Lebesgue Integration

By J.H. Williamson

This concise introduction to Lebesgue integration is geared toward advanced undergraduate math majors and may be read by any student possessing some familiarity with real variable theory and elementary calculus. The self-contained treatment features exercises at the end of each chapter that range from simple to difficult.
The approach begins with sets and functions and advances to Lebesgue measure, including considerations of measurable sets, sets of measure zero, and Borel sets and nonmeasurable sets. A two-part exploration of the integral covers measurable functions, convergence theorems, convergence in mean, Fourier theory, and other topics. A chapter on calculus examines change of variables, differentiation of integrals, and integration of derivatives and by parts. The text concludes with a consideration of more general measures, including absolute continuity and convolution products.

• Language: English
• Publisher: Dover Publications
• Release date: Jul 28, 2014
• ISBN: 9780486796734

Book preview

LEBESGUE INTEGRATION

J. H. WILLIAMSON

DOVER PUBLICATIONS, INC.

Mineola, New York

Copyright

Copyright © 1962, 1990 by J. H. Williamson

All rights reserved.

Bibliographical Note

This Dover edition, first published in 2014, is an unabridged republication of the edition originally published in the Athena Series: Selected Topics in Mathematics by Holt, Rinehart and Winston, New York, in 1962.

Library of Congress Cataloging-in-Publication Data

Williamson, J. H. (John Hunter), 1926–

Lebesgue integration / J.H. Williamson.—Dover edition.

pages cm

Originally published: New York : Holt, Rinehart and Winston, 1962.

Includes index.

eISBN-13: 978-0-486-79673-4

1. Lebesgue integral. 2. Measure theory. I. Title.

QA312.W53 2014

515′.43—dc23

2014011311

Manufactured in the United States by Courier Corporation

78977201     2014

www.doverpublications.com

Preface

This is an introductory text on Lebesgue integration, and may be read by any student who knows a little about real variable theory and elementary calculus. Most of the material on sets and functions required in the book is collected together in Chapter 1. The reader who is not already acquainted with these topics should find there enough for an understanding of the chapters that follow. At one or two points a slight acquaintance with linear algebra (matrices and determinants) is helpful. No knowledge of the Riemann integral is required.

It is widely agreed that the Lebesgue integral should be developed in the general setting of n-dimensional Euclidean space, rather than in the special case n = 1. This is done here. Apart from one or two initial complications, the more general approach is no more difficult, and is indeed essential for the formulation of some important theorems. Nothing essential will be lost if the reader takes n = 2 throughout the book.

The basic material is developed in detail; in later chapters (in particular, Chapter 6) more is left to the reader. The exercises at the ends of the chapters range from the trivial to the difficult, and include one or two substantial theorems. One exercise is, of course, implicit throughout the book: verify in detail all assertions made in the text and not proved there.

There are many excellent advanced texts on integration theory, to which this book may serve as an introduction—for example, P. R. Halmos, Measure. Theory or A. C. Zaanen, An Introduction to the Theory of Integration.

J. H. W.

Cambridge, England

January, 1962

Contents

Chapter 1.      Sets and Functions

1.1.   Generalities

1.2.   Countable and Uncountable Sets

1.3.   Sets in Rn

1.4.   Compactness

1.5.   Functions

Chapter 2.      Lebesgue Measure

2.1.   Preliminaries

2.3.   Measurable Sets

2.4.   Sets of Measure Zero

2.5.   Borel Sets and Nonmeasurable Sets

Chapter 3.      The Integral I

3.1.   Definition

3.2.   Elementary Properties

3.3.   Measurable Functions

3.4.   Complex and Vector Functions

3.5.   Other Definitions of the Integral

Chapter 4.      The Integral II

4.1.   Convergence Theorems

4.2.   Fubini’s Theorems

4.3.   Approximations to Integrable Functions

4.4.   The Lp Spaces

4.5.   Convergence in Mean

4.6.   Fourier Theory

Chapter 5.      Calculus

5.1.   Change of Variables

5.2.   Differentiation of Integrals

5.3.   Integration of Derivatives

5.4.   Integration by Parts

Chapter 6.      More General Measures

6.1.   Borel Measures

6.2.   Signed Measures and Complex Measures

6.3.   Absolute Continuity

6.4.   Measures, Functions, and Functionals

6.5.   Norms, Fourier Transforms, Convolution Products

Index

[ 1 ]

Sets and Functions

1.1. Generalities

We take the usual simple-minded analyst’s view of set theory. The terms set, collection, aggregate, class, and so forth, are looked on as the same and will not be defined. The elements that make up a set may be called points, irrespective of their nature (often they will be points, in the usual sense, in Euclidean space). A collection of sets will be called a class rather than a set. We shall tend to use lower-case italic letters for points, italic capitals for sets, and script capitals for classes.

As usual, the property that the element a is in the set A (a belongs to A, A contains a) is denoted by a ∈ A. The negation of this is written a ∉ A. If a ∈ A implies a ∈ B (every element in A is also in B), then A is a subset of B, and we write A ⊂ B or B ⊃ A. If A ⊂ B and B ⊂ A, then the sets A and B are equal; A = B. If A ⊂ B and A ∉ B, then A is a proper subset of B. If A consists of the finite number of points a, b, · · ·, k, we write A = {a, b, · · ·, k}. More generally, if A consists of all points a for which the statement P(a) is true, we write A = {a: P(a)}.

It is convenient to introduce the empty set, denoted by Ø, which contains no points; it is a subset of every set. Also, all sets under consideration at any time will be subsets of some large set, the universal set, whole space · · ·. This will usually be left to be understood from the context; it will almost always be Euclidean space of the appropriate dimension, or a suitable subset of this.

A map or mapping f of A into B, or a function f from A to B, is a correspondence which assigns to each a ∈ A an element f(a) ∈ B; f(a) is the image of a under f. The set A is the domain (of definition) of f; in order to specify a function completely, its domain should be given, but often we will leave this to be understood from the context; no confusion should result. If E is a subset of A, the set f (E) = {f(a): a ∈ E} is the image of E under f. The set f (A) is the range of f. If F ⊂ B, the subset of A defined by {a: f (a) ∈ F} is called the inverse image of F under f, and it is denoted by f−1 (F). If f (A) = B, then f is a map on to B. If f (a) = f {a′) implies a = a′, then f is one to one (1-1).

Let A be a set. A family in A is a set I (the index set) and a mapping f of I into A. The element f (i) may be written ai, and the family may be denoted by (ai)i∈I or simply (ai). The most familiar example, of course, is where I is the set of integers from 1 to n, or the set of all positive integers; the family is then a finite or infinite sequence. A family in A is thus something more complicated than a subset of A; naturally, to each family (ai) there corresponds a subset f (I) of A. In the particular case where the map f is 1-1, the family (ai) contains no repetitions, and is then an indexed set. Any set can be regarded as an indexed set (take the set itself as index set), and hence as a family. The set corresponding to the family (ai) may be denoted {ai}, {ai: i ∈ I}, or {ai}i∈I.

If A and B are any sets, their union A ∪ B is the set of points which are in at least one of A, Bis a class or {Ai)i∈I when is no risk of confusion. The intersection of A and B, A ∩ B, is the set of all points which are in both A and Bin the obvious way. If A ∩ B = Ø, A and B are said to be disjointis disjoint.

Unions and intersections have various algebraic properties; for instance, the associative properties A ∪ (B ∪ C) = (A ∪ B) ∪ C = A ∪ B ∪ C and A ∩ (B ∩ C) = (A ∩ B) ∩ C = A ∩ B ∩ C; the obvious commutative relations, and the two distributive laws (A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C) and (A ∪ B) ∪ C = {A ∪ C) ∩ (B ∪ C). The proof in each case is immediate, and, of course, very substantial generalizations are equally easy. One consequence of the identities

, it is seen that we must have

where X is the universal set. A rather similar situation arises with empty sets of real numbers; we must have

The difference A \ B is the set of points of A which are not in B; B need not be a subset of A. If B is a subset of A, then A \ B is also called the complement of B with respect to A. If A is the universal set X, we speak simply of the complement of B. For any A, B. It is clear that A ⊂ B . From this there follows the very useful principle of duality: from any set-theoretic relation, we can obtain another by reversing inclusions, replacing unions by intersections and intersections by unions, and replacing each set by its complement. In particular, from any formal identity—such as (A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C)—we can obtain another, equally true, by interchanging unions and intersections; in the present case the result is (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C), the other distributive law. The principle can be extended to cover topological situations; the complement of an open set is closed and the complement of a closed set is open (see Theorem 1.3d).

Fig. 1

The Cartesian product A × B of A and B is the set of all pairs (a, b) with a ∈ A, b ∈ B; this is true also for any finite number of factors. The product is not commutative; it is, however, associative: (A × B) × C = A × (B × C) = A × B × C in the sense that there is a natural correspondence between ((a, b), c), (a, (b, c)) and (a, b, c). The product R × R of the real line with itself is the Euclidean plane R² (hence the name Cartesian, after Descartes) and the product of n factors R is n-dimensional Euclidean space Rn. The relation Rm × Rn = Rm+n holds A very convenient pictorial representation of the Cartesian product of two sets is obtained by taking each to be a segment of the real line, in which case their product is a subset of the plane.

Let C be a subset of A × B, and a ∈ A. By the section of C by a we mean the set {b: (a, b) ∈ C}; it is a subset of B . The projection of C on B is the set {b: (a, b) ∈ C for some a ∈ A}. It is the union of all the sections of C by points of A (Fig. 1).

1.2. Countable and Uncountable Sets

The sets A and B are equivalent (or similar) if there is a 1-1 map of one on to the other; we write A ∼ B. It is clear that A ∼ A, that A ∼ B if and only if B ∼ A, and that, A ∼ B, B ∼ C imply A ∼ C. If A = Ø, or A ∼ {1, 2, · · ·, n} for some n, then A is finite; if A is finite, or equivalent to the set of all positive integers, it is countable.

Infinite sets have some surprising properties; for instance, two sets may be equivalent even though one is obviously larger than the other. (It is proved below that the rational numbers are countable.) The relation A ∼ B does not exlude the possibility that A ∼ C, where C is a proper subset of B; indeed, it is characteristic of infinite sets that they are equivalent to proper