann operator on ², is known as Liouville’s theorem.)

26. On n × , let G(x, t) = .

a. is the tempered function . (Use Proposition 8.24 and Exercise 18.)

b. Deduce that . (Cf. Exercise 15.)

27. Suppose that 0 < Re α < n.

a. For any ϕ S,

(Hint: By Proposition 8.24 and Lemma 8.25, if t > 0 we have

Multiply both sides by t−1+(n-α)/2 dt and integrate from 0 to ∞.)

b. Let Rα(x) = Γ((n − α)/2)[Γ(α/2)2απn/2]⁻¹|x|α-n. Then Rα is a tempered function and αa is the tempered function α(ξ) = (2π|ξ|)−α.

c. If n > 2, then ΔR₂ = −δ. (Cf. Exercise 14. See the next exercise for the case n = 2.)

28. Suppose n = 2. For 0 < Re α < 2, let cα = Γ((2 − α)/2)[Γ(α/2)2απ]⁻¹ and Qα(x) = cα(|ξ|α−2 − 1). (Note that Qα differs by a constant from the Rα in Exercise 27.)

a. limα→2 Qα(x) = −(2π)⁻¹ log |x|, pointwise and in S’.

b. By (a), limα→2 α exists in S, and by Exercise 27b, α(ξ) = (2π|ξ|)−α − cαδ. Noting that (2π|ξ|)⁻² is not integrable near the origin and that , find an explicit formula for . (Exercise 12 may help.)

29. For 1 be the set of all F S′ for which there exists C ≥ 0 such that ||F * ϕ||p ≤ C||ϕ||p for all ϕ S, so that the map extends to a bounded operator on Lp.

a. . (If F , consider F * ϕt where {ϕt} is an approximate identity, and apply Alaoglu’s theorem.)

b. . (Use the Plancherel theorem.)

c. If p and q are conjugate exponents, then . (Hint: .)

d. If 1 ≤ p ≤ 2 and q is the conjugate exponent to p, then for all r (p, q). (Use the Riesz-Thorin theorem.)

e. for all .

9.3 SOBOLEV SPACES

One of the most satisfactory ways of measuring smoothness properties of functions and distributions is in terms of L² norms. There are two reasons for this: L² has the advantage of being a Hilbert space, and the Fourier transform, which converts differentiation into multiplication by the coordinate functions, is an isometry on L².

As a first step, suppose k and let Hk be the space of all functions f L²(n) whose distribution derivatives ∂αf are L² functions for |α| ≤ k. One can make Hk into a Hilbert space by imposing the inner product

However, it is more convenient to use an equivalent inner product defined in terms of the Fourier transform. Theorem 8.22e and the Plancherel theorem imply that f Hk iff ξα L² for |α| ≤ k. A simple modification of the argument in the proof of Proposition 8.3 shows that there exist C₁, C₂ > 0 such that

from which it follows that f Hk iff (1 + |ξ|²)k/2 L² and that the norms

are equivalent. The latter norm, however, makes sense for any k , and we can use it to extend the definition of Hk to all real k.

We proceed to the formal definitions. For any s the function is and slowly increasing (Exercise 30), so the map λs defined by

is a continuous linear operator on S′ — actually an isomorphism, since λs⁻¹ = λ-s. If s , we define the Sobolev space Hs to be

and we define an inner product and norm on Hs by

(The equality of the two formulas for and for ||f||(s) follows from the Plancherel theorem.) Note that the inner products are conjugate linear in the second variable, but we are continuing to use the notation for the bilinear pairing between S′ and S. This will cause no confusion, since we shall not be using the inner products ·, ·(s) explicitly.

The following properties of Sobolev spaces are simple consequences of the definitions and the preceding discussion:

i. The Fourier transform is a unitary isomorphism from Hs to L²(n, μs) where dμs(ξ) = (1 + |ξ|²)s dξ. In particular, Hs is a Hilbert space.

ii. S is a dense subspace of Hs for all s . (This follows easily from (i) and Proposition 8.17.)

iii. If t < s, Hs is a dense subspace of Ht in the topology of Ht, and ||·||(t) ≤ ||·||(s).

iv. λt is a unitary isomorphism from Hss to Hs-t for all s, t .

v. H₀ = L² and ||·||₍₀₎ = ||·||₂ (by Plancherel).

vi. ∂α is a bounded linear map from Hs to Hs-|α| for all s, α (because |ξ|α ≤ (1 + |ξ|²)|α|/2).

By (iii) and (v), for s ≥ 0 the distributions in Hs are L² functions. For s < 0 the elements of Hs are generally not functions. For example, the point mass δ is in Hs iff s < −1/2n, for is the constant function 1, and iff s < −1/2n. Another example: The distribution Wt whose Fourier transform is (2π|ξ|)⁻¹ sin 2πt|ξ|, which arose in the discussion of the wave equation in §8.7, is in Hs iff s < 1 − 1/2n; it is in L¹ L² when n = 1 and in L¹ \ L² for n = 2, but is not a function for n ≥ 3.

9.16 Proposition. If s , the duality between S’ and S induces a unitary isomorphism from H-s to (Hs)*. More precisely, if f H-s, the functional on 8 extends to a continuous linear functional on Hs with operator norm equal to ||f||(−s), and every element of (Hs)* arises in this fashion.

Proof. If f H-s and ϕ S,

since f(ξ) = (−ξ) is a tempered function. Thus by the Schwarz inequality,

so the functional extends continuously to HS, with norm at most ||f||(−s). In fact, its norm equals ||f||(−s). since if g S′ is the distribution whose Fourier transform is we nave g Hs and

Finally, if G (Hs)*, then G o F⁻¹ is a bounded linear functional on L²((μs) where dμs(ξ) = (1 + |ξ|²)s dξ, so there exists g L²(μs) such that

But then where , and f H-s since

For s > 0, the elements of Hs are L² functions that are “L²-differentiable up to order s,” and it is natural to ask what is the relationship between this notion of smoothness and ordinary differentiability. Of course, if one thinks of elements of Hs as distributions or elements of L², there is no distinction among functions that agree almost everywhere; from this perspective, when one says that a function in Hs is of class Ck, one means that it agrees a.e. with a Ck function. With this understanding, the question just posed has a simple and elegant answer. We introduce the notation

C₀k is a Banach space with the Ck norm .

9.17 The Sobolev Embedding Theorem. Suppose s .

a. If f Hs then (∂α f)^ L¹ and ||(∂α f)^||₁H ≤ C||f||(s) for |α| ≤ k, where C depends only on k − s.

b. Hs ⊂ C₀k, and the inclusion map is continuous.

Proof. By the Schwarz inequality,

The first factor on the right is ||f||(s), and the second one is finite by Corollary 2.52 since 2(k − s) < −n. This proves (a), and (b) follows by the Fourier inversion theorem and the Riemann-Lebesgue lemma.

9.18 Corollary. If f Hs for all s, then f C∞.

An example may help to elucidate this theorem. Let fλ(x) = ϕ(x)|x|λ, where λ and ϕ Cc∞ with ϕ = 1 on a neighborhood of 0. Then the (classical) derivative ∂α fλ is C∞ except at 0 and is homogeneous of degree λ − |α| near 0, so that |∂α fλ| ≤ Cα, λ|x|λ-|α|, and in particular ∂α fλ L¹ provided λ − |α| > −n. In this case ∂α fλ, as an L¹ function, is also the distribution derivative of fλ. (To see this, replace fλ by the C∞ function ϕ(x)(|x|² + ²)λ/2 and consider the limit as → 0.) Moreover, ∂α fλ L² iff λ − |α| > −1/2n, so f Hk (k = 0,1,2….) iff λ > k − 1/2n, whereas fλ C₀k iff λ > k. See also Exercises 33–35 for some related results.

Next, we show that multiplication by suitably smooth functions preserves the Hs spaces. We need a lemma:

9.19 Lemma. For all and s ,

Proof. Since , we have and hence

If s ≥ 0, we have merely to raise both sides to the sth power. If s < 0, we interchange ξ and and replace s by − s, obtaining

which is again the desired result.

9.20 Theorem. Suppose that ϕ C₀(n) and that is a function that satisfies

for some a > 0. Then the map Mϕ(f) = ϕf is a bounded operator on Hs for |s| ≤ a.

Proof. Since Λs is a unitary map from Hs to H₀ = L², it is equivalent to show that ΛsMϕΛ−s is a bounded operator on L². But

where

By Lemma 9.19,

so if |s| ≤ a, then are bounded by 2a/2C. That ΛsMϕΛ−s is bounded on L² therefore follows from the Plancherel theorem and Theorem 6.18.

9.21 Corollary. If ϕ S, then Mϕ is a bounded operator on Hs for all s

Our next result is a compactness theorem that is of great importance in the applications of Sobolev spaces.

9.22 Rellich’s Theorem. Suppose that {fk} is a sequence of distributions in Hs that are all supported in a fixed compact set K and satisfy . Then there is a subsequence {fkj} that converges in Ht for all t < s.

Proof. First we observe that by Proposition 9.11, k is a slowly increasing C∞ function. Pick ϕ Cc∞ such that ϕ = 1 on a neighborhood of K. Then fk = ϕ fk, so where the convolution is denned pointwise by an absolutely convergent integral. By Lemma 9.19 and the Schwarz inequality,

Likewise, since , we see that (1 + || is bounded by a constant independent of ξ, j, and k. In particular, the k’s and their first derivatives are uniformly bounded on compact sets, so by the mean value theorem and the Arzelà-Ascoli theorem there is a subsequence {kj} that converges uniformly on compact sets.

We claim that {fkj} is Cauchy in Ht for all t < s. Indeed, for any R > 0 we can write the integral

as the sum of the integrals over the regions |ξ| ≤ R and |ξ| > R. For |ξ| ≤ R we use the estimate

and for |ξ| > R we use the estimate

which yield

Given > 0, the second term will be less than 1/2 provided R is chosen sufficiently large, since t − s < 0; once such an R is fixed, the first term will less than 1/2 provided i and j are sufficiently large. The proof is therefore complete.

Although the definition of Sobolev spaces in terms of the Fourier transform entails their elements being defined on all of n, these spaces can also be used in the study of local smoothness properties of functions. The key definition is as follows: If U is an open set in n, the localized Sobolev space Hsloc(U) is the set of all distributions f D’(U) such that for every precompact open set V with there exists g Hs such that g = f on V.

9.23 Proposition. A distribution f D’(U) is in Hsloc(U) iff ϕf Hs for every ϕ Cc∞(U).

Proof. If f Hsloc(U) and ϕ Cc∞(U), then f agrees with some g Hs on a neighborhood of supp(ϕ); hence ϕf = ϕg Hs by Corollary 9.21. For the converse, given a precompact open V with U, we can choose ϕ Cc∞(U) with on a neighborhood of by the C∞ Urysohn lemma; then ϕf Hs and ϕf = f on V. (We have implicitly used Proposition 4.31 to obtain compact neighborhoods of supp ϕ and in U.)

We conclude this section with one of the classic applications of Sobolev spaces, a regularity theorem for certain partial differential operators.

If L = ∑₀m aj (d/dx)j is an ordinary differential operator with C∞ coefficients such that am never vanishes, it is not hard to show that smooth data give smooth solutions. More precisely, if Lu = f and f is Ck on an open interval I, then u is Ck+m on I. No such result holds for partial differential operators in general. For example, for any f Lloc¹() the function u(x, t) = f(x − t) satisfies the wave equation ()u = 0, but u has only as much smoothness as f. However, there is a large class of differential operators for which a strong regularity theorem holds. We restrict attention to the constant-coefficient case, although the results are valid in greater generality.

Let P(D) = (notation as in §8.7) be a constant-coefficient operator. We assume that m is the true order of P(D), i.e., that cα ≠ 0 for some α with |α| = m. The principal symbol Pm is the sum of the top-order terms in its symbol:

P(D) is called elliptic if Pm(ξ) ≠ 0 for all nonzero ξ n. Thus, ellipticity means that, in a formal sense, P(D) is genuinely mth order in all directions. (For example, the Laplacian Δ is elliptic on n, whereas the heat and wave operators ∂t − Δ and ∂t² − Δ are not elliptic on n+1.)

9.24 Lemma. Suppose that P(D) is of order m. Then P(D) is elliptic iff there exist C, R > 0 such that |P(ξ)| ≥ C|ξ|m when |ξ| ≥ R.

Proof. If P(D) is elliptic, let C₁ be the minimum value of the principal symbol Pm on the unit sphere |ξ| = 1. Then C₁ > 0, and since Pm is homogeneous of degree m, we have |Pm(ξ)| ≥ C₁|ξ|m for all ξ. On the other hand, P − Pm is of order m − 1, so there exists C₂ such that |P(ξ) − Pm(ξ)| ≤ C₂|ξ|m−1. Therefore,

Conversely, if P(D) is not elliptic, say Pm(ξ₀) = 0, then |P(ξ)| ≤ C|ξ|m−1 for every scalar multiple ξ of ξ₀.

9.25 Lemma. If P(D) is elliptic of order m, u Hs, and P(D)u Hs, then u Hs+m.

Proof. The hypotheses say that (1 + |ξ|²)s/2 L² and (1 + |ξ|²)s/2 P L². By Lemma 9.24, for some R > 1 we have

and . If follows that

that is, u Hs+m.

9.26 The Elliptic Regularity Theorem. Suppose that L is a constant-coefficient elliptic differential operator of order m, Ω is an open set in n, and u D’(Ω). If Lu Hsloc(Ω) for some s , then u Hs+mloc(Ω); and if Lu C∞(Ω), then u C∞(Ω).

Proof. The second assertion follows from the first in view of Corollary 9.18, so by Proposition 9.23 we must show that if Lu Hsloc(Ω) and ϕ Cc∞(Ω), then ϕu Hs+m. Let V be a precompact open set such that supp(ϕ) , and choose Cc∞(Ω) such that = 1 on . Then u ε’, so it follows from Proposition 9.11 that u Hσ for some σ . By decreasing σ we may assume that s + m − σ is a positive integer k. Set ₀ = and k = ϕ, and choose recursively ∞ such that j = 1 on a neighborhood of supp(ϕ) and supp(j) is contained in the set where j−1 = 1. We shall prove by induction that ju Hσ+j. When j = k, we obtain ϕu = ku Hσ+k = Hm, which will complete the proof.

The crucial observation is that for any the operator [L, ζ] defined by

is a differential operator of order m − 1 whose coefficients are linear combinations of derivatives of ζ; in particular, these coefficients are C∞ functions that vanish on any open set where ζ is constant. (This follows from the product rule for derivatives.) Thus, if f Ht, we have ∂α f Ht-(m-1) for |α| ≤ m − 1 and hence [L, ζ]f Ht-(m-1) by Theorem 9.20.

For j = 0 we have ₀u Hσ by assumption. Suppose we have established that ju Hσ+j, where 0 ≤ j < k. Then by the preceding remarks,

Since j+1u = j+1ju Hσ+j, Lemma 9.25 (with P(D) = L) implies that j+1u Hσ+j+1, and we are done.

Two classical special cases of this theorem are particularly noteworthy. First, every distribution solution of Laplace’s equation Δu = 0 is a C∞ function. (This fact is known as Weyl’s lemma.) Second, if on ², the equation Lu = 0 is the Cauchy-Riemann equation, whose solutions are the holomorphic (or analytic) functions of z = x₁ + ix₂. We thus recover the fact that holomorphic functions are C∞.

Exercises

30. Let fs(ξ) = (1 + |ξ|²)s/2. Then ||.

31. If k , Hk is the space of all f L² that possess strong L² derivatives ∂α f, as defined in Exercise 8 in §8.2, for |α| ≤ k; and these strong derivatives coincide with the distribution derivatives.

32. Suppose r < s < t. For any > 0 there exists C > 0 such that ||f||(s) ≤ ||f||(t) + C||f||(r) for all f Ht.

33. (Converse of the Sobolev Theorem) If Hs C₀k, then s > k + 1/2n. (Use the closed graph theorem to show that the inclusion map Hs → C₀k is continuous and hence that ∂αδ (Hs)* for |α| ≤ k.)

34. (A Sharper Sobolev Theorem) For 0 < α < 1, let

a. If s = n + α where 0 < α < 1, then . (We have . Write the integral defining ||τxδ − τyδ||(−s)²) as the sum of the integrals over the regions |ξ| ≤ R and |ξ| > R, where R = |x − y|⁻¹, and use the mean value theorem to estimate on the first region.)

b. If s = n + α where 0 < α < 1, then .

c. If s = n + k + α where k and 0 < α < 1, then

35. The Sobolev theorem says that if s > 1/2n, it makes sense to evaluate functions in Hs at a point. For 0 ≤ s ≤ 1/2n, functions in Hs are only defined a.e., but if s > 1/2k with k < n, it makes sense to restrict functions in Hs to subspaces of codimension k. More precisely, let us write , and define R : S(n) → S(n-k) by R f(y) = f(y, 0).

a. (See Exercise 20 in §8.3.)

b. If s > k,

c. R extends to a bounded map from Hs(n) to Hs-(k/2)(n-k) provided s > 1/2k.

36. Suppose that 0 ≠ ϕ Cc∞ and {aj} is a sequence in n with |aj| → , and let ϕj(x) = ϕ(x − aj). Then {ϕj} is bounded in Hs for every s but has no convergent subsequence in Ht for any t.

37. The heat operator ∂t − Δ is not elliptic, but a weakened version of Theorem 9.26 holds for it. Here we are working on n+1 with coordinates (x, t) and dual coordinates (ξ, τ), and ∂t − Δ = P(D) where P(ξ, τ) = 2πiτ + 4π²|ξ|².

a. There exist C, R > 0 such that |ξ||(ξ, τ)|1/2 ≤ C|P(ξ, τ)| for |(ξ, τ)| > R. (Consider the regions |τ| ≤ |τ|² and |τ| ≥ |τ|² separately.)

b. If f Hs and , then f Hs+1 and ∂xif Hs+(1/2) for 1 ≤ i ≤ n.

c. If ζ Cc∞(n+1), we have

d. If Ω is open in n+1, u D’(Ω), and (∂t − Δ)u Hsloc(Ω), then u Hs+1loc(Ω). (Let j be as in the proof of Theorem 9.26. Show inductively that if ₀u Hσ, then ju Hσ+(j/2) and provided σ + 1/2j ≤ s.)

38. Suppose s₀ ≤ s₁ and t₀ ≤ t₁, and for 0 ≤ λ ≤ 1 let

If T is a bounded linear map from Hs₀ to Ht₀ whose restriction to Hs₁ is bounded from Hs₁ to Ht₁, then the restriction of T to Hsλ is bounded from Hsλ to Htλ for 0 ≤ λ ≤ 1. (T is bounded from Hs to Ht iff λsTλ-t is bounded on L². Observe that λz is well defined for all z and λz is unitary on every Hs if Re z = 0. Let s(z) = (1 − z)s₀ + zs₁, t(z) = (1 − z)t₀ + zt₁, and for 0 ≤ Re z ≤ 1 and ϕ, S let F(z) = ]. Apply the three lines lemma as in the proof of the Riesz-Thorin theorem.)

39. Let Ω be an open set in n, and let G : Ω → n be a C∞ diffeomorphism. For any ϕ Cc∞(G(Ω)), the map Tf = (ϕf) is bounded on Hs for all s; consequently, whenever f Hsloc(G(Ω)). Proceed as follows:

a. If s = 0, 1, 2, …, use the chain rule and the fact that f Hs iff ∂α f L² for |α| ≤ s.

b. Use Exercise 38 to obtain the result for all s > 0.

c. For s < 0, use Proposition 9.16 and the fact that the transpose of T is another operator of the same type, namely, T’ f = (f) where H = G⁻¹ and = (Jϕ) G with J(x) = | det DxG|.

40. State and prove analogues of the results in this section for the periodic Sobolev spaces

9.4 NOTES AND REFERENCES

The mathematical foundations for the theory of distributions were largely laid in the 1930s. On the one hand, several researchers in partial differential equations arrived at the notion of “weak derivatives” of functions; to wit, if f, g Lloc¹(U), g is the derivative ∂αf in the weak sense if for all ϕ in some suitable space of test functions. On the other, various attempts were made to extend the domain of the Fourier transform beyond L¹ + L². The idea of defining “generalized functions” as linear functionals on certain function spaces goes back to Sobolev [136], but it was Laurent Schwartz who systematically developed the theory of distributions and who introduced the spaces S and S’ as natural domains for the Fourier transform. (See Dieudonné [33] for a more detailed historical account.)

Rudin [126] contains a good concise introduction to the theory of distributions that includes some functional-analytic points we have elided, such as the definition of the topology on Cc∞(U) and the properties of convolution on D’ × ε’. More extensive treatments can be found in Gelfand and Shilov [55], Schwartz [132], and Treves [150]. Hörmander [77] contains an excellent full-scale treatment of distribution theory with a view toward its applications to differential equations.

§9.2: See Folland [49] for a case study of the analytic techniques used in manipulating distributions and their Fourier transforms and applying them to differential equations.

§9.3: The spaces originally considered by Sobolev [137] are the spaces Hkp of functions f Lp whose distribution derivatives ∂αf are in Lp for |α| ≤ k. When 1 , it turns out that Hkp = {f : λkf Lp} (although this is far from obvious when p ≠ 2), and this characterization of Hkp can be used to define Hsp for all s . Sobolev’s embedding theorem, in this setting, is that if s < n/p then Hsp Lq where q⁻¹ = p⁻¹ − n⁻¹s, and if s > k + n/p then Hsp Ck. See Stein [140, §§V.2–3]. Further results on Sobolev space and their applications can be found in Adams [1] and Lieb and Loss [93].

In Rellich’s theorem the hypothesis that K is compact can be replaced by the hypothesis that when s ≥ 0; see Lair [89].

A differential operator L = with C∞ coefficients is called elliptic on if for all x Ω and all nonzero ξ n. The elliptic regularity theorem remains true as stated for such operators; see Folland [48]. The Lp version of this theorem is also valid for but not for p = 1 or p = ∞. It is not true, except in dimension 1, that if Lu Ck(Ω) then u Ck+m(Ω), but if ∂αLu is not just continuous but Holder continuous of exponent λ (0 < λ < 1) for |α| = k, then u Ck+m(Ω) and ∂βu satisfies the same Hölder condition for |β| = k + m. See Taylor [147, Chapter XI].

Chapter 10

Topics in Probability Theory

Probability theory, originally conceived to analyze games of chance, has developed into a broadly useful discipline with deep connections to other branches of mathematics and many applications to other subjects such as physics, statistics, and economics. The mathematical study of probability began some two and a half centuries ago, but until the advent of modern analytic tools the theory was limited to combinatorial theorems involving discrete sample spaces and a few other results of somewhat doubtful rigor. It is now recognized that the fundamental datum in probability theory is a measure space (X, M, μ) such that μ(X) = 1; such a measure μ is called a probability measure. X is to be considered as the set of all possible outcomes of some process, such as an experiment or a gambling game, and the measure of a set E M is interpreted as the probability that the outcome lies in E. Although measure spaces are the natural setting for the study of probability, it is hardly accurate to say that probability theory is a branch of measure theory, for its central ideas and many of its techniques are distinctively its own.

This brief chapter is intended not as a systematic introduction to probability theory but rather as an advertisement for the subject; it also serves to illustrate further some results of previous chapters.

10.1 BASIC CONCEPTS

Probability theory has its own vocabulary, which is partly a legacy of its development before the connection with measure theory was made explicit and partly a result of the fact that the probabilistic point of view is different. We therefore begin by presenting a brief dictionary of probabilists’ dialect.

Analysts’ Term | Probabilists’ Term
Measure space (X, M, μ) (μ(X) = 1) | Sample space (Ω, B, P)
(σ)algebra | (σ)field
Measurable set | Event
Measurable real-valued function f | Random variable X
Integral of f, f dμ | Expectation or mean of X, E(X)
Lp [as adjective] | Having finite pth moment
Convergence in measure | Convergence in probability
Almost every(where), a.e. | Almost sure(ly), a.s.
Borel probability measure on | Distribution
Fourier transform of a measure | Characteristic function of a distribution
Characteristic function | Indicator function

Probabilists have an aversion to displaying the arguments of random variables. For example, {ω : X(ω) > a} and P({ω : X(ω) > a}) are commonly written as {X > a} and P(X > a).

Henceforth we shall, for the most part, adopt probabilistic language in this chapter, although we shall use the term “Lp random variable” in preference to the more cumbersome “random variable with finite pth moment.” One more standard piece of terminology, which has no equivalent in classical analysis, is the following: If X is a random variable, its variance σ²(X) and standard deviation σ(X) are defined by

If X L², then σ²(X) = ∞. If X L², then E[(X−a)²] = E(X²)-2aE(X)+a² is a quadratic function of a whose minimum occurs when a = E(X); hence

σ(X) is a measure of how widely X deviates from its mean E(X).

At this point we must discuss a general measure-theoretic construction. Let (Ω, , P) be a probability space (or, for that matter, an arbitrary measure space), let (Ω’ ′) be another measurable space, and let ϕ : Ω → Ω’ be a (, B′)-measurable map. Then the measure P induces an image measure Pϕ on Ω’ by

That this is indeed a measure follows from the fact that ϕ⁻¹ commutes with unions and intersections.

10.1 Proposition. With notation as above, if f : Ω’ → is a measurable function, then whenever either side is defined.

Proof. When f = χE with E B′, this is just the definition of Pϕ, since χE o ϕ = χϕ⁻¹ (E). The general result follows by taking linear combinations and limits.

If X is a random variable on Ω, then Pχ is a probability measure on , called the distribution of X, and the function

(which determines Pχ by Theorem 1.16) is called the distribution function of X. If {Xα}αA is a family of random variables such that Pχα = Pχβ for all α, β A, the Xα’s are said to be identically distributed.

More generally, for any finite sequence X₁, …, Xn of random variables, we can consider (X₁, …, Xn) as a map from Ω to n, and the measure P(X₁, …, Xn) on n is called the joint distribution of X₁, …, Xn. It is a general principle that all properties of random variables that are relevant to probability theory can be expressed in terms of their joint distributions. For example, by Proposition 10.1,

In fact, given a Borel probability measure λ on , one can simply speak of the mean and variance σ² of λ,

which are the mean and variance of any random variable with distribution λ.

One of the most important concepts in probability theory, and the one that most clearly sets it apart from general measure theory, is that of (stochastic) independence. To motivate this idea, consider a probability space (Ω, , P) and an event E such that P(E) > 0. Then the set function is a probability measure on Ω called the conditional probability on E; PE(F) represents the probability of the event F given that E occurs. If PE(F) = P(F), that is, if the probability of F is the same whether or not we restrict to E, then F is said to be independent of E. Thus, F is independent of E iff moreover, the latter condition is clearly symmetric in E and F and makes sense even if P(E) = 0.

With this in mind, we define a collection {Eα}αA of events in Ω to be independent if

(For the events Eα to be independent it does not suffice for them to be pairwise independent; see Exercise 1.)

A collection {Xα}αA of random variables on Ω is called independent if the events {Xα α} = Xα⁻¹(α) are independent for all Borel sets . This condition can be neatly rephrased as follows. Observe that if α₁, …, αn A and we write Xj = Xαj, we have

whereas

These quantities are equal for all Borel sets iff

That is, {Xα}αA is an independent set of random variables iff the joint distribution of any finite set of Xα’s is the product of their individual distributions.

The following proposition expresses the fact that functions of independent random variables are independent.

10.2 Proposition. Let {Xnj : 1 ≤ j ≤ J(n), 1 ≤ n ≤ N} be independent random variables, and let fn : J(n) ≤ be Borel measurable for 1 ≤ n ≤ N. Then the random variables Yn = fn(Xn1, …, XnJ(n)), 1 ≤ n ≤ N, are independent.

Proof. Let Xn = (Xn1, …, XnJ(n)). If ₁, …, N are Borel subsets of , we have Yn⁻¹ (n) = n⁻¹ (fn⁻¹ (n)) and hence

Therefore, by the independence of the Xnj’s and Fubini’s theorem,

We now present some fundamental properties of independent random variables. For the first one we need the notion of convolutions of measures on developed in §8.6. An easy induction on (8.47) shows that if λ₁, …, λn M(), then λ₁ * … * λn is given by

(10.3) equation

10.4 Proposition. If {Xj}₁n are independent random variables, then

and by (10.3), the last expression equals P₁ * … * Pn.

10.5 Proposition. Suppose that are independent random variables. If Xj L¹ for all j, then and E.

Proof. We have .

Hence

This proves the first assertion, and once this is known, the same argument (with the absolute values removed) proves the second one.

10.6 Corollary. If {Xj}₁n are independent and in L², then σ²(X₁ + · · · + Xn) = .

Proof. Let Yj = Xj − E(Xj). Then {Yj}n₁ are independent and have mean zero, so

Therefore,

These results show that independence is a very stringent property. For one thing, it is usually not the case that the product of two L¹ functions is in L¹. For another, suppose that X and Y are independent and E(X) = 0. Then for any Borel measurable function f on such that f o Y L¹ we have

In other words, X is orthogonal (in the L² sense) to every function of Y. This indicates that, for example, if one tries to construct a sequence of independent random variables on [0, 1] with Lebesgue measure by using the familiar functions of calculus, one will probably not succeed. (Perhaps the simplest example is Xn(x) = the nth digit in the decimal expansion of x; see Exercise 23.) Rather, the natural setting for independence is product spaces.

Indeed, suppose

Then any random variables X₁, …, Xn on Ω such that Xj depends only on the jth coordinate are independent, for if Xj = fj o πj where πj : Ω → Ωj is the coordinate map,

and hence

The same idea can be made to work for infinitely many factors; see §10.4.

As an application of these ideas, we present Bernstein’s constructive proof of the Weierstrass approximation theorem.

10.7 Theorem. Given f C([0, 1]), let

Then Bn → f uniformly on [0, 1] as n → ∞.

Proof. Given x [0, 1], let λ = xδ₁ + (1 − x)δ₀ where δt is the point mass at t. Let Ω = n, P = λ × · · · × λ, and Xj = the jth coordinate function on n. Then X₁, …, Xn are independent and have the common distribution λ. It is easy to check (Exercise 7) that

and hence, in view of Proposition 10.4,

Now, = 1 by the binomial theorem, so

(10.8) equation

Given ε > 0, by the uniform continuity of f on [0, 1] there exists δ > 0, independent of x and y, such that |f(x) − f(y)| ≤ ε whenever |x − y| ≤ δ. The sum of the terms in (10.8) such that |x − (k/n)| ≤ δ is at most ε, while the sum of the remaining terms is at most

But

so by Corollary 10.6,

and Chebyshev’s inequality therefore gives

which is less than 2ε provided n is sufficiently large.

Exercises

1. Let Ω consist of four points, each with probability 1/4. Find three events that are pairwise independent but not independent. Generalize.

2. Let {Xj} be a sequence of independent identically distributed positive random variables such that E(Xj) = α < ∞ and E(1/Xj) = b < ∞, and let Sn =

a. E(Xj/Sn) = 1/n if j ≤ n, and E(Xj/Sn) = aE(1/Sn) if j > n.

b. E(Sm/Sn) = m/n if m ≤ n and E(Sm/Sn) = 1 + (m − n)aE(1/Sn) if m > n.

3. Suppose that {Eα}αA is a collection of events in Ω.

a. If Eα₁, …, Eαn are independent, so are Eα₁, …, Eαn−1, Eαnc.

b. If {Eα} is an independent set, so is {Fα}, where each Fα is either Eα or Eαc.

c. {Eα} is an independent set of events iff {χEα} is an independent set of random variables.

4. Let X, Y, Z be positive independent random variables with a common distribution λ, and let F(t) = λ((0, t]). The probability that the polynomial Xt² + Yt + Z has real roots is

5. If X is a random variable with distribution dPX(t) = f(t)dt where f(t) = f(−t), then the distribution of .

6. For a, u > 0, let , the gamma distribution with parameters a and u.

a. The mean and variance of γa, u are a/u and a/u², respectively.

b. γa, u * γb,u = γa+b, u. (Use Exercise 60 in §2.6.)

c. If X₁, …, Xn are independent and all have the distribution dPX (t) = , then X₁² + · · · + Xn² has the distribution γn/2,1/2. (Use (b) and Exercise 5. γn/2,1/2 is called the chi-square distribution with n degrees of freedom.)

7. Let δt denote the point mass at t . Given 0 < p < 1, let βp = pδ₁ + (1 − p)δ₀, and let βp*n be the nth convolution power of βp. Then

and the mean and variance of βp*n are np and np(1 − p). βp*n is called the binomial distribution on {0, …, n} with parameter p.

8. Let δt denote the point mass at t . Given a > 0, let the Poisson distribution with parameter a.

a. The mean and variance of λa are both equal to a.

b. λa * λb = λa+b.

c. The binomial distribution (Exercise 7) converges vaguely to λa as n → ∞. (Use Proposition 7.19.)

9. Suppose that is a sequence of random variables. If Xn → X in probability, then PXn → PX vaguely. (Use Proposition 7.19.)

10. (The Moment Convergence Theorem) Let X₁, X₂, …,, X be random variables such that PXn → PX vaguely and sup, where r > 0. Then E(|Xn|s) → E(|X|s) for all s (0, r), and if also s , then . (By Chebyshev’s inequality, if ε > 0, there exists a > 0 such that P(|Xn| > a) < ε for all n. Consider where ϕ Cc() and ϕ(t) = 1 for |t| ≤ a.)

10.2 THE LAW OF LARGE NUMBERS

If one plays a gambling game many times, one’s average winnings or losses per game should be roughly the the expected winnings or losses in each individual game; more generally, if one plays a sequence of possibly different games, one’s average winnings or losses should be roughly the average of the expected winnings or losses in the individual games. In symbols: If is a sequence of independent random variables and E(Xj) = μj, then the average should be close to the constant when n is large.

The law of large numbers is a precise formulation of this idea. It comes in several versions, depending on the hypotheses one wishes to make. The first version, with the weakest hypotheses and conclusions, has a very simple proof.

10.9 The Weak Law of Large Numbers. Let {Xj}₁∞ be a sequence of independent L² random variables with means {μj} and variances as n → ∞, then in probability as n → ∞.

Proof. has mean 0 and variance (the latter by Corollary 10.6). Hence by Chebyshev’s inequality, for any > 0 we have

Under slightly stronger hypotheses, we can obtain the sharper conclusion that 0 almost surely. To establish this, we need the following two lemmas, which are of interest in their own right.

10.10 The Borel-Cantelli Lemma. Let be a sequence of events.

a. If , then P(lim sup An) = 0.

b. If the An’s are independent and , then P(lim sup An) = 1.

Proof. We recall that lim sup , so that

and the latter sum tends to zero as k → ∞ if Σ P(An) converges. On the other hand, suppose that Σ P(An) diverges and the An’s are independent. We must show that

and for this it is enough to show that for all k. But the Anc’s are independent (Exercise 3), so since 1 − t ≤ e−t,

The last expression tends to zero as K → ∞, which yields the desired result.

10.11 Kolmogorov’s Inequality. Let X₁, …, Xn be independent random variables with mean 0 and variances σ₁², …, σn², and let Sk = X₁ + · · · + Xk. For any > 0,

Proof. Let Ak be the set where |Sj| < for j < k and |Sk| ≥ . Then the Ak’s are disjoint and their union is the set where max |Sk| ≥ , so

because . On the other hand,

It will suffice to show that E(χAk Sk(Sn − Sk)) = 0 for all k, for then we have

by Corollary 10.6, since the Xk’s have mean zero. But χAk is a measurable function of S₁, …, Sk and hence of X₁, …, Xk, whereas Sn − Sk is a measurable function of Xk+1, …, Xn. Moreover, = for all k. Therefore, by Propositions 10.2 and 10.5,

10.12 Kolmogorov’s Strong Law of Large Numbers. If {Xn}₁∞ is a sequence of independent L² random variables with means {μn} and variances {σn²} such that then n⁻¹ almost surely as n → ∞.

Proof. Let . Given > 0, for k let Ak be the set where n⁻¹|Sn| ≥ for some n such that 2k−1 ≤ n < 2k. Then on Ak we have |Sn| ≥ 2k−1 for some n < 2k, so by Kolmogorov’s inequality,

Therefore,

so P(lim sup Ak) = 0 by the Borel-Cantelli lemma. But lim sup Ak is precisely the set where n⁻¹|Sn| ≥ for infinitely many n, so

Letting → 0 through a countable sequence of values, we conclude that n⁻¹ Sn → 0 almost surely.

The hypotheses of this theorem are a bit stronger than those of the weak law (Exercise 11). They are certainly satisfied when the Xn’s are identically distributed L² random variables, since then σn² is independent of n. However, in the identically distributed case the assumption that Xn L² can be weakened.

10.13 Khinchine’s Strong Law of Large Numbers. If {Xn}₁∞ is a sequence of independent identically distributed L¹ random variables with mean μ, then n⁻¹ Σ₁n Xj → μ almost surely as n → ∞.

Proof. Replacing Xn by Xn − μ, we may assume that μ = 0. Let λ be the common distribution of the Xj’s; we are thus assuming that

Let Yj = Xj on the set where |Xj| ≤ j and Yj = 0 elsewhere. Then

Since interchanging the order of summation yields

By the Borel-Cantelli lemma, then, with probability one we have Xj = Yj for j sufficiently large, and it therefore suffices to show that n⁻¹ Σ₁n Yj → 0 almost surely.

We have

and hence

Reversing the order of summation again and using the fact that (by comparison to ), we obtain

By Theorem 10.12, therefore, if μj = E(Yj) we have almost surely. However, by the dominated convergence theorem,

and it follows easily (Exercise 12) that also. Hence n⁻¹ Σ₁n Yj → 0 a.s., and the proof is complete.

Thus far we have not shown how to construct sequences of random variables that satisfy the hypotheses of the theorems in this section. We shall do so in §10.4.

Exercises

11. If , then (Estimate and separately.)

12. If and lim an = a, then lim

13. The weak law of large numbers remains valid if the hypothesis of independence is replaced by the (much weaker) hypothesis that = 0 for j ≠ k.

14. If {Xn} is a sequence of independent random variables such that E(Xn) = 0 and , then converges almost surely. (Apply Kolmogorov’s inequality to show that the partial sums are Cauchy a.s.) Corollary: If the plus and minus signs in are determined by successive tosses of a fair coin, the resulting series converges almost surely.

15. If {Xn} is a sequence of independent identically distributed random variables that are not in L¹, then lim almost surely. (Let An = {|Xn| > n}. Using an idea from the proof of Theorem 10.13, show that and apply the Borel-Cantelli lemma.)

16. (Shannon’s Theorem) Let {Xi} be a sequence of independent random variables on the sample space Ω, having the common distribution λ = Σ₁r pjδj where 0 < , and δj is the point mass at j. Define random variables Y₁, Y₂, … on Ω by

a. . (The notation is peculiar but correct: Xi(·) {1, …, r} a.s., so pX, is well-defined a.s.)

b. n⁻¹ log almost surely. (In information theory, the Xi’s are considered as the output of a source of digital signals, and – Σ₁r pj log pj is called the entropy of the signal.)

17. A collection or “population” of N objects (such as mice, grains of sand, etc.) may be considered as a smaple space in which each object has probability N⁻¹. Let X be a random variable on this space (a numerical characteristic of the objects such as mass, diameter, etc.) with mean μ, and variance σ². In statistics one is interested in determining μ and σ² by taking a sequence of random samples from the population and measuring X for each sample, thus obtaining a sequence {Xj} of numbers that are values of independent random variables with the same distribution as X. The nth sample mean is Mn = n⁻¹ Σ₁n Xj and the nth sample variance is Show that E(Mn) = μ, E(Sn²) = σ², and Mn → μ and Sn² → σ² almost surely as n → ∞. Can you see why one uses (n − 1)⁻¹ instead of n⁻¹ in the definition of Sn²?

10.3 THE CENTRAL LIMIT THEOREM

Suppose μ . and σ > 0. By Proposition 2.53 and some elementary calculus, the measure vμσ² on defined by

is a probability measure that satisfies

It is called the normal or Gaussian distribution with mean μ and variance σ². The special case v₀¹ is called the standard normal distribution.

It is a matter of empirical observation that normal and approximately normal distributions are extremely common in applied probability and statistics. The theoretical explanation for this phenomenon is the central limit theorem, the idea of which is as follows. Suppose that {Xj} is a sequence of independent identically distributed random variables with mean 0 and variance σ². Then n⁻¹ Σ₁n Xj has mean 0 and variance n⁻¹ σ², so there is a high probability that it is close to 0 when n is large; this is the content of the weak law of large numbers. On the other hand, n−1/2 Σ₁n Xj has mean 0 and variance σ² for all n, so one might ask if its distribution approaches some nontrivial limit as n → ∞. The remarkable answer is that no matter what the distribution of the Xj’s is, this limit exists and equals the normal distribution with mean 0 and variance σ².

The central limit theorem is really a theorem in Fourier analysis. We shall state it as such and then translate it into probablity theory.

10.14 Theorem. Let λ be a Borel probability measure on such that

(The finiteness of the first integral implies the existence of the second.) For n let λ*n = λ * · · · * λ (n factors) and define the measure λn by , where . Then λn → v₀¹ vaguely as n → ∞.

Proof. The hypotheses on the measure λ imply that its Fourier transform is of class C² and satisfies , and (Differentiate the integral twice as in Theorem 8.22d.) Thus by Taylor’s theorem,

where o(α) denotes a quantity that satisfies α⁻¹o(α) → 0 as α → 0. Moreover, , so by the obvious change of variable,

Thus, since log(1 + z) = z + o(z),

which tends to −2π²ξ² as n → ∞. In other words, for all ξ, so the conclusion follows from Propositions 8.24 and 8.50.

10.15 The Central Limit Theorem. Let {Xj} be a sequence of independent identically distributed L² random variables with mean μ and variance σ². As n → ∞, the distribution of converges vaguely to the standard normal distribution v₀¹ and for all a ,

Proof. Replacing Xj by σ⁻¹(Xj − μ), we may assume that μ = 0 and σ = 1. If λ is the common distribution of the Xj’s, then λ satisfies the hypotheses of Theorem 10.14, and in the notation used there, λn is the distribution of . The first assertion thus follows immediately, and the second one is equivalent to it by Proposition 7.19.

As the reader may readily verify, the same argument yields the following more general result. Under the hypotheses of the central limit theorem, if {Kn} is any sequence of finite subsets of such that kn = card(Kn) → ∞, then the distribution of (Xj − μ) converges vaguely to v₀¹.

Exercises

18. A fair coin is tossed 10,000 times; let X be the number of times it comes up heads. Use the central limit theorem and a table of values (printed or electronic) of erf(x) = dt to estimate

a. the probability that 4950 ≤ X ≤ 5050;

b. the number k such that |X − 5000| ≤ k with probability 0.98.

19. If {Xj} satisfies the hypotheses of the central limit theorem, the sequence does not converge in probability. (Use the remarks following the central limit theorem to show that {Y2k} is not Cauchy in probability.)

20. If {Xj} is a sequence of independent identically distributed random variables with mean 0 and variance 1, the distributions of

both converge vaguely to the standard normal distribution.

21. Let {Xn} be a sequence of independent random variables, each having the Poisson distribution with mean 1 (Exercise 8). Let

a. E(Yn) = (For me first equation, use Exercise 8b. As for the second, the sum telescopes.)

b. PYn converges vaguely to (Use Proposition 7.19.)

c. ≤ 1 for all n.

d. E(Yn) = . (Use Exercise 10.) Combining this with (a), one obtains Stirling’s formula:

22. In this exercise we consider random variables with values in the circle , regarded as {z : |z| = 1}. The distribution of such a random variable is a measure on .

a. If X₁, …, Xn are independent, then PX₁ X₂…Xn = PX₁ * · · ·* PXn.

b. If {Xj} is a sequence of independent random variables with a common distribution λ, the distribution of π₁n Xj converges vaguely to the uniform distribution on (= arc length over 2π) unless λ is supported on a finite subgroup Zm = of , in which case it converges to the uniform distribution on Zm. (Use Exercise 8.39.)

10.4 CONSTRUCTION OF SAMPLE SPACES

The preceding two sections have dealt with sequences of random variables whose joint distributions have certain properties. We now address the question of finding examples of such sequences, and more generally of constructing families {Xα}αA of random variables indexed by an arbitrary set A whose finite subfamilies have prescribed joint distributions.

If the index set A is finite, this is easy: given any Borel probability measure P on n, P is by definition the joint distribution of the coordinate functions X₁, …, Xn on the space (n, Bn, P). If A is infinite, however, the problem is more delicate. Suppose to begin with that {Xα}αA is a family of random variables on some sample space (Ω, B, P), and for each ordered n-tuple (α₁, …, αn) of distinct elements of A (n ) let Pα1,…,αn) be the joint distribution of Xα1, …, Xαn. Then the measures P(α₁, …, αn) satisfy the following consistency conditions:

(10.16) equation

(10.17) equation

Conversely, given any family of measures P(α₁, …, αn) satisfying (10.16) and (10.17), we shall show that there exist a sample space (Ω, , P) and random variables {Xα} on Ω such that P(α₁, …, αn) is the joint distribution of Xα₁, …, Xαn. To do this, it is convenient to make one minor technical modification: We replace by its one-point compactification * = {∞}. Any Borel measure on n can be regarded as a Borel measure on (*)n that assigns measure zero to (*)n \ n, and vice versa. In other words, we allow our random variables to assume the value ∞, although they will do so with probability zero. The point of this is that the space (*)A is compact for any A, by Tychonoff’s thoerem.

With this modification, the construction of the sample space (Ω, , P) in the case where the random variables Xα are independent, so that P should be the product of the Pα’s, is contained in Theorem 7.28. The general case is achieved by a simple adaptation of the argument given there, which we review in detail for the convenience of the reader.

10.18 Theorem. Let A be an arbitrary nonempty set, and suppose that for each ordered n-tuple of distinct elements of A(n ) we are given a Borel probability measure P(α₁, …, αn) on n, or equivalently on (*)n, satisfying (10.16) and (10.17). Then there is a unique Radon probability measure P on the compact Hausdorff space Ω = (*)A such that P(α₁, …, αn) is the joint distribution of Xα₁, …, Xαn, where Xα : Ω → * is the αth coordinate function.

Proof. Let CF (Ω) be the set of all f C(Ω) that depend only on finitely many coordinates. If f CF(Ω), say f(x) = F(xα₁, … xαn), let

I(f) is well defined because of (10.16) and (10.17): If we permute the variables or add some extra ones, the result is the same. Clearly I(f) ≥ 0 if f ≥ 0, and |I(f)| ≤ ||f||u with equality when f is constant.

Now, CF(Ω) is clearly an algebra that separates points, contains constant functions, and is closed under complex conjugation, so by the Stone-Weierstrass theorem it is dense in C(Ω). Hence, the functional I extends uniquely to a positive linear functional on C(Ω) with norm 1, and the Riesz representation theorem therefore yields a unique Radon measure P on Ω such that for all f C(Ω). Let Xα be the αth coordinate function on Ω and let P′(α₁ …, αn) be the joint distribution of Xα₁, …, Xαn on (*)n. If F C((*)n) and as above, then

But P′(α₁, …, αn) and P(α₁, …, αn) are both Radon measures by Theorem 7.8, so they are equal by the uniqueness of the Riesz representation.

The only property of * used in this proof is that it is a compact Hausdorff space in which every open set is σ-compact, so the theorem admits an obvious generalization. In particular, if for each α there is a compact set such that P(α₁ …, αn) is supported in Kα₁ × · · · × Kαn for all α₁, …, αn, we could take and thus avoid introducing the point at infinity.

Of special interest is the independent case, in which P(α₁, …, αn) = Pα₁ × · · · × Pαn. We state it as a corollary:

10.19 Corollary. Suppose {Pα}αA is a family of probability measures on . Then there exist a sample space (Ω, B, P) and independent random variables {Xα}αA on Ω such that Pα is the distribution of Xα for every α A. Specifically, we can take Ω to be (*)A and Xα to be the αth coordinate function; if Pα is supported in the compact set K for each α, we can take Ω to be .

Exercises

23. Given b \ {1}, let B = {0, 1, …. b − 1}, and let P₀ be the probability measure on B (or ) that assigns measure b⁻¹ to each point in B. Let P be the measure on Ω = B given by Corollary 10.19, where A = and Pn = P₀ for all n , and let {Xn}₁∞ be the coordinate functions on Ω.

a. If A₁, …, An ⊂ B,

and P({ω}) = 0 for all ω Ω.

b. Let for infinitely many n}. Then is countable and P(Ω’) = 1.

c. Define F : Ω → [0, 1] by (so F(ω) is the number such that {Xn(ω)} is the sequence of digits in its base b decimal expansion). Then F|Ω’ is a bijection from Ω’ to (0, 1] that maps BΩ’ bijectively onto B(0,1]. (BΩ’ is generated by sets of the form Xn⁻¹(A) . The image under F of such a set is a finite union of intervals of the form , and these sets generate )

d. The image measure of P under F is Lebesgue measure.

e. (Borel’s Normal Number Theorem) A number x (0, 1] is called normal in base b if the digits 0, 1, …, b − 1 occur with equal frequency in its base b decimal expansion, that is, if

Almost every x (0,1] (with respect to Lebesgue measure) is normal in base b for every b.

10.5 THE WIENER PROCESS

It is observed that small particles suspended in a fluid such as water or air undergo an irregular motion, known as Brownian motion, due to the collisions of the particles with the molecules of the fluid. A physical derivation of the statistical properties of Brownian motion was developed independently by Einstein and Smoluchowski, but the rigorous mathematical model for Brownian motion — in the limiting case where the motion is assumed to result from an infinite number of collisions with molecules of infinitesimal size — is due to Wiener. This model, called the Wiener process or Brownian motion process, has turned out to be of central importance in probability theory and its applications to physics and mathematical analysis.

One can consider Brownian motion in any number of space dimensions. We shall describe the theory in dimension one and indicate how to generalize it.

The position of a particle undergoing Brownian motion on the line at time t > 0 is considered to be a random variable Xt (on a sample space to be specified later) satisfying the following conditions. First, as a matter of normalization, we assume that the particle starts at the origin at time t = 0:

(10.20) equation

Second, since any given collision affects the particle by only an infinitesimal amount, it has no long-term effect, so the motion of the particle after time t should depend on its position Xt at that time but not on its previous history. Thus we assume:

(10.21) equation

Third, since the physical processes underlying Brownian motion are homogeneous in time, we shall postulate that the distribution of Xt − Xs depends only on t − s. If we divide the interval [s, t] into n equal subintervals [t₀, t₁], …, [tn−1, tn] (t₀ = s, tn = t) and write Xt − Xs = it then follows from (10.21) that Xt − Xs is a sum of n independent identically distributed random variables. Since n can be taken arbitrarily large, the central limit theorem suggests that the distribution of Xt − Xs should be normal, and this conclusion is also supported by experimental evidence. Moreover, by Corollary 10.6, and it follows that whenever t − s = r(t’ − s’) and r is rational; this strongly indicates that σ²(Xt − Xs) should be proportional to t − s. Finally, since the particle is as likely to move to the left as to the right, the mean of Xt − Xs should be 0. Putting this all together, we are led to the third assumption:

(10.22) equation

The constant C, which expresses the rate of diffusion, is of course related to the physical parameters of the system. For simplicity, we shall henceforth take C = 1.

A family {Xt}t≥0 of random variables satisfying (10.20)–(10.22), with C = 1, is called an abstract Wiener process. The generalization to n dimensions can now be described easily: an n-dimensional abstract Wiener process is a family of n-valued random variables {Xt}t≥0, where Xt = {Xt¹, …, Xtn), such that (i) to is a one-dimensional abstract Wiener process for each j, and (ii) if Yj is any function of the variables for j = 1, …, n, then Y₁, …, Yn are independent. In other words, an n-dimensional abstract Wiener process is just a Cartesian product of n one-dimensional abstract Wiener processes. In particular, Xt − Xs has the n-dimensional normal distribution for t > s:

which has the appropriate sort of spherical symmetry.

We return to the one-dimensional case. The conditions (10.20)–(10.22) completely determine the joint distributions of the Xt’s as follows. If t₁ < · · · < tn, then are independent (since X₀ = 0 a.s.), so their joint distribution is the product measure

But

where

Since detT = 1, Theorem 2.44 implies that the joint distribution P(t₁ …, tn) of Xt₁, …, Xtn is given by

(10.23) equation

where t₀ = x₀ = 0. We thus know Pt₁, …, tn when t₁ < · · · < tn, and we obtain it in the general case by permuting the variables according to (10.16). Also, it follows easily from (10.23) that (10.17) is satisfied. Therefore, by Theorem 10.18, abstract Wiener processes exist.

This situation leaves something to be desired, however. Physically, one expects the position of a particle to be a continuous function of time, so one would like the sample space for the Wiener process to be C([0, ∞),) (or some subset thereof) and the random variable Xt to be evaluation at t. Actually, Theorem 10.18 yields something along these lines: The sample space it provides is the space (*)[0,∞) of all functions from [0, ∞) into die compactified line, and Xt(ω) is indeed ω(t). We can therefore achieve our goal by showing that the measure P of Theorem 10.18 is concentrated on C([0, ∞),), considered as a subset of (*)[0, ∞). The resulting realization of the abstract Wiener process on C([0, ∞), ) is what is usually called the Wiener process.

Henceforth we shall use the notation

and P will denote the Radon measure on Ω whose finite-dimensional projections are given by (10.23).

To begin with, we need to make a few comments about the role of the point at infinity. The function f(t, s) = |t − s| maps ² to [0, + ∞) (we write + ∞ to distinguish it from the point at infinity in *), and we extend f to a map from (*)² to [0, + ∞] by declaring that . When thus extended, f is of course discontinuous at ∞, but it is lower semicontinuous, as the reader may verify (Exercise 24). Thus for a, t, s [0, ∞) the sets {ω Ω : |ω(t) − ω(s)| > a} are open and the sets {ω; Ω: |ω(t) − ω(s)| ≤ a} are closed in Ω.

Next, we need to make some estimates in terms of the quantity

These estimates are contained in the following four lemmas, after which we come to the main theorem.

10.24 Lemma. For each ε > 0, = 0.

Proof. We have

whence . The exponential term tends to zero faster than any power of δ, so the result follows.

10.25 Lemma. Suppose ε > 0, δ > 0, 0 ≤ t₁ < · · · < tk, tk − t₁ ≤ δ, and

Then P(A) ≤ 2p(, δ).

Proof. For j = 1, …, k, let

If ω A, then ω Dj for some j ≥ 2; if ω is not also in Bj, then ω must be in B₁, for ε whenever . In other words,

But P(Bj) ≤ ρ(ε, δ) by (10.22), and Bj and Dj are independent events by (10.21); also, the Dj’s are clearly disjoint. Therefore,

10.26 Lemma. With the notation of Lemma 10.25, let

Then P(E) ≤ 2ρ(, δ).

Proof. If |ω(ti) − ω(tj)| > 4, we have either |ω(ti) − ω(t₁)| > 2ε or |ω(tj) − ω(t₁)| > 2. Thus E ⊂ A, so the result follows from Lemma 10.25.

10.27 Lemma. Suppose > 0, 0 ≤ a < b, and b − a ≤ δ. Let

Then P(V) ≤ 2ρ(, δ).

Proof. If S is a finite subset of [a, b], let

Then the sets V(S) are open in Ω by the remarks preceding Lemma 10.24, and their union is V. Also, by Lemma 10.26, P(V(S)) ≤ 2ρ(ε, δ). Since the family {V(S) : S is a finite subset of [a, b]} is closed under finite unions, if K is any compact subset of V we have K ⊂ V(S) for some S and hence P(K) ≤ 2ρ(, δ). But then P(V) ≤ 2ρ(, δ) by the inner regularity of P.

10.28 Theorem. Let Ω = (*)[0,∞) and Ωc = C([0,∞),), and let P be the Radon measure on Ω whose finite-dimensional projections are given by (10.23), according to Theorem 10.18. Then Ωc is a Borel subset of Ω and P(Ωc) = 1.

Proof. A real-valued function ω on [0, ∞) is continuous iff it is uniformly continuous on [0, n] for each n, and it is uniformly continuous on [0, n] iff for every j there exists k such that (note the use of ≤ rather than) for all s [0, n] and all Moreover, even if we only assume that ω is *-valued, this last condition implies that it is real-valued unless it is identically ∞. Therefore, if ω∞ denotes the function whose value is identically ∞, we have

(10.29) equation

By the remarks preceding Lemma 10.24, is closed for all s, t, and j. Hence is an Fσδ set, and Ωc is therefore a Borel set.

Moreover, if for , δ > 0 and n we set

by (10.29) we have

Clearly = 0, so in order to show that , it will suffice to show that lim = 0 for all ε and n.

The interval [0, n] is the union of the subintervals [0, k⁻¹], [k⁻¹, 2k⁻¹], …, [n − k⁻¹, n]. If ω U(n, ε, k⁻¹),then |ω(t) − ω(s)| > 8ε for some t, s lying in the same subinterval or in adjacent subintervals, and hence, in the notation of Lemma 10.27, ω ε V where δ = k⁻¹ and [a, b] is one of the subintervals. (In the case of adjacent subintervals, use their common endpoint as an intermediate point.) As there are nk subintervals, Lemma 10.27 implies that . Lemma 10.24 then shows that P(U(n,,k⁻¹)) → 0 as k → ∞, which completes the proof.

Exercises

24. The function f : (*)² → [0, + ∞] defined by f(t, s) = [t − s| for t, s , f(∞, t) = f(t, ∞) = +∞ for t , and f(∞, ∞) = 0 is lower semicontinuous.

25. Let the coordinate functions on Ω, and for any = the σ-algebra generated by {Xt}t≥A. (Thus is the product σ-algebra on Ω, corresponding to the Borel σ-algebras on the factors.)

a. Suppose V A and ω, ω′ Ω. If ω V and ω′(t) = ω(t) for all t A, then ω′ V.

b. If V [0, ∞), then V MA for some countable set A. (Use Exercise 5 in §1.2.)

c. The set Ωc = C([0, ∞), ) is not in [0,∞).

26. Let Ωc and P be as in Theorem 10.30. If ω Ωc, it can be shown that ω is almost surely not of bounded variation, so if f is a Borel measurable function on [0, ∞), the integral apparently makes no sense. However:

a. If f = is a step function, define

Then If is an L² random variable on Ωc with mean 0 and variance (Hint: The intervals [aj, bj) may be assumed disjoint.)

b. The map f → If extends to an isometry from L²([0, ∞), m) to L²(Ωc, P).

c. If f BV([0, ∞)) is right continuous and supp(f) is compact, there is a sequence {fn} of step functions such that fn → f in L² and dfn → df vaguely, where df denotes the Lebesgue-Stietjes measure defined by f. (By Exercise 41 in §8.6, there is a sequence {μn} of linear combinations of point masses such that dμn → df vaguely. Consider fn(x) = μn((0, x]) + f(0).)

d. If f BV([0, ∞)) is right continuous and supp(f) is compact, then If(ω) = − almost surely. (Check this directly when f is a step function and apply (b) and (c).)

10.6 NOTES AND REFERENCES

The development of probability theory as a rigorous mathematical discipline began in the early part of the 20th century, when the tools of measure theory and Lebesgue-Stieltjes integrals became available. In 1933 Kolmogorov [85] put the subject on a solid foundation by explicitly identifying sample spaces and random variables with measure spaces and measurable functions. Since then it has grown extensively.

More detailed accounts of probability theory on a level comparable to that of this book can be found in Billingsley [17], Chung [25], and Lamperti [90].

§ 10.3: An account of the long history of the central limit theorem can be found in Adams [2]. More general versions of this result exist in which the random variables are not assumed to be identically distributed; see the references given above. The form of Taylor’s theorem used in the proof is explained in Folland [45].

The proof of Stirling’s formula outlined in Exercise 21 is due to Wong [162]; see Blyth and Pathak [18] for some other probabilistic proofs of Stirling’s formula.

There is one more major result about the asymptotic behavior of sums of independent identically distributed random variables that should be mentioned along with the law of large numbers and the central limit theorem:

The Law of the Iterated Logarithm: Suppose that {Xn}₁∞ is a sequence of independent identically distributed L² random variables with mean μ and variance σ², and let Sn = . Then

This theorem is due to Kolmogorov in the case where Xn L∞; the result as stated is due to Hartman and Wintner [67]. Proofs can also be found in Chung [25] (for Xn L³) and Lamperti (for Xn L∞).

§10.4: Theorem 10.18 is a variant due to Nelson [103] of the fundamental existence theorem of Kolmogorov [85]. In Kolmogorov’s original construction, the sample space is A (which could be replaced by (*)A) and the σ-algebra on which the measure P is constructed is . Theorem 10.18 is a decided improvement on Kolmogorov’s theorem, both in the simplicity of its proof and in the fact that the Borel σ-algebra on (*)A properly includes when A is uncountable. (The significance of the latter fact is evident from Exercise 25.)

§10.5: Wiener constructed his measure P on C([0, ∞), ) in [159] and [160]; his approach is quite different from ours. Our proof of Theorem 10.30 follows Nelson [104]. See also Nelson [105] for some related material, including the derivation of the postulates (10.20)–(10.22) from physical principles.

A discussion of the many interesting properties of the Wiener process is beyond the scope of this book. We shall mention only one, as a complement to Theorem 10.30: The sample paths of the Wiener process are almost surely nowhere differentiable; in fact, with probability one, at each point they are Hölder continuous of every exponent α < 1/2 but not of exponent 1/2. This fact may be startling at first, but it seems almost inevitable when one reflects that |ω(t) − ω(s)|/|t − s|1/2 has the standard normal distribution for all t, s.

Knight [84] is a good source for further information about the Wiener process.

Chapter 11

More Measures and Integrals

In this chapter we discuss some additional examples of measures and integrals that are of importance in analysis and geometry: invariant measures on locally compact groups, geometric measures of lower-dimensional sets in n, and integration of densities and differential forms on manifolds. Although we have grouped these topics together in one chapter, they are substantially independent of one another.

11.1 TOPOLOGICAL GROUPS AND HAAR MEASURE

A topological group is a group G endowed with a topology such that the group operations and are continuous from G × G and G to G. Examples include topological vector spaces (the group operation being addition), groups of invertible n × n real matrices (with the relative topology induced from n×n), and all groups equipped with the discrete topology. If G is a topological group, we denote the identity element of G by e, and for A, B G and x G we define

We say that A ⊂ G is symmetric if A = A⁻¹.

Here are some of the basic properties of topological groups:

11.1 Proposition. Let G be a topological group.

a. The topology of G is translation invariant: If U is open and x G, then Ux and xU are open.

b. For every neighborhood U of e there is a symmetric neighborhood V of e with V ⊂ U.

c. For every neighborhood U of e there is a neighborhood V of e with VV ⊂ U.

d. If H is a subgroup of G, so is .

e. Every open subgroup of G is also closed.

f. lf K₁, K₂ are compact subsets of G, so is K₁ K₂.

Proof. (a) is equivalent to the continuity in each variable of the map , and (b) and (c) are equivalent to the continuity of at the identity. (Details are left to the reader.) For(d), if x, y , there exist nets in H that converge to x and y. Then xα⁻¹ → x⁻¹ and xαyβ → xy (with the usual product ordering on A x B), so x⁻¹ and xy belong to . For (e), if H is an open subgroup, the cosets xH are open for all χ so that is open and hence H is closed. Finally, (f) is true because K₁K₂ is the image of the compact set K₁ x K₂ under the continuous map .

If f is a continuous function on the topological group G and y G, we define the left and right translates of f through y by

(The point of using y⁻¹ on the left and y on the right is to make Lyz = LyLz and Ryz = RyRz.) f is called left (resp. right) uniformly continuous if for every > 0 there is a neighborhood V of e such that ||Lyf − f||u < (resp. ||Ryf − f||u < ) for y V. (Some authors reverse the roles of Ly and Ry in this definition.)

11.2 Proposition. If f Cc(G), then f is left and right uniformly continuous.

Proof. We shall consider right uniform continuity; the proof on the left is the same. Let K = supp(f) and suppose > 0. For each x K there is a neighborhood Ux of e such that |f(xy)−f(x)| < 1/2 for y Ux, and by Proposition 11.1 (b, c) there is a symmetric neighborhood Vx of e with . Then {xVx}xK covers K, so there exist x₁, …, xn K such that . Let V = ₁n Vxj; we claim that On the one hand, if x K, then for some j we have xj⁻¹ x Vxj and hence ; therefore,

On the other hand, if x K, then f(x) = 0, and either or xj⁻¹xy Vxj for some j (if xy K); in the latter case , so that and hence |f(xy)| < .

One usually assumes that the topology of a topological group is Hausdorff. The following proposition shows that this is not much of a restriction.

11.3 Proposition. Let G be a topological group.

a. If G is T₁, then G is Hausdorff.

b. If G is not T₁, let H be the closure of {e}. Then H is a normal subgroup, and if G/H is given the quotient topology (i.e., a set in G/H is open iff its inverse image in G is open), G/H is a Hausdorff topological group.

Proof. (a) If G is T₁ and x ≠ y G, by Proposition 11.1(b, c) there is a symmetric neighborhood V of e such that xy⁻¹ VV. Then Vx and Vy are disjoint neighborhoods of x and y, for it z = v₁x = v₂y for some v₁, v₂ V, then .

(b) H is a subgroup by Proposition 11.1d; it is clearly the smallest closed subgroup of G. It follows that H is normal, for if H’ were a conjugate of H with H′ ≠ H, H′ H would be a smaller closed subgroup. It is routine to verify that the group operations on G/H are continuous in the quotient topology, so that G/H is a topological group. If is the identity element in G/H, then {} is closed since its inverse image in G is H. But then every singleton set in G/H is closed by Proposition 11.1a, so G/H is T₁ and hence Hausdorff.

In the context of Proposition 11.3b, it is easy to see that every Borel measurable function on G is constant on the cosets of H and hence is effectively a function on G/H. Thus for most purposes one may as well work with the Hausdorff group G/H. We shall be interested in the case where G is locally compact, and we henceforth use the term locally compact group to mean a topological group whose topology is locally compact and Hausdorff.

Suppose that G is a locally compact group. A Borel measure μ on G is called left-invariant (resp. right-invariant) if μ(xE) = μ(E) (resp. μ(Ex) = μ(E)) for all x G and E G. Similarly, a linear functional I on Cc(G) is called left- or right-invariant if I(Lxf) = I(f) or I(Rxf) = I(f) for all f. A left (resp. right) Haar measure on G is a nonzero left-invariant (resp. right-invariant) Radon measure μ on G. For example, Lebesgue measure is a (left and right) Haar measure on n, and counting measure is a (left and right) Haar measure on any group with the discrete topology. (Other examples will be found in the exercises.) The following proposition summarizes some elementary properties of Haar measures; in it, and in the sequel, we employ the notation

11.4 Proposition. Let G be a locally compact group.

a. A Radon measure μ on G is a left Haar measure iff the measure defined by (E) = μ(E⁻¹) is a right Haar measure.

b. A nonzero Radon measure μ on G is a left Haar measure iff for all f Cc⁺ and y G.

c. If μ is a left Haar measure on G, then μ(U) > 0 for every nonempty open U O for all f Cc⁺.

d. If μ is a left Haar measure on G, then μ(G) < iff G is compact.

Proof. (a) is obvious. The “only if” implication of (b) follows by approximating f by simple functions, and the converse is true by (7.3). As for (c), since μ 0, by the regularity of μ there is a compact with μ(K) > 0. If U is open and nonempty, K can be covered by finitely many left translates of U, and it follows that μ(U) > 0. If f Cc⁺, let U = {x : f(x) > 1/2||f||u}. Then .

Finally, we prove (d). If G is compact, then μ(G) < ∞ since μ is Radon. If G is not compact and V is a compact neighborhood of e, then G cannot be covered by finitely many translates of V, so by induction we can find a sequence {xn} such that for all n. By Proposition 11.1(b, c) there is a symmetric neighborhood U of e such that . If m > n and is nonempty, then , a contradiction. Hence is a disjoint sequence, and μ{xnU) = μ(U) > 0 by (c), whence μ(G) ≥ .

Our aim now is to prove the existence and uniqueness of Haar measures. In view of Proposition 11.4a, one can pass from left Haar measures to right Haar measures at will, so for the sake of definiteness we shall concentrate on left Haar measures. We begin with some motivation for the existence proof.

If E G and V is open and nonempty, let (E : V) denote the smallest number of left translates of V that cover E, that is,

where #(A) = card(A) if A is finite and #(A) = ∞ otherwise. Thus (E : V) is a rough measure of the relative sizes of E and V. If we fix a precompact open set E₀, the ratio (E : V)/{E₀ : V) gives a rough estimate of the size of E when the size of E₀ is normalized to be 1. This estimate becomes the more accurate the smaller V is, and it is obviously left-invariant as a function of E. We might therefore hope to obtain a Haar measure as a limit of the “quasi-measures” (E : V)/(E₀ : V) as V shrinks to {e}.

This idea can be made to work as it stands, but it is simpler to carry out if we think of integrals of functions instead of measures of sets. If f, ϕ Cc⁺, then {x : ϕ(x) > 1/2||ϕ||u} is open and nonempty, so finitely many left translates of it cover supp(f), and it follows that

It therefore makes sense to define the “Haar covering number” of f with respect to ϕ:

Clearly (f : ϕ) > 0; in fact, .

11.5 Lemma. Suppose that f, g, ϕ Cc⁺.

a. (f : ϕ) = (Lxf : ϕ) for any x G.

b. (cf : ϕ) = c(f : ϕ)for any c > 0.

c. (f + g : ϕ) ≤ (f : ϕ) + (g : ϕ).

d. (f : ϕ) ≤ (f : g)(g : ϕ).

Proof. We have this proves (a), and (b) is equally obvious. If and , then follows by minimizing . Similarly, if f then f ≤ , (d) follows.

At this point we make a normalization by choosing f₀ Cc⁺ once and for all and defining

By Lemma 11.5(a–c), for each fixed ϕ the functional Iϕ is left-invariant and bears some resemblance to a positive linear functional except that it is only subadditive. Moreover, by Lemma 11.5d it satisfies

(11.6) equation

We now show that, in a certain sense, Iϕ is approximately additive when supp(ϕ) is small.

11.7 Lemma. If f₁, f₂ Cc⁺ and ε > 0, there is a neighborhood V of e such that Iϕ(f₁) + Iϕ(f₂) ≤ Iϕ(f₁ + f₂) + ε whenever supp(ϕ) ⊂ V.

Proof. Fix g Cc⁺ such that g = 1 on supp(f₁ + f₂), and let δ be a positive number to be specified later. Set h = f₁ + f₂ + δg and hi = fi/h (i = 1, 2), where it is understood that hi = 0 outside supp(fi). Then hi Cc⁺, so by Proposition 11.2 there is a neighborhood V of e such that . If , and h ≤ , then δ whenever supp(ϕ), so

But then and since hi + h₂ ≤ 1,

Now, Σ cj can be made arbitrarily close to (h : ϕ), so by Lemma 11.5(b,c),

In view of (11.6), therefore, it suffices to choose δ small enough so that

11.8 Theorem. Every locally compact group G possesses a left Haar measure.

Proof. For each f Cc⁺ let Xf be the interval [(f₀ : f)⁻¹, (f : f₀)], and let X = Then X is a compact Hausdorff space by Tychonoff’s theorem, and by (11.6), every Iϕ is an element of X. For each compact neighborhood V of e, let K(V) be the closure in X of . Clearly so by Proposition 4.21 there is an element I in the intersection of all the K(V)’s. Every neighborhood of I in X intersects {Iϕ : supp(ϕ) V} for all V; in other words, for any neighborhood V of e and any f₁, …, fn Cc⁺ and > 0 there exists Cc⁺ with supp such that |I(fj) – Iϕ(fj)| < for j = 1, …, n. Therefore, in view of Lemmas 11.5 and 11.7, I is left-invariant and satisfies I(af + bg) = aI(f) + bI(g) for all f, g Cc⁺ and a, b > 0. It follows easily, as in the proof of Lemma 7.15, that if we extend I to Cc by setting I(f) = I(f⁺) − I(f⁻), then I is a left-invariant positive linear functional on Cc(G). Moreover, I(f) > 0 for all f Cc⁺ by (11.6). The proof is therefore completed by invoking the Riesz representation theorem.

11.9 Theorem. If μ and v are left Haar measures on G, there exists c > 0 such that μ = cu.

Proof. We first present a simple proof that works when μ is both left- and right-invariant — in particular, when G is Abelian. Pick h Cc⁺ such that h Cc⁺ and h(x) = h(x⁻¹) (e.g., h(x) = g(x) + g(x⁻¹) where g is any element of Cc⁺). Then for any f Cc(G),

so that μ = cv where by Proposition 11.4c, and Fubini’s theorem is applicable since the functions in question are supported in sets which are compact and hence of finite measure. The same remarks apply below.)

Now, another proof for the general case. The assertion that μ = cv is equivalent to the assertion that the ratio is independent of f Cc⁺. Suppose, then, that f, g Cc⁺; we shall show that τj = τg.

Fix a symmetric compact neighborhood V₀ of e and set

Then A and B are compact by Proposition 11.1f, and for y V₀ the functions and are supported in A and B, respectively. Next, given > 0, by Proposition 11.2 there is a symmetric compact neighborhood such that supsx|f(xy) − f(yx)| < and supx |g(xy) − g(yx)\ < for y V. Pick h Cc⁺ with supp(h) . Then

and since h(x) = h(x⁻¹),

Thus,

By the same reasoning,

Dividing these inequalities by and , respectively, and adding them, we obtain

Since is arbitrary, we are done.

We conclude this section by investigating the relationship between left and right Haar measures. If μ is a left Haar measure on G and x G, the measure μx(E) = μ(Ex) is again a left Haar measure, because of the commutativity of left and right translations (i.e., the associative law). Hence, by Theorem 11.9 there is a positive number Δ(x) such that μx = Δ(x)μ. The function thus defined is independent of the choice of μ by Theorem 11.9 again; it is called the modular function of G.

11.10 Proposition. Δ is a continuous homomorphism from G to the multiplicative group of positive real numbers. Moreover, if μ is a left Haar measure on G, for any f L¹ (μ) and y in G we have

(11.11) equation

Proof. For any x, y G and E G,

so Δ is a homomorphism from G to (0, ∞). Also, since χE(xy) = χEy⁻¹ (x)

This proves (11.11) when f = χE, and the general case follows by the usual linearity and approximation arguments. Finally, it is an easy consequence of Proposition 11.2 that the map is continuous for any f Cc(G) (Exercise 2), so the continuity of Δ follows from (11.11).

Evidently, the left Haar measures on G are also right Haar measures precisely when Δ is identically 1, in which case G is called unimodular. Of course, every Abelian group is unimodular; remarkably enough, groups that are highly noncommutative are also unimodular. To be precise, let [G, G] denote the smallest closed subgroup of G containing all elements of the form [x, y] = xyx⁻¹y⁻¹. [G, G] is called the commutator subgroup of G; it is normal because z[x, y]z⁻¹ = [zxz⁻¹, zyz⁻¹], and it is trivial precisely when G is Abelian.

11.12 Proposition. If G/[G, G] is finite, then G is unimodular.

Proof. Every continuous homomorphism (such as Δ) from G into an Abelian group must annihilate [x, y] for all x, y and must therefore factor through G/[G, G). If the latter group is finite, Δ(G) is a finite subgroup of (0, ∞); but (0, ∞) has no finite subgroups except {1}.

11.13 Proposition. If G is compact, then G is unimodular.

Proof. For any x G, obviously G = Gx. Hence if μ is a left Haar measure, we have μ(G) = μ(Gx) = Δ(x)μ(G), and since 0 < μ(G) < ∞ we conclude that Δ(x) = 1.

We observed above that if μ is a left Haar measure, is a right Haar measure. We now show how to compute it in terms of μ and Δ.

11.14 Proposition. .

Proof. By (11.11), if f Cc(G),

Thus the functional is right-invariant, so its associated Radon measure is a right Haar measure. However, this Radon measure is simply Δ⁻¹ dμ by Exercise 9 in §7.2; hence, by Theorem 11.9, for some c > 0. If c ≠ 1, we can pick a symmetric neighborhood U of e in G such that |Δ(x)⁻¹ − 1| < 1/2|c − 1| on U. But , so

a contradiction. Hence c = 1 and .

11.15 Corollary. Left and right Haar measures are mutually absolutely continuous.

Exercises

1. If G is a topological group and , then = {EV : V is a neighborhood of e}.

2. If μ is a Radon measure on the locally compact group G and f Cc(G), the functions and , are continuous.

3. Let G be a locally compact group that is homeomorphic to an open subset U of n in such a way that, if we identify G with U, left translation is an affine map — that is, xy = Ax (y) + bx where Ax is a linear transformation of n and bx n. Then |det Ax|⁻¹ dx is a left Haar measure on G, where dx denotes Lebesgue measure on n. (Similarly for right translations and right Haar measures.)

4. The following are special cases of Exercise 3.

a. If G is the multiplicative group of nonzero complex numbers z = x + iy, (x² + y²)⁻¹ dx dy is a Haar measure.

b. If G is the group of invertible n × n real matrices, | det A|−n dA is a left and right Haar measure, where dA = Lebesgue measure on nxn. (To see that the determinant of the map is (det A)n, observe that if X is the matrix with columns X¹, …, Xn, then AX is the matrix with columns AX¹, …, AXn.)

c. If G is the group of 3 × 3 matrices of the form

then dx dy dz is a left and right Haar measure.

d. If G is the group of 2 × 2 matrices of the form

then x⁻²dx dy is a left Haar measure and x⁻¹ dx dy is a right Haar measure.

5. Let G be as in Exercise 4d. Construct a Borel set in G with finite left Haar measure but infinite right Haar measure, and a left uniformly continuous function on G that is not right uniformly continuous.

6. Let {Gα}αA be a family of topological groups and G = ΠαA Gα.

a. With the product topology and coordinatewise multiplication, G is a topological group.

b. If each Gα is compact and μα is the Haar measure on Gα such that μα(Gα) = 1, then the Radon product of the μα’s, as constructed in Theorem 7.28, is a Haar measure on G.

7. In Exercise 6, for each α let Gα be the multiplicative group {-1, 1} with the discrete topology. Let μ be a Haar measure on G.

a. If πα : G → {-1, 1} is the αth coordinate function, then = 0 for α ≠ β.

b. If A is uncountable, L²(μ) is not separable even though .

8. Let have the relative topology induced from . Then is a topological group that is not locally compact, and there is no nonzero translation-invariant Borel measure on that is finite on compact sets.

9. Let G be a locally compact group with left Haar measure μ.

a. G has a subgroup H that is open, closed, and σ-compact. (Let H be the subgroup generated by a precompact open neighborhood of e.)

b. The restriction of μ to subsets of H is a left Haar measure on H.

c. μ is decomposable in the sense of Exercise 15 in §3.2.

d. If the topology of G is not discrete, then μ({x}) = 0 for all x G. In this case, μ is regular iff μ is semifinite iff μ is σ-finite iff G is σ-compact. (See Exercises 12 and 14 in §7.2.)

11.2 HAUSDORFF MEASURE

In geometric problems it is important to have a method for measuring the size of lower-dimensional sets in n, such as curves and surfaces in ³. Differential-geometric techniques provide such a method that applies to smooth submanifolds of n; see §11.4. However, there is also a measure-theoretic approach to the problem that applies to more general sets. Indeed, the basic ideas can be carried out just as easily in arbitrary metric spaces, so we begin by working in this generality.

Let (X, ρ) be a metric space. (See §0.6 for the relevant terminology.) An outer measure μ* on X is called a metric outer measure if

11.16 Proposition. If μ* is a metric outer measure on X, then every Borel susbset of X is μ*−measurable.

Proof. Since the closed sets generate the Borel σ-algebra, it suffices to show that every closed set -measurable. Thus, given with , we wish to show that

Let Bn = {x A\F : ρ(x, F) ≥ n⁻¹}. Then Bn is an increasing sequence of sets whose union is A \ F (since F is closed), and ρ(Bn, F) ≥ n⁻¹. Therefore,

so it will be enough to show that μ*(A \ F) = lim μ*(Bn). Let Cn = Bn+1 \ Bn. If x Cn+1 and ρ(x, y) < [n(n + 1)]⁻¹, then

so that . A simple induction therefore shows that

and similarly . Since , it follows that the series and are convergent. But by subadditivity we have

As n → ∞, the last sum vanishes and we obtain

as desired.

We are now ready to define Hausdorff measure. Suppose that (X, ρ) is a metric space, p ≥ 0, and λ > 0. For A ⊂ X, let

with the convention that inf decreases the infimum is being taken over a smaller family of coverings of A, so Hp, δ(A) increases. The limit

is called the p-dimensional Hausdorff (outer) measure of A.

Several comments on this definition are in order:

The sets Bj in the definition of Hp,δ are arbitrary subsets of X. However, one obtains the same result if one requires the Bj’s to be closed (because diam Bj = diam j), or if one requires the Bj’s to be open (because one can replace Bj by the open set Uj = {x : ρ(x, Bj) < 2−j−1}, whose diameter is at most (diam Bj) + 2−J. Similarly, if X = , one can restrict the Bj’s to be closed or open intervals.

The intuition behind the definition of Hp is that if p is an integer and A is a “p-dimensional” subset of n such as a relatively open set in a p-dimensional linear subspace of n, the amount of A that is contained in a region of diameter r should be roughly proportional to rp.

The restriction to coverings by sets of small diameter is necessary to provide an accurate measure of irregularly shaped sets; otherwise one could simply cover a set by itself, with the result that its measure would be at most the pth power of its diameter. Consider, for example, the curve Am = {(x, sin mx) : |x| ≤ 1} in ². Clearly diam Am ≤ 23/2 for all m, but the length of Am tends to along with m. One needs to take δ m⁻¹ before H1,δ(A) becomes an accurate estimate of the length of Am.

We now derive the basic properties of Hp.

11.17 Proposition. Hp is a metric outer measure.

Proof. Hp,δ is an outer measure by Proposition 1.10, and it follows that Hp is an outer measure. If ρ(A, B) > 0 and {Cj} is a covering of AB such that (diam Cj) ≤ δ < ρ(A, B) for all j, then no Cj can intersect both A and B. Splitting Σ (diam Cj)p into two parts according to whether shows that Σ(diam Cj)p ≥ Hp,δ(A) + Hp,δ(B), and hence Hp,δ(AB) ≥ Hp,δ(A) + Hp,δ(B). As this inequality is valid whenever δ < ρ(A, B), the desired result follows by letting δ → 0.

In view of Propositions 11.16 and 11.17, the restriction of Hp to the Borel sets is a measure, which we still denote by Hp and call p-dimensional Hausdorff measure.

11.18 Proposition. Hp is invariant under isometries of X. Moreover, if Y is any set and f, g : Y → X satisfy ρ(f(y), f(z)) ≤ C ρ(g(y), g(z)) for all y, z Y, then Hp(f(A)) ≤ Cp Hp(g(A)) for all A ⊂ Y

Proof. The first assertion is evident from the definition of Hp. As for the second, given , δ > 0, cover g(A) by sets Bj such that diam Bj ≤ C⁻¹ δ and Σ(diam Bj)p ≤ Hp (g(A)) + . Then the sets B’j = f(g⁻¹(Bj)) cover f(A), and diam , so that

The proof is completed by letting δ → 0 and → 0.

11.19 Proposition. If Hp(A) < ∞, then Hq(A) = 0 for all q > p. If Hp(A) > 0, then Hq(A) = ∞ for all q < p.

Proof. It suffices to prove the first statement, as the second one is its contra-positive. If Hp(A) < ∞, for any δ > 0 there exists {Bj}₁∞ with A ∞ Bj, diam Bj ≤ δ, and Σ(diam Bj)p ≤ Hp(A) + 1. But if q > p,

so Hq,δ(A) ≤ δq−p(Hp(A) + 1). Letting δ → 0, we see that Hq(A) = 0.

According to Proposition 11.19, for any A ⊂ X the numbers

are equal. Their common value is called the Hausdorff dimension of A.

From now on we restrict attention to the case X = n. Our object is to show that for p = 1, …, n, Hp gives the geometrically correct notion of measure (up to a normalization constant) for p-dimensional submanifolds of n. We begin with the case p = n.

11.20 Proposition. There is a constant γn > 0 such that γn Hn is Lebesgue measure on n.

Proof. Hn is a translation-invariant Borel measure on n. If Q ∞ n is a cube, it is easily verified that 0 < Hn(Q) < ∞ (Exercise 10). It follows that Hn ≠ 0 and that Hn is finite on compact sets, whence Hn is a Radon measure by Theorem 7.8. The desired result is therefore a consequence of Theorem 11.9. (The simple argument given there for the Abelian case can be read without going through the rest of §11.1: Simply read x + y for xy and −x for x⁻¹.)

The normalization constant γn turns out to be the volume of a ball of diameter 1, which by Corollary 2.55 is πn/2/2nΓ(1/2n + 1). We shall not give the proof here, as the value of γ is irrelevant for our purposes. (The hard part is proving the intuitively obvious fact that among all sets of diameter 1, the ball has the largest volume.) Many authors build γn into the definition of Hausdorff measure; that is, they define p-dimensional Hausdorff measure, for p [0, ∞), to be γpHp where γp =

We now consider lower-dimensional sets in n. If 1 ≤ k ≤ n, a k-dimensional C¹ submanifold of n is a set M ⊂ n with the following property: For each x M there exist a neighborhood U of x in n, an open set V ⊂ κ, and an injective map f : V → U of class C¹ such that f(V) = M U and the differential Dxf — i.e., the linear map from κ to n whose matrix is is injective for each x V. Such an f is called a parametrization of M U. Every submanifold M can be covered by countably many U’s for which M U has a parametrization, so for our purposes it will suffice to assume that M = M U has a global parametrization.

(There are other common definitions of “submanifold”: M is a κ-dimensional C¹ submanifold if it is locally the set of zeros of a C¹ map g : n → n−κ such that Dxg is surjective at each x M, or if it is locally the image of a ball in a κ-dimensional linear subspace of n under a C¹ diffeomorphism of n. The equivalence of these definitions with the one given above is a standard exercise in the use of the implicit function theorem.)

We begin our study of submanifolds with the linear case. If T is a linear map from κ to n and T* : n → k is its transpose, then T*T is a positive semidefinite linear operator on n. Its determinant is therefore nonnegative, and we may define

11.21 Proposition. If , and T : k → n is linear, then Hk(T(A)) = J(T)Hk(A).

Proof. If κ = n, then det(T*T) = (det T)², so J(T) = |det T| and the assertion reduces to Theorem 2.44 because of Proposition 11.20. If k < n, let R be a rotation of n that maps the range of T to the subspace k × {0} = {y n : yj = 0 for j > k}, and let S = RT. Then S* S = T* R*RT = T*T, so that J(S) = J(T), and Hk(S(A)) = Hk(T(A)) since Hk is rotation-invariant. But if we identify k × {0} with k, S becomes a map from k to itself, and the definition of S*S is unchanged when this identification is made. We are therefore back in the equidimensional case, which was disposed of above.

It is now easy to guess what the corresponding formula must be for a general smooth injection f : k → n, since locally every smooth map is approximately linear. Our next lemma makes this idea precise.

11.22 Lemma. Suppose M is a k-dimensional C¹ submanifold of n parametrized by f : V → n. For any α > 1 there is a sequence {Bj} of disjoint Borel subsets of V such that , and a sequence {Tj} of linear maps from k to n, such that

(11.23) equation

and

(11.24) equation

Proof. Let us fix > 0 and β > 1 such that

and let be a countable dense subset of the set of linear maps from k to n (e.g., the set of matrices with rational entries). For and m let E(T, m) be the set of all x V such that

The definition of E(T, m) is unaffected if y and z are restricted to lie in countable dense subsets of V and k; hence E(T, m) is defined by countably many inequalities involving continuous functions, so it is a Borel set. It will therefore suffice to show that the sets E(T, m) cover V. Indeed, each E(T, m) is a countable union of sets Ei(T, m) of diameter less than m⁻¹, so by disjointifying the countable collection {Ei(T, m) : T T, i, m } we obtain the desired sets Bj and the associated maps Tj.

Suppose, then, that x V, and let δ₀ = inf{|(Dxf)z| : |z| = 1}. Since Dxf is injective, δ₀ is positive. Choose δ > 0 so that δ ≤ (β − 1)δ₀ and δ ≤ (1 − β⁻¹)δ₀, and then pick T T such that ||T − Dxf|| < δ. Then

and similarly . This establishes the first inequality and also shows that T is injective, so η = is positive. Since f is differentiable at x, there exists m such that

But then

and similarly . In short, x E(T, m), so we are done.

11.25 Theorem. Let M be a κ-dimensional C¹ submanifold of n parametrized by f : V → n. If A is a Borel subset of V, then f (A) is a Borel subset of n, and

(11.26) equation

Moreover, if ϕ is a Borel measurable function on M that is either nonnegative or in L¹(M, Hκ), then

(11.27) equation

Proof. Since V is an open subset of κ, it is σ-compact. It follows that if A is closed in V, then A is σ-compact, and hence f(A) is σ-compact since f is continuous. The collection of all such that f(A) is Borel is therefore a σ-algebra that contains all closed sets, hence all Borel sets. We shall prove (11.26); this establishes (11.27) when ϕ = χf(A) and the general result follows from the usual linearity and approximation arguments.

Given α > 1, let {Bj}, {Tj} be as in Lemma 11.22, and let It follows from (11.23) and Proposition 11.18 that

and hence by Proposition 11.21,

But it also follows from (11.24) and Proposition 11.18 that

and from Proposition 11.21 that Hk(Tj(Aj)) = J(Tj)Hk(Aj). Therefore,

Summing over j, we obtain

so the proof is completed by letting α → 1.

If both sides of the identities (11.26) and (11.27) are multiplied by the normalizing constant γk in Proposition 11.20, the integrals on the right become ordinary Lebesgue integrals, and we obtain the formula for measures and integrals on M given by Riemannian geometry (see §11.4). Moreover, if k = n we have J(Dxf) = |det Dxf|, so the result reduces to Theorem 2.47. See also Exercises 11 and 12.

There remains the question of whether p-dimensional Hausdorff measure is of any interest when . An affirmative answer will be provided in the next section.

Exercises

10. Show directly from the definition of Hn that if Q is a cube in n, then 0 < Hn(Q) < ∞. (Hint: There is a constant C such that if , the Lebesgue measure of E is at most C(diam E)n.)

11. If f : (a, b) → n is a parametrization of a smooth curve (i.e., a 1-dimensional C¹ submanifold of n), the Hausdorff 1-dimensional measure of the curve is .

12. If ϕ : k → is a C¹ function, the graph of ϕ is a k-dimensional C¹ submanifold of k+1 parametrized by , the k-dimensional volume of the portion of the graph lying above A is

(First do the linear case, ϕ(x) = a · x. Show that if T : k → k+1 is given by Tx = (x, a · ×), then T*T = I + S where Sx = (a · ×)a, and hence det(T*T) = 1 + |a|². Hint: The determinant of a matrix is the product of its eigenvalues; what are the eigenvalues of S?)

13. In any metric space, zero-dimensional Hausdorff measure is counting measure.

14. If Am is a subset of a metric space X of Hausdorff dimension pm for m , then has Hausdorff dimension supmpm.

15. If A ⊂ n has Hausdorff dimension p, then A × A ⊂ 2n has Hausdorff dimension ≥ 2p.

11.3 SELF-SIMILARITY AND HAUSDORFF DIMENSION

In this section we produce some geometrically interesting examples of sets of fractional Hausdorff dimension. The sets we consider are “self-similar,” which means roughly that each small part of the set looks like a shrunken copy of the whole set. We begin by establishing the terminology necessary to discuss such sets. Our definitions will be more restrictive than is really necessary, since we aim only to give the flavor of the theory and to display some examples.

For r > 0, a similitude with scaling factor r is a map S : n → n of the form S(x) = rO(x) + b, where O is an orthogonal transformation (a rotation or the composition of a rotation and a reflection) and b n. Suppose that S = (S₁, …, Sm) is a finite family of similitudes with a common scaling factor , we define

E is called invariant under S if S(E) = E. In this case, Sk(E) = E for all k, which means that for every k ≥ 1, E is the union of mk copies of itself that have been scaled down by a factor of rk. If, in addition, these copies are disjoint or have negligibly small overlap, E can be said to be “self-similar.”

Before proceeding with the theory, let us examine some of the standard examples of self-similar sets. They are all obtained by starting with a simple geometric figure, applying a family of similitudes repeatedly to it, and passing to the limit.

Given β (0, 1/2), let Cp be the Cantor set obtained from [0, 1] by successively removing open middle (1 − 2β) ths of intervals, as discussed at the end of §1.5.

That is, where

See Figure 11.1a.

Fig. 11.1 The first three approximations to (a) the Cantor set C3/s, (b) the Sierpiński gasket, and (c) the snowflake curve.

The Sierpiński gasket Γ is the subset of ² obtained from a solid triangle by dividing it into four equal subtriangles by bisecting the sides, deleting the middle subtriangle, and then iterating. Thus, if we take the initial triangle Δ to be the closed triangular region with vertices (0,0), (1,0), and (1/2, 1), then Γ = , where S = (S₁, S₂, S₃) with

See Figure 11.1b.

The snowflake curve Σ is the subset of ² obtained from a line segment by replacing its middle third by the other two legs of the equilateral triangle based on that middle third (there are two such triangles; make a definite choice), and then iterating. That is, let L be the broken line joining , and let

where Rθ denotes the rotation through the angle θ. Then Σ = lim(L); more precisely, Σ = (M), where M is the union of the open middle thirds of the line segments that constitute L. See Figure 11.1c. (The actual “snowflake” is made by joining three rotated and reflected copies of Σ in the same way in which one can join three copies of the initial figure L to make a six-pointed star.)

The Cantor sets, the Sierpiński gasket, and the snowflake curve are all clearly invariant under the families of similitudes used to generate them. The condition of negligible overlap of the rescaled copies is also satisfied, for in all cases is either empty or a single point.

We now return to the general theory. Suppose that S = (S₁, …, Sm) is a family of similitudes with scaling factor r < 1. We introduce some notation for the actions of the iterations of S on points, sets, and measures: For x n, n, μ M(n), and ii, …, ik {1, …, m}, we set

It is an important property of compact sets that are invariant under a family of similitudes that they carry measures with a corresponding invariance property.

11.28 Theorem. Suppose that S = (S₁, …, Sm) is a family of similitudes with scaling factor r < 1 and that X is a nonmempty compact set that is invariant under S. Then there is a Borel measure μ on n such that μ(n) = 1, supp(μ) = X, and for all κ ,

(11.29) equation

Proof. We shall construct μ as a measure on X, extending it to n by setting μ(Xc) = 0. Pick x X, let δx be the point mass at x, and for k define μk M(X) by

that is, for f C(X),

Thus each μk is a probability measure on X. We claim that the sequence {μk} converges vaguely as k → ∞. Indeed, given f C(X) and ε > 0, there exists K > 0 such that |f(x) − f(y)| < ε whenever x, y X and |x − y| < rK(diam X). Suppose l > k ≥ K. Since xi₁ … il Xi₁ … ik and (diam X), we have

Summing over ik+1, …, il gives

and then summing over yields . Thus the sequence converges for every f, and the limit defines a positive linear functional on C(X).

Let μ be the associated Radon measure, according to the Riesz representation theorem. Clearly . Also, we have xi₁ … xk diam = rk(diam X), and X = , so me points (k ) are dense in X; it follows that supp(μ) = X. Also, from the definition of μk we have

As and μl both tend vaguely to μ, and composition with similitudes preserves vague convergence, so (11.29) follows.

The existence of an invariant measure on the invariant set X requires no special hypotheses on the similitudes Sj, but in order to be able to compute the Hausdorff dimension of X, we need to impose an extra condition which (as we shall see in Theorem 11.33b) guarantees that the sets Sj(X) have negligibly small overlap. Namely, we require S to possess a separating set: a nonempty bounded open set U such that

(11.30) equation

The existence of a separating set is more delicate than it might seem at first; the first condition in (11.30) will fail if U is too small, and the second one will fail if U is too big. However, all the examples considered above admit separating sets. As the reader may easily verify, for the Cantor sets Cβ one; can take U = (0, 1), for the Sierpiński gasket one can take U to be the interior of the initial triangular region Δ, and for the snowflake curve one can take U to be the interior of the triangular region with vertices (0, 0), (1, 0), and (1/2, 1/6 ), i.e., the interior of the convex hull of the initial figure L.

11.31 Proposition. Suppose that S is a family of similitudes with scaling factor r < 1 that admits a separating set U. Then there is a unique nonempty compact set that is invariant under S, namely,.

Proof. Since S(U) ⊂ U and the Sj’s are continuous, we have

It follows that is a compact invariant set for S, and it is nonempty by Proposition 4.21. If Y is another such set, let d(Y, X) = maxyY ρ(y, X) be the maximum distance from points in Y to X. Since Sj decreases distances by a factor of r, we have d(Sj(Y), Sj(X)) = rd(Y, X). But , so d(Y, X) ≤ maxj d(Sj(Y), X) ≤ rd(Y, X). It follows that d(Y, X) = 0, which means that Y ⊂ X since X and Y are compact. By the same reasoning, X ⊂ Y, so X is the unique compact invariant set.

11.32 Lemma. Let c, C, δ be positive numbers. Suppose that {Uα}αA is a collection of disjoint open sets in n such that each Uα contains a ball of radius cδ and is contained in a ball of radius Cδ. Then no ball of radius δ intersects more than (1 + 2C)nc−n of the sets α.

Proof. If B is a ball of radius δ and , then α is contained in the ball concentric with B with radius (1 + 2C)δ. Hence, if N of the α’s intersect B, there are N disjoint balls of radius cδ contained in a ball of radius (1+2C)δ. Adding up their Lebesgue measures, we see that .

11.33 Theorem. Suppose that S = (S¹,…, Sm) is a family of similitudes with scaling factor r < 1 that admits a separating set U, let X be the unique nonempty compact set that is invariant under S, and let p = log1/r m. Then

a. 0 < Hp(X) < ∞; in particular, X has Hausdorff dimension p.

b. Hp(Si(X) Sj(X)) = 0 for i ≠ j.

Proof. For any k we have X = Sk(X) = Xi₁…ik and diam Xi₁,…ik = rk(diam X), so if δk = rk(diam X),

Since δk → 0 as k → ∞, it follows that Hp(X) ≤ (diam X)p < ∞.

Next we show that Hp(X) > 0. Choose positive numbers c and C so that the separating set U contains a ball of radius cr⁻¹ and is contained in a ball of radius C, and let N = (1 + 2C)nc−n. We shall prove that Hp(X) ≥ N⁻¹ by showing that if is any covering of X by sets of diameter ≤ 1, then Σ(diam Ej)p ≥ N⁻¹. Since any set E with diameter δ is contained in a (closed) ball of radius δ, it is enough to show that if where Bj is a ball of radius δj ≤ 1, then .

Let μ be the measure on X given by Theorem 11.28. We claim that if B is any ball of radius δ ≤ 1, then μ(B) ≤ Nδp. The desired conclusion is an immediate consequence:

To prove the claim, let k be the integer such that rk < δ ≤ rk−1. By (11.29),

Since by Proposition 11.31, we have = 0 unless B intersects . On the other hand, iteration of (11.30) shows that the sets Ui₁…ik are all disjoint, and each of them contains a ball of radius crk−1 ≤ cδ and is contained in a ball of radius Crk < Cδ. By Lemma 11.32, B can intersect at most N of the ’s. Therefore,

as claimed.

Finally, since Sj decreases distances by a factor of r, we have Hp(Sj(X)) = rp Hp(X) = m⁻¹ Hp(X) and hence . But since , this can happen only if Hp(Si(X) Sj(X)) = 0 for i ≠ j.

11.34 Corollary. The Cantor sets Cβ, the Sierpiński gasket, and the snowflake curve have Hausdorff dimension log1/β 2, log₂ 3, and log₃ 4, respectively.

Exercises

16. Modify the construction of the snowflake curve by using isosceles triangles rather than equilateral ones. That is, given < β < , let L be the broken line connecting (0, 0) to . Proceeding inductively, let Lk be the figure obtained by replacing each line segment in Lk−1 by a copy of L, scaled down by a factor of βk, and let Σβ be the limiting set. (Thus is the ordinary snowflake curve.) Find the family of similitudes under which is invariant, show that it possesses a separating set, and find the Hausdorff dimension of Σβ.

17. Investigate analogues of the Sierpiński gasket constructed from squares, or higher-dimensional cubes, rather than triangles. (There are various possibilities here.)

18. Given n and p (0, n), construct Borel sets of Hausdorff dimension p, with the following properties.

a. 0 < Hp(E₁) < ∞. (Exercise 17 could be used here.)

b. Hp(E₂) = ∞. (Use (a).)

c. Hp(E₃) = 0. (Use Exercise 14.)

19. The measure μ in Theorem 11.28 is unique.

11.4 INTEGRATION ON MANIFOLDS

This section is a brief essay on integration on manifolds for the benefit of those who are familiar with the language of differential geometry and wish to see how the geometric notions of integration fit into the measure-theoretic framework. The discussion and the notation will both be quite informal.

Let M be a C∞ manifold of dimension m. (We work in the C∞ category for convenience; C¹ would actually suffice.) Given a coordinate system x = (x₁,…, xm) on an open set U ⊂ M, one can consider Lebesgue measure dx = dx₁ … dxm on U. This has no intrinsic geometric significance, for if y = (y₁,…, ym) is another coordinate system on U, by Theorem 2.47 we have where denotes the matrix . However, dy and dx are mutually absolutely continuous, and the Radon-Nikodym derivative is a function. It therefore makes sense to define a smooth measure on M to be a Borel measure μ which, in any local coordinates x, has the form dμ = ϕx dx where ϕx is a nonnegative C∞ function. The representations of μ in different coordinate systems are then related by

(11.35) equation

(This conveniently sloppy notation is in the same spirit as the formula du/dt = (du/dx)(dx/dt) for the chain rule. More precisely, if y = F(x), then

Equation (11.35) may be interpreted in the language of geometry as follows. The functions are the transition functions for a line bundle on M; a section of this bundle, called a density on M, is an object that is represented in each local coordinate system x by a function ϕx, such that the functions for different coordinate systems are related by (11.35). In short, smooth measures can be identified with nonnegative densities on M. More generally, any density ϕ defines, at least locally, a smooth signed or complex measure μ on M, so the integral is well defined for any compact , as is for any f Cc(M).

Suppose now that M is equipped with a Riemannian metric. In any coordinate system x, the metric is represented by a positive definite matrix-valued function gx = (gijx). The matrix gy for another coordinate system is related to gx by

so that

It follows that is a positive density on M canonically associated to the metric g; it is called the Riemannian volume density on M. In particular, if M is a submanifold of n (n ≥ m), it inherits a Riemannian structure from the ambient Euclidean structure. If M is parametrized by f : V → n as in §11.2, the metric g is given in the coordinates induced by f by

Theorem 11.25 therefore asserts that integration of the Riemannian volume density gives m-dimensional Hausdorff measure on M, up to the factor γm.

These ideas yield an easy construction of a left Haar measure on any Lie group, that is, any topological group that is a C∞ manifold and whose group operations are C∞. Namely, choose an inner product on the tangent space at the identity element and transport it to every other point by left translation. The result is a left-invariant Riemannian metric, and the associated volume density defines a left Haar measure.

The most popular things to integrate on manifolds are differential forms. For our purposes it will suffice to describe a differential m-form on an m-dimensional manifold M as a section of the line bundle whose transition functions are . That is, a differential m-form ω is given in local coordinates x by a function ωx, and the function ωy for a different coordinate system is related to ωx by ωx = (The usual notation is ω = ωx dx₁ Differential m-forms thus look just like densities if one restricts oneself to coordinate systems whose Jacobian matrices have positive determinant. If it is possible to do this consistently on all of M, M is called orientable. In this case, assuming that M is connected, the coordinate systems on subsets of M fall into two classes such that within each class one always has 0; a choice of one of these classes is an orientation of M. (In ³, for example, one speaks of left-handed or right-handed coordinates.) If M is equipped with an orientation, therefore, differential m-forms may be identified with densities and as such may be integrated over compact subsets of M.

The notion of density can be generalized. If 0 ≤ θ ≤ 1, a θ-density on M is a section of the line bundle whose transition functions are . (Thus, a 1-density is a density, and a 0-density is just a smooth function.) Suppose θ > 0 and p = θ⁻¹. If ϕ is a θ-density, |ϕ|p is well defined as a nonnegative density and so can be integrated over M. The set of θ-densities ϕ such that is a normed linear space whose completion Lp(M) is called the intrinsic Lp space of M. The duality results of Chapter 6 work in this setting: If ϕj is a θj-density for j = 1, 2, where θ₁ + θ₂ = 1, then ϕ₁ϕ₂ is a density and where pj = θj⁻¹, and

11.5 NOTES AND REFERENCES

§11.1: The existence and uniqueness of Haar measure were first proved by Haar [59] and von Neumann [155], respectively, for groups whose topology is second countable; the general case is due to Weil [158]. Our proof of existence and uniqueness follows Weil [158] and Loomis [94]. There is another proof, due to H. Cartan, which yields existence and uniqueness simultaneously and avoids the use of the axiom of choice (which we invoked via Tychonoff’s theorem). This proof, as well as further references and historical remarks, can be found in Hewitt and Ross [75, §15].

Haar measure is the foundation for harmonic analysis on locally compact groups. The articles by Graham, Weiss, and Sally in Ash [7] provide a good introduction to this field; a more extensive treatment can be found in Folland [47].

§ 11.2: Proposition 11.16 is due to Carathéodory [22], and the theory of Hausdorff measure was developed in Hausdorff [69]. The computation of the constant γn can be found in Billingsley [17] or Falconer [39, §1.4]. There are other ways of defining lower-dimensional measures on n, all of which agree on smooth submanifolds but sometimes differ on more irregular sets; see Federer [41].

The concept of Hausdorff measure can be generalized. If λ is any strictly increasing continuous function on [0, ∞) such that λ(0) = 0, for a subset A of a metric space X one can define

and Hλ(A) = limδ→0 Hλ, δ(A). Thus Hp = Hλ where λ(t) = tp. Rogers [119] contains a systematic treatment of these generalized Hausdorff measures.

§11.3: The computation of the Hausdorff dimension of the Cantor sets Cβ goes back to Hausdorff [69]. The arguments presented here are due to Hutchinson [78]; they can easily be extended to families of similitudes with different scaling factors, that is, S = (S₁,…, Sm) where Sj has scaling factor rj < 1. Such a family always has a unique nonempty compact invariant set X, and if it possesses a separating set, the Hausdorff dimension of X is the number p such that Σ₁m rjp = 1. See Hutchinson [78] or Falconer [39, §8.3].

Self-similar sets are among the simplest examples of “fractals.” Falconer [39] is a good reference for the geometric measure theory of fractals; see also Edgar [37], Falconer [40], and Mandelbrot [96] for other aspects of the theory of fractals.

Continuous curves of Hausdorff dimension > 1 can be constructed from non-differentiable functions. For example, if f : [a, b] → is Hölder continuous of exponent α, 0 < α < 1, the graph of f in ² can have Hausdorff dimension as large as 2 − α, and the range of a sample path of the n-dimensional Wiener process (n ≥ 2) almost surely has Hausdorff dimension 2. See Falconer [39, §§8.2,7].

§ 11.4: The theory of integration of differential forms can be found in a number of books such as Warner [157] and Loomis and Sternberg [95]; the latter book also has a discussion of densities.

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Index of Notation

For the basic notation used throughout the book for sets, mappings, numbers, and metric and topological spaces, see Chapter 0. Notation used only in the section in which it is introduced is, for the most part, not listed here.

Analysis on Euclidean space: x · y (dot product). ∂α, xα, α!, |α| (multi-index notation). n (n-torus).

Functions and operations on functions: f± (positive and negative parts). sgn. χE (characteristic function). fx, fy (sections). Γ. supp(f) (support). λf (distribution function). τyf (translation). f * g (convolution). ϕt (dilation). , F f (Fourier transform). F, ϕ>. (reflection).

Integrals: The basic notation is developed in §2.2. f(x) dx (Lebesgue integral). f dμ dv (iterated integral). g dF (Stieltjes integral).

Measures: μF (Lebesgue-Stieltjes measure). m, mn (Lebesgue measure). μ x v (product). σ (surface measure on sphere). v± (positive and negative variations). |v| (total variation). μ v (mutual singularity). μ << v (absolute continuity). f dμ. dμ/dv (Radon-Nikodym derivative). supp(μ) (support). μ v (Radon product). μ * v (convolution).

Norms and seminorms: ||f||u (uniform norm). ||T|| (operator norm). ||f||p (Lp norm). . [f]p (weak Lp quasi-norm). ||μ|| (measure norm). ||ϕ||(N,α) (Schwartz space norm). ||f||(s) (Sobolev norm).

Probability theory: E(X) (expectation). σ²(X) (variance). Pϕ, (image measure, distribution). vμσ² (normal distribution).

Sets: Fσ, Fσδ, Gδ, Gδσ. Ex, Ey (sections).

σ-algebras: (ε) (σ-algebra generated by ε). BX (Borel sets). Mα, M N (products). (Lebesgue measurable sets). (Baire sets).

Spaces of functions, measures, etc.: L⁺. L¹. Lloc¹. BV. N BV. C(X, Y). B(X, ). BC(X, ). B(X). C(X). BC(X). Cc(X). C₀(X). L(X, Y). X*. L². l². Lp. lp. Lp. weak Lp. M(X). Ck. C∞. Cc∞. δ. D′. ’. CS′. Hs. Hsloc.

Index

A

Abel mean

Abel summation

Absolute continuity

of a function

of a measure

Absolute convergence

Accumulation point

Adjoint

A.e.

Alaoglu’s theorem

Alexandroff compactification

Algebra

Banach

of functions

of sets

Almost every(where)

Almost sure(ly)

Almost uniform convergence

Approximate identity

Arcwise connected set

Arzelà-Ascoli theorem

A.s.

Axiom

of choice

of countability

of separation

B

Baire category theorem

Baire classes

Baire set

Ball

Banach algebra

Banach space

Banach-Tarski paradox

Base for a topology

Bessel’s inequality

Bijective mapping

Binomial distribution

Bochner integral

Bochner-Riesz means

Bolzano-Weierstrass property

Borel isomorphism

Borel measurable function

Borel measure

Borel set

Borel σ-algebra

Borel’s normal number theorem

Borel-Cantelli lemma

Boundary

Bounded linear map

Bounded set

Bounded variation

C

Cantor function

Cantor set

Cantor-Lebesgue function

Carathéodory’s theorem

Cartesian product

Cauchy net

Cauchy sequence

in measure

Cauchy-Riemann equation

Central limit theorem

Cesàro mean

Cesàro summation

Characteristic function

of a probability distribution

Chebyshev’s inequality

Chi-square distribution

Closed graph theorem

Closed linear map

Closed set

Closure

Closure operator

Cluster point

Coarser topology

Cofinal net

Cofinite topology

Commutator subgroup

Compact set

Compact space

countably

locally

sequentially

Compactification

one-point

Stone-ech

Compactly supported function

Complement

Complete measure

Complete metric space

Complete orthonormal set

Complete topological vector space

Completely regular algebra

Completely regular space

Completion

of a measure

of a normed vector space

of a σ-algebra

Complex measure

Composition

Condensation of singularities

Conditional expectation

Conjugate exponent

Connected component

Connected set

Constant-coefficient operator

Content

Continuity

absolute

Hölder

Lipschitz

of linear maps on Cc∞

of measures

uniform

Continuous measure

Continuum

Continuum hypothesis

Convergence

absolute

almost uniform

in C∞c

in L¹

in measure

in probability

of a filter

of a net

of a sequence

of a series

Convex function

Convolution

of distributions

of functions

of measures

Coordinate

Coordinate map

Countable additivity

Countable ordinals

Countable set

Countably compact space

Counting measure

Cover

Cube

D

Daniell integral

Decomposable measure

Decreasing function

Decreasing rearrangement

DeMorgan’s laws

Dense set

Density

of a set

Riemannian volume

Denumerable set

Derivative

Lp

of a distribution

Radon-Nikodym

Diameter

Diffeomorphism

Difference of sets

Differential operator

Dirac measure

Directed set

Dirichlet kernel

Dirichlet problem

Disconnected set

Discrete

Discrete measure

Discrete topology

Disjoint sets

Distribution function

Distribution

homogeneous

joint

of a random variable

periodic

tempered

Domain

Dominated convergence theorem

Dual space

E

Egoroff’s theorem

Elementary family

Elliptic differential operator

Elliptic regularity theorem

Embedding

Entropy

Equicontinuity

Equivalence class

Equivalence relation

Equivalent metric

Equivalent norm

Essential range

Essential supremum

Event

Eventually

Expectation

conditional

Extended integrable function

Extended real number system

F

Fσ and Fσδ sets

Fatou’s lemma

Fejér kernel

Field of sets

Filter

Finer topology

Finite intersection property

Finite measure

Finite signed measure

Finitely additive measure

First category

First countable space

First uncountable ordinal

Fourier integral

Fourier inversion theorem

Fourier series

Fourier transform

of a measure

of a tempered distribution

Fourier-Stieltjes transform

Fractional integral

Frequently

Fréchet space

Fubini’s theorem

Fubini-Tonelli theorem

Function

Fundamental theorem of calculus

G

Gδ and Gδσ sets

Gamma distribution

Gamma function

Gauge

Gauss kernel

Gaussian distribution

Generalized Cantor set

Gibbs phenomenon

Gram-Schmidt process

Graph of a linear map

H

H-interval

Haar measure

Hahn decomposition

Hahn decomposition theorem

Hahn-Banach theorem

Hardy’s inequalities

Hardy-Littlewood maximal function

Hausdorff dimension

Hausdorff maximal principle

Hausdorff measure

Hausdorff space

Hausdorff-Young inequality

Heat equation

Heine-Borel property

Heisenberg’s inequality

Henstock-Kurzweil integral

Hermite function

Hermite operator

Hermite polynomial

Hilbert space

Hilbert’s inequality

Homeomorphism

Homogeneous distribution

Hull of an ideal

Hölder continuity

Hölder’s inequality

I

Ideal in an algebra

Identically distributed random variables

Iff

Image

Image measure

Increasing function

Independent events

Independent random variables

Indicator function

Indiscrete topology

Inequality

Bessel’s

Chebyshev’s

Hardy’s

Hausdorff-Young

Heisenberg’s

Hilbert’s

Hölder’s

Jensen’s

Kolmogorov’s

Minkowski’s

Schwarz

triangle

Wirtinger’s

Young’s

Infimum

Initial segment

Injective mapping

Inner measure

Inner product

Inner regular measure

Integrable function

weakly

extended

locally

Integral

Bochner

Daniell

fractional

Henstock-Kurzweil

Lebesgue

Lebesgue-Stieltjes

of a complex function

of a nonnegative function

of a real function

of a simple function

Riemann

Interior

Invariant set

Inverse

Inverse image

Invertible linear map

Isometry

Isomorphism Borel

linear

order

unitary

J

Jensen’s inequality

Joint distribution

Jordan content

Jordan decomposition

of a function

of a signed measure

Jordan decomposition theorem

K

Kernel of a closed set

Kolmogorov’s inequality

Krein extension theorem

L

Lp derivative

Lp norm

Lp space

intrinsic

weak

Laplacian

Lattice of functions

Law of large numbers

strong

weak

Law of the iterated logarithm

LCH space

Lebesgue decomposition

Lebesgue differentiation theorem

Lebesgue integral

Lebesgue measurable function

Lebesgue measurable set

Lebesgue measure

Lebesgue set

Lebesgue-Radon-Nikodym theorem

Lebesgue-Stieltjes integral

Lebesgue-Stieltjes measure

Left continuous function

Left-invariant measure

Lemma

Borel-Cantelli

Fatou’s

monotone class

Riemann-Lebesgue

three lines

Urysohn’s

Weyl’s

Zorn’s

Limit inferior

Limit of a net

Limit superior

Linear functional

positive

Linear ordering

Liouville’s theorem

Lipschitz continuity

Localized Sobolev space

Locally compact group

Locally compact space

Locally convex space

Locally finite cover

Locally integrable function

Locally measurable set

Locally null set

Lower bound

Lower semicontinuous function

LSC function

Lusin’s theorem

M

Map

Mapping

Marcinkiewicz interpolation theorem

Maximal element

Maximal function

Maximal theorem

Meager set

Mean ergodic theorem

Mean

Measurable function

Measurable mapping

Measurable set

Lebesgue

locally

with respect to an outer measure

Measurable space

Measure

Borel

complete

complex

continuous

counting

decomposable

dicrete

Dirac

finitely additive

Hausdorff

inner

inner regular

Lebesgue

Lebesgue-Stieltjes

outer

outer regular

positive

Radon

regular

semifinite

c-finite

signed

singular

smooth

Measure space

Metric

Metric outer measure

Metric space

Minimal element

Minkowski’s inequality

for integrals

Modular function

Moment convergence theorem

Monotone class

Monotone class lemma

Monotone convergence theorem

Monotone function

Monotonicity of measures

Multi-index

Mutually singular measures

N

Negative part of a function

Negative set

Negative variation

of a function

of a signed measure

Neighborhood

Neighborhood base

Net

Norm

Lp

operator

product

quotient

uniform

Norm topology

Normal distribution

Normal number

Normal space

Normed linear space

Normed vector space

Nowhere dense set

Null set

locally

O

One-point compactification

Open map

Open mapping theorem

Open set

Operator norm

Order isomorphism

Order topology

Ordinal

Orientable manifold

Orientation

Orthogonal projection

Orthogonal set

Orthonormal basis

Orthonormal set

Outer measure

metric

Outer regular measure

P

Paracompact space

Parallelogram law

Parametrization

Parseval’s identity

Partial ordering

Partition

of an interval

of unity

tagged

Periodic distribution

Periodic function

Plancherel theorem

Point mass

Pointwise bounded family

Poisson distribution

Poisson kernel

Poisson summation formula

Polar coordinates

Polar decomposition

Positive definite function

Positive linear functional

Positive measure

Positive part of a function

Positive set

Positive variation

of a function

of a signed measure

Pre-Hilbert space

Precompact set

Predecesor

Premeasure

Principal symbol

Probability measure

Product measure

Product metric

Product norm

Product σ-algebra

Product topology

Projection

orthogonal

Proper map

Pythagorean theorem

Q

Quotient norm

Quotient space

Quotient topology

R

Radon measure

complex

Radon product

Radon-Nikodym derivative

Radon-Nikodym theorem

Random variable

Range

Real-analytic function

Rectangle

Refinement of a cover

Reflexive space

Regular measure

Regular space

Relation

Relative topology

Rellich’s theorem

Residual set

Reverse inclusion

Riemann integrable function

Riemann integral

Riemann-Lebesgue lemma

Riemannian volume density

Riesz representation theorem

Riesz-Thorin interpolation theorem

Right continuous function

Right-invariant measure

Ring of seta

S

Sample mean

Sample space

Sample variance

Sampling theorem

Saturated measure

Saturation of a measure

Scalar product

Schröder-Bernstein theorem

Schwartz space

Schwarz inequality

Second category

Second countable space

Section of a set or function

Semifinite measure

Semifinite part

Seminorm

Separable space

Separating set

Separation

of points

of points and closed sets

Sequence

Sequentially compact space

Shannon’s theorem

Shrink nicely

Sides of a rectangle

Sierpiński gasket

σ-algebra

Borel

generated by a family of functions

generated by a family of sets

of countable or co-countable sets

product

σ-compact space

σ-field

σ-finite measure

σ-finite set

σ-finite signed measure

σ-ring

Signed measure

Similitude

Simple function

Singular measure

Slowly increasing function

Smooth measure

Snowflake curve

Sobolev embedding theorem

Sobolev space

localized

Standard deviation

Standard normal distribution

Standard representation of a simple function

Stirling’s formula

Stone-ech compactification

Stone-Weierstrass theorem

Strong law of large numbers

Strong operator topology

Strong type

Stronger topology

Subadditivity

Subbase for a topology

Sublinear functional

Sublinear map

Submanifold

Subnet

Subordination

Subsequence

Subspace of a vector space

Support

of a distribution

of a function

of a measure

Supremum

Surjective mapping

Symbol

principal

Symmetric difference

Symmetric neighborhood

T

T₀ …, T₄ space

Tagged partition

Tempered distribution

Tempered function

Theorem

Alaoglu’s

Arzelà-Ascoli

Baire category

Carathéodory’s

central limit

closed graph

dominated convergence

Egoroff’s

Fourier inversion

Fubini-Tonelli

Hahn decomposition

Hahn-Banach

Jordan decomposition

Krein extension

Lebesgue differentiation

Lebesgue-Radon-Nikodym

Liouville’s

Lusin’s

Marcinkiewicz interpolation

maximal

moment convergence

monotone convergence

open mapping

Plancherel

Pythagorean

Radon-Nikodym

Rellich’s

Riesz representation

Riesz-Thorin interpolation

sampling

Schröder-Bernstein

Shannon’s

Sobolev embedding

Stone-Weierstrass

Tietze extension

Tychonoff’s

Urysohn metrization

Vitali convergence

Vitali covering

Weierstrass approximation

Three lines lemma

Tietze extension theorem

Tonelli’s theorem

Topological group

Topological space

Topological vector space

Topology

cofinite

generated by a family of sets

indiscrete

norm

of uniform convergence

of uniform convergence on compact sets

product

quotient

relative

strong operator

trivial

vague

weak operator

weak

weak*

Zariski

Torus

Total ordering

Total variation

of a complex measure

of a function

of a signed measure

Totally bounded set

Transfinite induction

Transpose

Triangle inequality

Trivial topology

Tychonoff space

Tychonoff’s theorem

U

Uniform boundedness principle

Uniform continuity

Uniform integrability

Uniform norm

Unimodular group

Unitary map

Upper bound

Upper semicontinuous function

Urysohn metrization theorem

Urysohn’s lemma

USC function

V

Vague topology

Vanish at infinity

Variance

Vitali convergence theorem

Vitali covering theorem

W

Wave equation

Weak convergence

Weak Lp

Weak law of large numbers

Weak operator topology

Weak topology

Weak type

Weak* topology

Weaker topology

Weierstrass approximation theorem

Weierstrass kernel

Well ordering principle

Well ordering

Weyl’s lemma

Wiener process

abstract

Wirtinger’s inequality

Y

Young’s inequality

Z

Zariski topology

Zorn’s lemma