The Lebesgue-Stieltjes Integral: A Practical Introduction (Undergraduate Texts in Mathematics)

Undergraduate Texts in Mathematics Editors

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(continued after index)

M. Carter

B. van Brunt

The Lebesgue­ Stieltjes Integral

A Practioal Introduction

With 45 Illustrations

Springer

M. Carter B. van Brunt Institute of Fundamental Sciences Palmerston North Campus Private Bag 11222

Massey University

Palmerston North 5301

New Zealand

Editorial Board S. Axler

F.W. Gehring

K.A. Ribet

Mathematics Department

Mathematics Department

Mathematics Department

San Francisco State

East Hall

University of California

University

San Francisco, CA 94132 USA

University of Michigan

Ann Arbor, MI 48109 USA

at Berkeley

Berkeley, CA 94720-3840

USA

Mathematics Subject Classification (2000): 28-01

Library of Congress Cataloging-in-Publication Data Carter, M. (Michael), 1940-

The Lebesgue-Stieltjes integral : a practical introduction I M. Carter, B. van Brunt. p.

em.- (Undergraduate texts in mathematics)

Includes bibliographical references and index. ISBN 0-387-95012-5 (alk. paper) 1. Lebesgue integral. (tA312.C37

I. van Brunt, B. (Bruce)

II. Title. III. Series.

2000

515'.43-dc21

00-020065

Printed on acid-free paper.

© 2000 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Usc in connection with any form of information storage and retrieval, electronic adaptation,

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Printed and bound by R.R. Donnelley and Sons, Harrisonburg, VA. Printed in the United States of America. 9 8 7 6 5 4 3 2 ] ISBN 0-387-95012-5 Springer-Verlag New York Berlin Heidelberg

SPIN 10756530

Preface

It is safe to say that for every student of calculus the first encounter with integration involves the idea of approximating an area by sum­ ming rectangular strips, then using some kind of limit process to obtain the exact area required. Later the details are made more precise, and the formal theory ofthe Riemann integral is introduced. The budding pure mathematician will in due course top this off with a course on measure and integration, discovering in the process that the Riemann integral, natural though it is, has been superseded by the Lebesgue integral and other more recent theories of integra­ tion. However, those whose interests lie more in the direction of applied mathematics will in all probability find themselves needing to use the Lebesgue or Lebesgue-Stieltjes integral without having the necessary theoretical background. Those who try to fill this gap by doing some reading are all too often put offby having to plough through many pages of preliminary measure theory. It is to such readers that this book is addressed. Our aim is to introduce the Lebesgue-Stieltjes integral on the real line in a nat­ ural way as an extension of the Riemann integral. We have tried to make the treatment as practical as possible. The evaluation of Lebesgue-Stieltjes integrals is discussed in detail, as are the key the­ orems of integral calculus such as integration by parts and change of v

vi

Preface

variable, as well as the standard convergence theorems. Multivariate integrals are discussed briefly, and practical results such as Fubini's theorem are highlighted. The final chapters of the book are devoted to the Lebesgue integral and its role in analysis. Specifically, func­ tion spaces based on the Lebesgue integral are discussed along with some elementary results. While we have developed the theory rigorously, we have not striven for completeness. Where a rigorous proof would require lengthy preparation, we have not hesitated to state important theo­ rems without proof in order to keep the book reasonably brief and accessible. There are many excellent treatises on integration that provide complete treatments for those who are interested. The book could also be used as a textbook for a course on in­ tegration for nonspecialists. Indeed, it began life as a set of notes for just such a course. We have included a number of exercises that extend and illustrate the theory and provide practice in the tech­ niques. Hints and answers to these problems are given at the end of the book. We have assumed that the reader has a reasonable knowledge of calculus techniques and some acquaintance with basic real analy­ sis. The early chapters deal with the additional specialized concepts from analysis that we need. The later chapters discuss results from functional analysis. It is intended that these chapters be essen­ tially self-contained; no attempt is made to be comprehensive, and numerous references are given for specific results. Michael Carter Bruce van Brunt Palmerston North, New Zealand

Contents

Preface I

2

3

Real 1.1 1. 2 1.3

v

Numbers Rational and Irrational Numbers . The Extended Real Number System Bounds . . . . . . . . . . . . . . . .

Some Analytic Preliminaries 2. 1 Monotone Sequences 2. 2 Double Series . . . . 2.3 One-Sided Limits . . 2.4 Monotone Functions 2.5 Step Functi0ns . . . . 2.6 Positive and Negative Parts of a Function 2. 7 Bounded Variation and Absolute Continuity The 3.1 3. 2 3.3

Riemann Integral Definition of the Integral Improper Integrals . . . A Nonintegrable Function

I

1 6 8 II

11 13 16 20 24 28 29

39

39 44 46

VII

VIII 4

Contents 49

The Lebesgue-Stieltjes Integral 4. 1 The Measure of an Interval . 4. 2 Probability Measures . . 4.3 Simple Sets . . . . . . . . 4.4 Step Functions Revisited 4.5 Definition of the Integral 4.6 The Lebesgue Integral

49 52 55 56 60 67

5 Properties of the Integral 5.1 Basic Properties . . . . 5.2 Null Functions and Null Sets 5.3 Convergence Theorems . 5.4 Extensions of the Theory 6

7

7I 71 75 79 81

Integral Calculus 6.1 Evaluation of Integrals . . . . . . . 6. 2 'TWo Theorems of Integral Calculus 6.3 Integration and Differentiation .

87 87 97 10 2

Double and Repeated Integrals 7.1 Measure of a Rectangle . . . . . . . . . . . . . . . . 7.2 Simple Sets and Simple Functions in Thro Dimensions . . . . . . . . . . . . 7.3 The Lebesgue-Stieltjes Double Integral . . 7.4 Repeated Integrals and Fubini's Theorem .

II3

8 The 8.1 8. 2 8.3 8.4 8.5 8.6 8.7 8.8 8.9

Lebesgue Spaces V Normed Spaces . . Banach Spaces . . . Completion of Spaces The Space L1 . . . The Lebesgue V . . Separable Spaces . . Complex V Spaces The Hardy Spaces HP Sobolev Spaces Wk,p .

113 114 115 115 I2 3

. . .

.

1 24 131 135 138 14 2 150 15 2 154 161

Contents

����--

--------------------------------------

9

Hilbert Spaces and L2 9 .1 Hilbert Spaces . . 9. 2 Orthogonal Sets . 9.3 Classical Fourier Series 9. 4 The Sturm-Liouville Problem 9.5 Other Bases for L2 •

•

.

.

.

•

1. X

1 65

165 17 2 180 188 199

.

10 Epilogue

203

10. 1 Generalizations of the Lebesgue Integral 10. 2 Riemann Strikes Back . 10.3 Further Reading . . . . .

.

Appendix: Hints and Answers

.

.

to

.

.

.

.

.

.

Selected Exercises

. 203 . 205 . 207 209

References

221

Index

225

Real Numbers CHAPTER

The field of mathematics known as analysis, of which integration is a part, is characterized by the frequent appeal to limiting processes . The properties of real numbers play a fundamental role in analysis . Indeed, it is through a limiting process that the real number system is formally constructed. It is beyond the scope of this book to recount this construction. We shall, however, discuss some of the properties of real numbers that are of immediate importance to the material that will follow in later chapters.

1.1

Rational and Irrational Numbers

The number systems of importance in real analysis include the nat­ ural numbers (N), the integers (Z), the rational numbers (Q), and the real numbers (JR). The reader is assumed to have some famil­ iarity with these number systems. In this section we highlight some of the properties of the rational and irrational numbers that will be used later. The set of real numbers can be partitioned into the subsets of rational and irrational numbers. Recall that rational numbers are

1

2

1. Real Numbers

numbers that can be expressed in the form min, where m and n are integers with n # 0 (for example �� �, -� (= -.,S ), 15 ( = \5), 0( = �)). Irrational numbers are characterized by the property that they cannot be expressed as the quotient of two integers. Numbers such as e, n, and v'2 are familiar examples of irrational numbers. It follows at once from the ordinary arithmetic of fractions that if r1 and r2 are rational numbers, then so are r1 + r2, r1 -r2, r1 r2, and r1/r2 (in the last case, provided that r2 =j:. 0). Using these facts we can prove the following theorem: Theorem 1 . 1 . 1 Ifr is a rational number and x. is an irrational number, then (i) r + x. is irrational; (ii) rx. is irrational, provided that r =j:. 0.

Proof See Exercises 1-1, No. 1.

0

A fundamental property of irrational and rational numbers is that they are both 11dense" on the real line. The precise meaning of this is given by the following theorem: Theorem 1 . 1 . 2 If a and b are real numbers with a < b, then there ex.ist both a rational number and an irrational number between a and b.

Proof Let a and b be real numbers such that a < b. Then b- a > 0, so v'2,!(b -a) > 0. Let k be an integer less than a, and let n be an integer such that n > v'il(b-a) . Then

,J2

- < b-a, n and so the succesive terms of each of the sequences 0

1 n

0 2m + 1 1 m > 0

so that a1 = 01 az = 1 1 a3 = 1 1 a4 = 2 1 and so on. The process of listing the elements of.Z as a sequence can be visualized by following the arrows in Figure 1.1 starting at 0. Much less obvious is the fact that the set Q is also countable. Figure 1. 2 depicts a scheme for counting the rationals. Th list the rationals as a sequence we can just follow the arrowed path in Figure 1. 2 starting at 0/1 = 01 and omitting any rational number that has already been listed. The set -

4

I.

Real Numbers I I ,I

I

-3/3

-2/3

-3/2

I

.f.

I

-1/3

+-

0/3 .-I/3

-2/2

-1/2

+-

0/2

+-

-311

-211

-Ill

011

-+

-3/-1

-2/-1

-1/-1

-3/-2

-2/-2-+ -1/-2

-3/-3

-2/-3

!

+-

!

!

!

!

!

-1/-3

FIGURE 1.2

2/3

3/3

1/2

2/2

3/2

Ill

211

311

3/-1

+--

t

t

t

t

0/-1

-+

11-1

-+

2/-1

-+

0/-2

-+

1/-2

-+

2/-2

1/-3

t

t

-+

0/-3

I I I

I

t

-+

2/-3

t

3/-2 3/-3

Counting the rationals

Q can thus be written as Q

=

{

0, 1,

�·- �· - 1, -2 , 2 , �' �'-�·-�·-�· -3 , 3, ... }.

The infinite sets N, Z, and Q are all countable, and one may won­ der whether in fact there are any infinite sets that are not countable. The next theorem settles that question:

Theorem 1 .1 .4 The set S ofall real numbers x such that 0
0 for all n, m E N. In this case we have the following result, which is stated without proof: •

•

•

I

Suppose that for all n, m E N we have amn > 0, where tlmn E IRe. Then the double series L� n=l amn and the two repeated series Ln=l CLm=l amn) and Lm=l (Ln=; amn ) either all converge to the same finite sum or are all properly divergent to oo. Theorem 2 .2 . I

More details on double series can be found in [6].

f(x)

f+e t t-e

I

----------------

• �

f(x) lies between 1-e and f+E for all X E (t-8, t)

I

t

FIGURE 2.3

2.3

One-Sided Limits

Let f : IR --* IR be a function, and t and .e real numbers. Recall that limx-H fCx) = .e if and only if for any positive real number E, however small, there exists a positive real number 8 such that 0

< lx-t I
0 such that f(x) > M whenever 0 < lx -tl < 8. A similar definition can be made for limx--+tf(x) = - oo. In these definitions x can be either to the left or the right oft, i.e., x is free to approach t from the left or right ( or for that matter oscillate on either side of t). Often it is of use to restrict the manner in which. x approaches t, particularly if no information about f is available on one side of t, or t lies at the end of the interval under consideration. For these situations it is useful to introduce the notion of limits from the left and from the right. Such limits are referred to as one-sided limits. The limit from the left is defined as follows: limx--+r- f(x) = .e if and only if for any positive real number E there exists a positive real number 8 such that t-8 < X < t

=>

lf(x)-ll < E

2.3.

One-Sided Limits

17

f(x)

-

-

-

---

-

1I I

I

I+E for all x e (t, t+8)

- - - _I

I I I I I

t+8

t

FIGURE 2 .4

( cf. Figure 2.3). In this case we say that f(x) tends to .e as x tends to t from the left. Similarly, the limit from the right is defined as limx-+t+ f(x) = .e if and only if for any positive real number E there exists a positive real number 8 such that t

<X
X ==>

lfC x) -a!