The Elements of Integration and Lebesgue Measure E-Book

Robert G. Bartle

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Wiley Classics Library
• Sprache: Englisch

Seitenzahl: 265

Veröffentlichungsjahr: 2014

Beschreibung

Consists of two separate but closely related parts. Originally published in 1966, the first section deals with elements of integration and has been updated and corrected. The latter half details the main concepts of Lebesgue measure and uses the abstract measure space approach of the Lebesgue integral because it strikes directly at the most important results--the convergence theorems.

Leseprobe

Contents

1. Introduction

2. Measurable Functions

COMPLEX-VALUED FUNCTIONS

FUNCTIONS BETWEEN MEASURABLE SPACES

3. Measures

4. The Integral

5. Integrable Functions

6. The Lebesgue Spaces Lp

7. Modes of Convergence

8. Decomposition of Measures

RIESZ REPRESENTATION THEOREM

9. Generation of Measures

THE EXTENSION OF MEASURES

LEBESGUE MEASURE

LINEAR FUNCTIONALS ON C

10. Product Measures

11. Volumes of Cells and Intervals

12. The Outer Measure

TRANSLATION INVARIANCE

13. Measurable Sets

14. Examples of Measurable Sets

BOREL SETS

NULL SETS

TRANSLATION INVARIANCE

NON-BOREL SETS

15. Approximation of Measurable Sets

16. Additivity and Nonadditivity

17. Nonmeasurable and Non-Borel Sets

References

Index

Copyright © 1966 by John Wiley & Sons, Inc.

Wiley Classics Library Edition Published 1995.

All rights reserved. Published simultaneously in Canada.

Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012.

Library of Congress Cataloging in Publication Data:

Library of Congress Catalog Card Number 66-21048

ISBN 0-471-04222-6

10 9

Preface

This book consists of two separate, but closely related, parts. The first part (Chapters 1-10) is subtitled The Elements of Integration; the second part (Chapters 11-17) is subtitled The Elements of Lebesgue measure. It is possible to read these two parts in either order, with only a bit of repetition.

The Elements of Integration is essentially a corrected reprint of a book with that title, originally published in 1966, designed to present the chief results of the Lebesgue theory of integration to a reader having only a modest mathematical background. This book developed from my lectures at the University of Illinois, Urbana-Champaign, and it was subsequently used there and elsewhere with considerable success. Its only prerequisites are a understanding of elementary real analysis and the ability to comprehend “ε-δ arguments”. We suppose that the reader has some familarity with the Riemann integral so that it is not necessary to provide motivation and detailed discussion, but we do not assume that the reader has a mastery of the subtleties of that theory. A solid course in “advanced calculus”, an understanding of the first third of my book The Elements of Real Analysis, or of most of my book Introduction to Real Analysis with D. R. Sherbert provides an adequate background. In preparing this new edition, I have seized the opportunity to correct certain errors, but I have resisted the temptation to insert additional material, since I believe that one of the features of this book that is most appreciated is its brevity.

There are many expositions of the Lebesgue integral from various points of view, but I believe that the abstract measure space approach used here strikes directly towards the most important results: the convergence theorems. Further, this approach is particularly well-suited for students of probability and statistics, as well as students of analysis. Since the book is intended as an introduction, I do not follow all of the avenues that are encountered. However, I take pains not to attain brevity by leaving out important details, or assigning them to the reader.

Readers who complete this book are certainly not through, but if this book helps to speed them on their way, it has accomplished its purpose. In the References, I give some books that I believe readers can profitably explore, as well as works cited in the body of the text.

I am indebted to a number of colleagues, past and present, for their comments and suggestions; I particularly wish to mention N. T. Hamilton, G. H. Orland, C. W. Mullins, A. L. Peressini, and J. J. Uhl, Jr. I also wish to thank Professor Roy O. Davies of Leicester University for pointing out a number of errors and possible improvements.

ROBERT G. BARTLE

Ypsilanti and UrbanaNovember 20, 1994

The Elements of Integration

CHAPTER 1

Introduction

The theory of integration has its ancient and honorable roots in the “method of exhaustion” that was invented by Eudoxos and greatly developed by Archimedes for the purpose of calculating the areas and volumes of geometric figures. The later work of Newton and Leibniz enabled this method to grow into a systematic tool for such calculations.

As this theory developed, it has become less concerned with applications to geometry and elementary mechanics, for which it is entirely adequate, and more concerned with purely analytic questions, for which the classical theory of integration is not always sufficient. Thus a present-day mathematician is apt to be interested in the convergence of orthogonal expansions, or in applications to differential equations or probability. For him the classical theory of integration which culminated in the Riemann integral has been largely replaced by the theory which has grown from the pioneering work of Henri Lebesgue at the beginning of this century. The reason for this is very simple: the powerful convergence theorems associated with the Lebesgue theory of integration lead to more general, more complete, and more elegant results than the Riemann integral admits.

Lebesgue’s definition of the integral enlarges the collection of functions for which the integral is defined. Although this enlargement is useful in itself, its main virtue is that the theorems relating to the interchange of the limit and the integral are valid under less stringent assumptions than are required for the Riemann integral. Since one frequently needs to make such interchanges, the Lebesgue integral is more convenient to deal with than the Riemann integral. To exemplify these remarks, let the sequence (fn) of functions be defined for x > 0 by . It is readily seen that the (improper) Riemann integrals

With a lntle effort one can show that F is continuous and that its derivative exists and is given by

which is obtained by differentiating under the integral sign. Once again, this inference follows easily from the Lebesgue Dominated Convergence Theorem.

At the risk of oversimplification, we shall try to indicate the crucial difference between the Riemann and the Lebesgue definitions of the integral. Recall that an interval in the set R of real numbers is a set which has one of the following four forms:

In each of these cases we refer to a and b as the endpoints and prescribe b – a as the length of the interval. Recall further that if E is a set, then the characteristic functionof E is the function xe defined by

A step function is a function φ which is a finite linear combination of characteristic functions of intervals; thus

If the endpoints of the interval Ej are aj,bj, we define the integral of φ to be

The Lebesgue integral can be obtained by a similar process, except that the collection of step functions is replaced by a larger class of functions. In somewhat more detail, the notion of length is generalized to a suitable collection X of subsets of R. Once this is done, the step functions are replaced by simple functions, which are finite linear combinations of characteristic functions of sets belonging to X. If

is such a simple function and if μ(E) denotes the “measure” or “generalized length” of the set E in X, we define the integral of φ to be

If f is a nonnegative function defined on R which is suitably restricted, we shall define the (Lebesgue) integral of f to be the supremum of the integrals of all simple functions φ such that φ(x) ≤ f(x) for all x in R. The integral can then be extended to certain functions that take both signs.

for each sequence (E,) of sets in X which are mutually disjoint. In this case an integral can be defined for a suitable class of real-valued functions on X, and this integral possesses strong convergence properties.

As we have stressed, we are particularly interested in these convergence theorems. Therefore we wish to advance directly toward them in this abstract setting, since it is more general and, we believe, conceptually simpler than the special cases of integration on the line or in Rn. However, it does require that the reader temporarily accept the fact that interesting special cases are subsumed by the general theory. Specifically, it requires that he accept the assertion that there exists a countably additive measure function that extends the notion of the length of an interval. The proof of this assertion is in Chapter 9 and can be read after completing Chapter 3 by those for whom the suspense is too great.

In this introductory chapter we have attempted to provide motivation and to set the stage for the detailed discussion which follows. Some of our remarks here have been a bit vague and none of them has been proved. These defects will be remedied. However, since we shall have occasion to refer to the system of extended real numbers, we now append a brief description of this system.

In integration theory it is frequently convenient to adjoin the two symbols – ∞, +∞ to the real number system R. (It is stressed that these symbols are not real numbers.) We also introduce the convention that –∞ < x < + ∞ for any x∈ R. The collection consisting of the set R ∪ { – ∞, +∞} is called the extended real number system.

If (xn) is a sequence of extended real numbers, we define the limit superior and the limit inferior of this sequence by

If the limit inferior and the limit superior are equal, then their value is called the limit of the sequence. It is clear that this agrees with the conventional definition when the sequence and the limit belong to R.

Finally, we introduce the following algebraic operations between the symbols ±∞ and elements x ∈ R:

It should be noticed that we do not define (+∞) + (−∞) or (−∞) + ( + ∞), nor do we define quotients when the denominator is ±∞.

CHAPTER 2

Measurable Functions

Given the set X, we single out a family X of subsets of X which are “well-behaved” in a certain technical sense. To be precise, we shall assume that this family contains the empty set ∅ and the entire set X, and that X is closed under complementation and countable unions.

2.1 DEFINITION. A family X of subsets of a set X is said to be a σ-algebra (or a σ-field) in case:

An ordered pair ( X, X) consisting of a set X and a σ-algebra X of subsets of X is called a measurable space. Any set in X is called an X-measurable set, but when the σ-algebra is fixed (as is generally the case), the set will usually be said to