Aspects of Integration: Novel Approaches to the Riemann and Lebesgue Integrals 1032481129, 9781032481128

Table of contents :
Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Contributors
Preface
To the Reader
Acknowledgement
Part I: A Novel Approach to Riemann Integration
CHAPTER 1. Preliminaries
1.1 Sums of Powers of Positive Integers
1.2 Bernstein Polynomials
CHAPTER 2. The Riemann Integral
2.1 Method of Exhaustion
2.2 Integral of a Continuous Function
2.3 Foundational Theorems of Integral Calculus
2.4 Integration by Substitution
CHAPTER 3. Extension to Higher Dimensions
3.1 Method of Exhaustion
3.2 Bernstein Polynomials in 2 Dimensions
3.3 Integral of a Continuous Function
Integrals Over More General Domains
Iterated Integration
CHAPTER 4. Extension to the Lebesgue Integral
4.1 Convergence and Cauchy Sequences
4.2 Completion of the Rational Numbers
Conundrum
Calculus
4.3 Completion of C in the 1-norm
Part II: Lebesgue Integration
Chapter 5. The Riesz-Nagy Approach to the Lebesgue Integral
5.1 Null Sets and Sets of Measure Zero
5.2 Lemma's A and B
5.3 The Class C1(I) of Riesz and Nagy
5.4 The Class C2 of Riesz and Nagy
5.5 Convergence Theorems
5.6 Completeness
5.7 The C2-Integral is the Lebesgue Integral
CHAPTER 6. Comparing Integrals
6.1 Properly Integrable Functions
6.2 Characterization of the Riemann Integral
6.3 Riemann vs. Lebesgue Integrals
6.4 The Novel Approach
Appendix A. Dini's Lemma
Appendix B. Semicontinuity
Appendix C. Completion of a Normed Linear Space
References
Index

Citation preview

Aspects of Integration Aspects of Integration: Novel Approaches to the Riemann and Lebesgue Integrals is comprised of two parts. The first part is devoted to the Riemann integral, and provides not only a novel approach but also includes several neat examples that are rarely found in other treatments of Riemann integration. Historical remarks trace the development of integration from the method of exhaustion of Eudoxus and Archimedes, used to evaluate areas related to circles and parabolas, to Riemann’s careful definition of the definite integral, which is a powerful expansion of the method of exhaustion and makes it clear what a definite integral really is. The second part follows the approach of Riesz and Nagy in which the Lebesgue integral is developed without the need for any measure theory. Our approach is novel in part because it uses integrals of continuous functions rather than integrals of step functions as its starting point. This is natural because Riemann integrals of continuous functions occur much more frequently than do integrals of step functions as a precursor to Lebesgue integration. In addition, the approach used here is natural because step functions play no role in the novel development of the Riemann integral in the first part of the book. Our presentation of the Riesz-Nagy approach is significantly more accessible, especially in its discussion of the two key lemmas upon which the approach critically depends, and is more concise than other treatments. Features • Presents novel approaches designed to be more accessible than classical presentations. • Is a welcome alternative approach to the Riemann integral in undergraduate analysis courses. • Makes the Lebesgue integral accessible to upper division undergraduate students. • Shows how completion of the Riemann integral leads to the Lebesgue integral. • Contains a number of historical insights. • Gives added perspective to researchers and postgraduates interested in the Riemann and Lebesgue integrals. About the cover image The cover of the book depicts a painting by Domenico Fetti (1620) of Archimedes of Syracuse (c. 287–212 BC), one of the greatest scientists of antiquity who made fundamental contributions to mathematics, physics, and engineering. The image was chosen because Archimedes was a master in the use of the method of exhaustion, the fundamental idea of integral calculus, and used it to find the areas under parabolic arcs and the area of circles. The method was discovered by Eudoxus and later discussed in Euclid’s Elements. Nearly 2,000 years passed before mathematicians, toward the end of the Renaissance, extended the method of exhaustion to develop the robust methods of integral calculus. Toward the end of this period, in 1854, Riemann gave the precise definition of a definite integral, the one used today. Essentially the same fundamental idea, with a twist, was used by Lebesgue when he developed the Lebesgue integral in the first decade of the 20th century.

Monographs and Research Notes in Mathematics Series Editors: John A. Burns, Thomas J. Tucker, Miklos Bona, and Michael Ruzhansky

About the Series This series is designed to capture new developments and summarize what is known over the entire field of mathematics, both pure and applied. It will include a broad range of monographs and research notes on current and developing topics that will appeal to academics, graduate students, and practi­ tioners. Interdisciplinary books appealing not only to the mathematical com­ munity, but also to engineers, physicists, and computer scientists are encouraged. This series will maintain the highest editorial standards, publishing welldeveloped monographs as well as research notes on new topics that are final, but not yet refined into a formal monograph. The notes are meant to be a rapid means of publication for current material where the style of exposition reflects a developing topic. Banach-Space Operators On C*-Probability Spaces Generated by Multi Semicircular Elements Ilwoo Cho Applications of Homogenization Theory to the Study of Mineralized Tissue Robert P. Gilbert, Ana Vasilic, Sandra Klinge, Alex Panchenko, Klaus Hackl Constructive Analysis of Semicircular Elements From Orthogonal Projections to Semicircular Elements Ilwoo Cho Inverse Scattering Problems and Their Applications to Nonlinear Integrable Equations, Second Edition Pham Loi Vu Generalized Notions of Continued Fractions Ergodicity and Number Theoretic Applications Juan Fernández Sánchez, Jerónimo López-Salazar Codes, Juan B. Seoane Sepúlveda, Wolfgang Trutschnig Aspects of Integration Novel Approaches to the Riemann and Lebesgue Integrals Ronald B. Guenther, John W. Lee For more information about this series please visit: https://www.routledge.com/ Chapman--HallCRC-Monographs-and-Research-Notes-in-Mathematics/bookseries/CRCMONRESNOT

Aspects of Integration Novel Approaches to the Riemann and Lebesgue Integrals

Ronald B. Guenther and John W. Lee Department of Mathematics, Oregon State University, Corvallis, Oregon USA

Designed cover image: Archimedes Thoughtful (also known as Portrait of a Scholar) by Domenico Fetti, 1620 First edition published 2024 by CRC Press 2385 NW Executive Center Drive, Suite 320, Boca Raton, FL 33431 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN CRC Press is an imprint of Taylor & Francis Group, LLC © 2024 Ronald B. Guenther and John W. Lee Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www. copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. ISBN: 978-1-032-48112-8 (hbk) ISBN: 978-1-032-48523-2 (pbk) ISBN: 978-1-003-38948-4 (ebk) DOI: 10.1201/9781003389484 Typeset in Minion by MPS Limited, Dehradun

Contents Contributors, vii Preface, ix Acknowledgement, xiii PART I

A Novel Approach to Riemann Integration

CHAPTER 1

■

Preliminaries

5

1.1

SUMS OF POWERS OF POSITIVE INTEGERS

5

1.2

BERNSTEIN POLYNOMIALS

9

CHAPTER 2

The Riemann Integral

15

2.1

METHOD OF EXHAUSTION

15

2.2

INTEGRAL OF A CONTINUOUS FUNCTION

21

2.3

FOUNDATIONAL THEOREMS OF INTEGRAL CALCULUS

26

INTEGRATION BY SUBSTITUTION

31

2.4

CHAPTER 3

■

■

Extension to Higher Dimensions

41

3.1

METHOD OF EXHAUSTION

42

3.2

BERNSTEIN POLYNOMIALS IN 2 DIMENSIONS

46

3.3

INTEGRAL OF A CONTINUOUS FUNCTION

49

CHAPTER 4

■

Extension to the Lebesgue Integral

59

4.1

CONVERGENCE AND CAUCHY SEQUENCES

60

4.2

COMPLETION OF THE RATIONAL NUMBERS

61 v

vi ▪ Contents

4.3

PART II

COMPLETION OF

IN THE 1-NORM

66

Lebesgue Integration

CHAPTER 5

■

The Riesz-Nagy Approach to the Lebesgue Integral

75

5.1

NULL SETS AND SETS OF MEASURE ZERO

77

5.2

LEMMA’S A AND B

82

5.3

THE CLASS C1(I) OF RIESZ AND NAGY

84

5.4

THE CLASS C2 OF RIESZ AND NAGY

91

5.5

CONVERGENCE THEOREMS

96

5.6

COMPLETENESS

105

5.7

THE C2-INTEGRAL IS THE LEBESGUE INTEGRAL

107

CHAPTER 6

■

Comparing Integrals

111

6.1

PROPERLY INTEGRABLE FUNCTIONS

115

6.2

CHARACTERIZATION OF THE RIEMANN INTEGRAL

118

6.3

RIEMANN VS. LEBESGUE INTEGRALS

122

6.4

THE NOVEL APPROACH

127

APPENDIX A

■

Dini’s Lemma

129

APPENDIX B

■

Semicontinuity

131

APPENDIX C

■

Completion of a Normed Linear Space

135

References, 139 Index, 143

Contributors Ronald B. Guenther is an emeritus professor in the Department of Mathematics at Oregon State University. His career began at the Marathon Oil Company where he served as an advanced research mathematician at its Denver Research Center. Most of his career was spent at Oregon State University, with visiting professorships at the Universities of Hamburg and Augsburg, and appointments at research laboratories in the United States and Canada, and at the Hahn-Meitner and Weierstrass Institutes in Berlin. His research interests include mathematically modeling deterministic systems and the ordinary and partial differential equations that arise from these models. John W. Lee is an emeritus professor in the Department of Mathematics at Oregon State University, where he spent his entire career with sabbatical leaves at Colorado State University, Montana State University, and many visits as a guest of Andrzej Granas at the University of Montreal. His research interests include topological methods used to study nonlinear differential and integral equations, oscillatory properties of problems of Sturm-Liouville type and related approximation theory, and various aspects of functional analysis and real analysis, especially measure and integration.

vii

Preface

T

HIS SMALL BOOK HAS TWO PARTS,

both with novel aspects; one part is on Riemann integration, and the other on Lebesgue integration. The reader is invited to explore certain aspects of the Riemann integral, the Lebesgue integral, improper integrals, and connections among them. The connections are often part of the mathematical folklore but tend not to make it into print. Riemann integrals are presented in a novel way that provides an attractive alternative to traditional developments that start with a heavy dose of properties of partitions and repeated uses of Riemann sum and limit passage arguments to establish many fundamental properties of the integral. Lebesgue integration is usually based on measure theory, but it doesn’t have to be as Riesz and Nagy pointed out in their classic text on functional analysis [8]. We find their approach intriguing, but difficult to follow. Others seem to have arrived at the same conclusion. An accessible version of the Riesz-Nagy approach is given in Part II of this book. We give a simple and straightforward development of the Lebesgue integral without the need to introduce any measure theory. The first part of the book takes a novel approach to the theory of Riemann integration. In a nutshell, it starts with a simple asymptotic formula for finite sums of p-th powers of the positive integers. These sums and the Method of Exhaustion due to Eudoxus and Archimedes lead naturally to the integrals of power functions, then to the integral of polynomials, and then, with the aid of the Weierstrass approximation theorem and of Bernstein polynomials, to the integral of any continuous function. It is relatively easy to show the integral of a continuous function defined in this way is what we now call a (proper) Riemann integral, as Riemann defined it in 1854. More effort is required to show that the integral defined here is coextensive with the proper Riemann integral. This is confirmed toward the end of the book.

ix

x ▪ Preface

The second part of the book follows the spirit of the approach of Riesz and Nagy to Lebesgue integration but with a novel twist at the start. Riesz and Nagy develop the Lebesgue integral by starting with the integral of a step function. They mention in passing that step functions and their integrals could be replaced by continuous functions and their (Riemann) integrals, but they do not pursue this possibility. We find the alternative of starting with continuous functions rather than step functions quite attractive and have used it in Part II. Step functions play little if any role in traditional introductory calculus courses, while continuous functions are front and center throughout the development of Riemann integration. So, it is natural to start with the Riemann integral when moving on to Lebesgue integration. This approach also fits well with the novel approach to the Riemann integral in Part I because step functions play no role in that approach. We compare the two types of integrals as we go along. For now, suffice it to say that both integrals play important but different roles in mathematics and its applications. The hallmarks of Lebesgue integration are its convergence theorems. The hallmarks of Riemann integration are the ease with which integrals can be evaluated, both exactly and numerically, and its simplicity, which makes meaningful applications almost suggest themselves.

TO THE READER Part I of the book, a novel approach to Riemann integration, can be profitably read by anyone who has had a proof-oriented course in Euclidean geometry and completed a two or three semester course in calculus, one that emphasizes the techniques of calculus and treats the limit concepts that lie at the foundation of calculus in an intuitive but insightful way. In particular, a course in advanced calculus or introductory real analysis is not a prerequisite. However, Part I, with its fresh approach to the Riemann integral, would be a nice complement to such a course – the novel approach to the Riemann integral can replace what amounts to a more precise version of the Riemann sum and limit passage reasoning already used in beginning calculus. Part II of the book, which follows the Riesz-Nagy approach to the Lebesgue integral, requires more mathematical maturity. A reader who has taken a course in advanced calculus will have the required mathematical training, which includes a precise understanding of the

Preface ▪ xi

limit concept and familiarity with sequences, series, open and closed sets, greatest lower bounds (infs), least upper bounds (sups), continuity of real-valued functions of a real variable, and compactness of sets of real numbers. Dini’s lemma is used in Chapter 5 at a pivotal point, and elementary properties of semicontinuous functions are used in one proof in Chapter 6. These topics are included in two appendices because they may not be part of typical advanced calculus courses. For the most part, standard mathematical conventions and symbols are used throughout the book. In particular, is the set of natural numbers 1, 2, 3, …; is the set of integers, positive, negative, and zero; is the set of rational numbers p/q where p and q are integers and q 0; and is the set of real numbers.

Acknowledgement The mathematicians whose methods and discoveries particularly influenced this book include Eudoxus, Archimedes, Dirichlet, Riemann, Weierstrass, Bernstein, Lebesgue and Riesz-Nagy. We owe them a great debt that we acknowledge here. We express our sincere appreciation to the editorial and production staffs that took great care to turn our manuscript into an attractive, inviting book. In particular, we thank Callum Fraser, Mathematics Books Editor at CRC Press, Mansi Kabra, Senior Editorial Assistant, Manmohan Negi, Production Manager, and their staffs.

xiii

I A Novel Approach to Riemann Integration

W

E BUILD UP THE THEORY of Riemann integration starting with a simple

approximation to sums of powers of natural numbers. At each step in the theory, we add a small bit of abstraction and go on to show how this leads to the integration of polynomials over a closed and bounded interval. The path we follow is one motivation for the Weierstrass approximation theorem and highlights the importance of Bernstein’s proof of that theorem. We then follow the outline provided by Riemann himself to extend the integral to discontinuous functions and improper integrals. The goal is to show that the principal results of Riemann integration follow from three elementary results: 1. An asymptotic formula for the sums of powers of the first n positive integers. 2. The integrals of the power functions xp for p a natural number or zero. 3. Bernstein’s version of the Weierstrass approximation theorem. This approach avoids the standard repeated use of Riemann sums and limit passage arguments to establish many of the fundamental properties of the integral.

DOI: 10.1201/9781003389484-1

1

2 ▪ A Novel Approach to Riemann Integration

We hasten to mention that our focus is on the foundational results of Riemann integration and not on the many topics and applications covered in standard one-year to two-year calculus courses. They can be found elsewhere. Before proceeding with the approach to Riemann integration described above, it is quite interesting to paraphrase what Riemann had b

to say about the meaning of a f (x ) dx in his 1854 paper, Ueber die Darstellbarkeit einer Functionen durch eine trigonometrishe Reihe [7]: The indeterminacy, which still exists in the understanding of the definite integral, makes it necessary for us, to bring forward some things about the concept of a definite integral and the extent of its validity. b

First: What is one to understand by a f (x ) dx? To answer this, take a series of numbers x1, x2, … , xn−1 that follow each other between a and b and denote the short paths x1 a by δ1, x2 x1 by δ2, … , b xn 1 by δn and by a positive proper fraction. Then the value of the sum

S=

1f

(a +

1 1)

+

2f

(x1 +

2 2)

+

+ n f (x n

1

+

n n)

depends on the choice of the intervals δ and of the quantities . If the sum has the property that, however δ and may be chosen, it approaches infinitesimally closely a fixed limit A, as soon as all the δ become infinitesimally small, then this value [A] is called does not have this property, then b a

b a

b a

f (x ) dx . If the sum

f (x ) dx has no meaning.

Riemann says next that f (x ) dx has no meaning if the function f (x ) becomes unbounded at a point in the interval because then the limit above b

cannot exist, but he observes that mathematicians also use a f (x ) dx in this case. He then gives the currently accepted definition of such an improper integral: if a function f (x ) becomes unbounded as x approaches a single point c in the interval [a, b], Riemann defines b a

f (x ) dx = lim 1

0

c a

1

f (x ) dx + lim

provided the indicated limits exist.

2

0

b c+

b a

f (x ) dx 2

f (x ) dx by ( 1,

2

> 0)

Aspects of Integration ▪ 3

He continues with a comment on what is now called a Cauchy principal value integral. Riemann’s comment is harder to paraphrase but says: other determinations by Cauchy about the concept of the definite integral in cases where there is no such thing according to the basic concept [the definition above] may be useful for individual classes of investigations; however, they have an arbitrariness that makes them less suitable for this purpose [a general definition of the integral]. b

We emphasize that Riemann views a f (x ) dx as defined when it satisfies either of the definitions he gave in his 1854 paper, and he calls such a function f (x ) integrable. Riemann does not distinguish between proper Riemann integrals and improper Riemann integrals, as we do today. A function is either integrable or no meaning is assigned to the symbol

b a

f (x ) dx .

CHAPTER

1

Preliminaries

T

background material that will be used in the novel approach to the Riemann integral coming later.

HIS CHAPTER CONTAINS

1.1 SUMS OF POWERS OF POSITIVE INTEGERS Let p {0, 1, 2, …} and n be a positive integer. Express the sum of the p-th powers of the first n positive integers by

Snp =

n

k p.

k =1

Familiar formulas from undergraduate mathematics are

Sn0 = n , Sn1 = n (n + 1)/2, Sn2 = n (n + 1)(2n + 1)/6. The first result is obvious. The second is attributed to Gauss, a result he obtained by inspection, reasoning as follows:

1 2 3 n (n 1) (n 2)

DOI: 10.1201/9781003389484-2

(n

2) (n 3

1) n 2 1

5

6 ▪ Aspects of Integration

Adding the vertically aligned pairs of summands yields

2Sn1 = (n + 1) + (n + 1) +

+(n + 1) = n (n + 1).

We will establish the results Sn2 and for Sn3 after we describe a natural geometric construction in the first quadrant of the xy -plane that gives a recursive formula for Snp in terms of S1p 1, …, Snp 1. As you read what follows, it will be helpful to sketch the construction step-by-step for the special case p = 2 and n = 4. On the x -axis mark the origin O and the points with x -coordinates S1p 1, S2p 1, …, Snp 1. These points divide the interval from O to Snp 1 on the x -axis into n horizontal line segments H1 to Hn and the k -th line segment Hk has length k p 1. On the y -axis mark the points with y -coordinates 1, 2, …, n. These points divide the interval from O to n on the y -axis into n vertical line segments V1 to Vn and the k -th segment Vk has length 1. Now, for k = 1 to n, construct a rectangle Rk with base the k -th horizontal segment Hk and with height k . The rectangle Rk has area k p, the union of the n rectangles, U, has area Snp , and U lies in the rectangle T = [0, Snp 1] × [0, n]. The part of the rectangle T outside U is comprised of n 1 rectangles. The k -th rectangle R˜k has for one of its vertical sides the segment Vk+1 for k = 1, …, n 1 on the y -axis, and its horizontal sides have length Skp 1. So R˜k has area Skp 1. Since the rectangle T is composed of the non-overlapping rectangles Rk for k = 1, …, n and R˜k for k = 1, …, n 1, the area of T satisfies

nSnp

1

= Snp + S1p

1

+ S2p

1

+

+ Snp 11,

which it is convenient to express as the recursion formula

Snp = nSnp

1

n 1 k =1

Skp 1.

Of course, once this formula is known, it also can be easily confirmed by mathematical induction on n.

Preliminaries ▪ 7

When p = 2 the recursion formula gives n 1

Sn2 = nSn1

k =1

Sk1,

and using Gauss’ result for p = 1 yields

Sn2

1 = n n (n + 1) 2

6Sn2

k (k + 1),

k =1

1 (n 1) n, 2 1 = n2 (n + 1) Sn2 + n2 (n 1) n 2 = 2n2 (n + 1) + 2n2 (n 1) n,

2Sn2 = n2 (n + 1) 2Sn2

n 1

1 2 Sn2

1

6Sn2 = n (2n2 + 3n + 1) = n (n + 1)(2n + 1), which confirms the result given earlier. When p = 3 the recursion formula gives n 1

Sn3 = nSn2

k =1

Sk2,

and using the result just found for p = 2 yields

Sn3

1 = n n (n + 1)(2n + 1) 6

6Sn3

= n2 (n + 1)(2n + 1)

8Sn3

= 2n3 + n2 (n + 1)(2n + 1)

16Sn3

1 6 2Sn3

n 1

k (k + 1)(2k + 1),

k =1 1

3Sn2

Sn1 1,

1

3 1 (n 1) n (2n 1) (n 1) n , 6 2 = 4n3 + 2n2 (n + 1)(2n + 1) (n 1) n (2n 1) (n 1) n.

A little patient arithmetic yields

Sn3 =

n (n + 1) 2

2

.

8 ▪ Aspects of Integration

The recursion relation enables us to find Snp for any positive integer p but, as a practical matter, the situation is different. There is no simple general formula that expresses Snp as a polynomial in n of degree p + 1. Fortunately, we do not need such a formula to develop the theory of Riemann integration. What is needed is an asymptotic formula for the sums Snp . A likely candidate for an asymptotic formula can be guessed from the expressions for Snp for p = 1, 2, and 3, which yield

( ), + O ( ), = + O ( ).

Sn1 n2

=

1 2

+

1+1/n n

=

1 2

Sn2 n3

=

1 3

+

3+1/n n

=

1 3

Sn3 n4

=

1 4

+

1 / 2 + 1 / 4n n

+O

1 n 1 n

1 4

1 n

()

()

1

1

M

In these results, O n stands for a function such that |O n | n for some constant M and all natural numbers n. We follow standard usage:

() 1

each use of the symbol O n may stand for a different function that is bounded by a constant times 1/n. With this convention,

rO

1 1 1 1 1 1 + sO =O and O O =O n n n n n n

()

1 0 as n for any real numbers r and s. Of course, O n for any O-function. The asymptotic results for p = 1, 2, and 3 suggest that

1 p 1 1 Sn = +O n p +1 p+1 n

(1.1)

for any fixed positive integer p. One way to prove this is by induction on n for each fixed p. If n = 1 then

p S1p 1 1 1 =1= + = +O . p +1 1 p+1 p+1 p+1 1 So, the basis for induction is established. Assume inductively that for an arbitrarily fixed positive integer n that Snp /n p+1 = 1/(p + 1) + O (1/n). Then

Preliminaries ▪ 9 Snp+1 (n + 1) p +1

=

1 + 2 p + n p + (n + 1) p (n + 1) p +1

=

1 + 2 p + + np n p +1

= =

p +1 n n+1

+

(n + 1) p (n + 1) p +1

(1 O ( ) )

+O 1 n

1 p+1 1 p+1

( )

1 n

p +1

+

1 n+1

() 1 n

+O

and the induction step is advanced. Thus,

Snp 1 1 = +O p+1 n p+1 n for any fixed positive integer p and any natural number n. The formula also holds when p = 0.

1.2 BERNSTEIN POLYNOMIALS Let f be a real-valued function defined for 0 polynomial of f is

Bn f (x ) =

n

f

k =0

k n

n k x (1 k

Bn is a linear operator on the vector space on [0, 1]; that is,

1. The n-th Bernstein

x

x )n k .

of all real-valued functions

Bn ( f + g ) = Bn f + Bn g for f and g in

and all real scalars α and β. Bn is also monotone,

f

0

Bn f

0.

Bernstein introduced the polynomials that now bare his name in the context just given. However, there are corresponding Bernstein polynomials defined on any closed bounded interval [a, b]. They will be introduced shortly.

10 ▪ Aspects of Integration

For p a natural number or 0 and x a real variable, xp denotes the polynomial function f (x ) = x p. In the initial discussion of Bernstein polynomials, x is restricted to 0 x 1. The Bernstein polynomials have the following properties, closely related to the fact that for each natural n k x (1 x )n k for k = 0, 1, …, n number n and each x [0,1], pk = k defines a probability distribution on {0, 1, …, n}. This is a direct consequence of the binomial theorem; see item 2 below. Bernstein introduced these polynomials in (1912) when he gave a new proof of the Weierstrass approximation theorem. Bernstein’s proof is expressed in probabilistic language. The proof given below, that does not use probabilistic reasoning, is due to Natanson [5]. Both proofs rely on the following properties of the Bernstein polynomials:

, Bn x p +1 = xBn x p +

{0}, and n

1. For p

x (1 x ) (Bn x p) n

.

2. Bn 1 = 1. 3. Bn x = x . x (1 x ) . n

4. Bn x 2 = x 2 +

Proof of 1. By the product rule

(Bn x p)

=

n k =0

=

n x +

n k =0

n p

k n

p +1

n

n 1

x

k =0 n

n 1 =

x n

x (1

[kx

p

k n

k =0

x)

k 1 (1

x )n

k

n k p x (1

x )n

k

k n

p +1

k n

p

Bn x p +1

n k p x (1

n k p x (1 n x (1

x)

k) x k (1

(n

x )n x )n

k

xBn x p

and solving for Bn x p+1 gives the desired result. Proof of 2, 3, and 4. By the binomial theorem

k

x )n

k 1

]

Preliminaries ▪ 11

1 = (x + (1

n

x ))n =

n k x (1 k

k =0

x )n

k

which is item 2. By items 1 and 2,

Bn x = xBn 1 +

x (1

x) n

(Bn 1) = x .

which is item 3. By items 1 and 3,

Bn x 2 = xBn x +

x (1

x)

(Bn x ) = x 2 +

n

x (1

x) n

.

Weierstrass’ original proof of his approximation theorem used what are now called reproducing properties of the fundamental solution of the heat equation. Bernstein used items 2, 3, and 4 in his proof of the Weierstrass Approximation Theorem. Theorem 1: (Bernstein) If f is a continuous function on, then the Bernstein polynomials Bn f converge uniformly to f on [0, 1] as n . Proof. Let N = {0, 1, …, n}. Given f (x ) f (x ) < when x x
0 there is a > 0 such that by the uniform continuity of k | n

N : |x

< }

and

set

Cn, x =

}. By continuity, there is a constant M such that

[0, 1]. Let

Bn f (x ) =

n k =0

f (x )

f

k n

n k x (1 k

x )n

k

12 ▪ Aspects of Integration

n (x )

+

f (x )

k A n, x

k n

f

k C n, x

n k x (1 k

x )n k .

The first sum on the right is bounded by by item 2 above. Use items 2, 3, and 4 to estimate the second sum: for k Cn, x we have

(x

k /n)2 /

2

1 and, hence,

f (x )

k n

f

k c n, x

=

2M 2

k 2 n

2nx + 2

k c n, x

2

()

k

x2

2M

n k x (1 k

n k =0

x2

x2

k k 2 x+ n n

2x 2 + x 2 +

x )n

f (x ) 2

k

k n

f

n k x (1 k

x (1 x ) n

n k x (1 k x )n

x )n

k

k

M 2n 2

for all x in [0, 1]. Therefore, n (x )

+

M 2n 2

0 uniformly on [0, 1] as for all x in [0, 1]. It follows that n (x ) n f uniformly on [0,1] as n ; equivalently, Bn f .■ The proof just given can be applied under weaker assumptions to obtain an interesting related result. Assume that a function f (x ) is bounded on [0, 1] by a constant M and is continuous at a point x. In this situation, given > 0 there exists x > 0 such that f (z ) f (x ) < provided z x < x and the reasoning in the foregoing proof gives

Preliminaries ▪ 13 n (x )

+

M . 2n x2

f (x ) as n Consequently, Bn f (x ) the function is continuous.

at each point x in [0, 1] where

Proposition 2: If a real-valued function f (x ) is bounded on the interval [0, 1], then Bn f (x ) f (x ) as n at each point x in [0, 1] where the function is continuous. The Bernstein polynomials Bn f defined above are associated with the interval [0, 1]. There are corresponding Bernstein polynomials Bn f associated with any interval [a, b] with a < b. The Bernstein polynomials on [a, b] inherit the properties established above for the interval [0, 1] via a linear change of variables. Let f be a real-valued function defined for a x b. The n-th Bernstein polynomial of f on [a, b] is

Bn f (x ) =

n

f a+

k (b

a) n

k =0

n k

x b

Bn is a linear operator on the vector space on [a, b]; that is,

a a

k

b b

x a

n k

.

of all real-valued functions

Bn ( f + g ) = Bn f + Bn g for f and g in

and all real scalars α and β. Bn is also monotone,

f

0

Bn f

0.

The change of variables x = a + t (b a) is a linear one-to-one map of the unit interval [0, 1] onto the interval [a, b]. The linear map t = (x a)/(b a) is its inverse. This change of variables sets up a oneto-one correspondence between real-valued functions F (t ) on [0, 1] with real-valued functions f (x ) on [a, b] given by f (x ) = F ((x a)/(b a)) or equivalently F (t ) = f (a + t (b a)). Under this change of variables, a

14 ▪ Aspects of Integration

function f (x ) on [a, b] and its Bernstein polynomial Bn f (x ) on [a, b] correspond to F (t ) and its Bernstein polynomial Bn F (t ) on [0, 1]. This leads immediately to Bernstein’s version of the Weierstrass approximation theorem for functions continuous on a closed bounded interval. Theorem 3: (Bernstein) If f is a continuous function on a closed bounded interval [a, b], then the Bernstein polynomials Bn f converge uniformly to f on [a, b] as n . Proof. Use the correspondence above. Given > 0 there is a index N = N such that n N implies that F (t ) Bn F (t ) < for all t in [0, 1], equivalently, f (x ) Bn f (x ) < for all x in [a, b]. ■ Similar reasoning transfers the four properties of the Bernstein polynomials established for the interval [0, 1] and Proposition 2 to their analogues for the interval [a, b]. For instance, Bn t 2 = t 2 + t (1 t )/n transfers to

Bn

x b

a a

2

=

x b

a a

2

+

(x

a)(b x ) (b a)2 n

for x in [a, b] and if f (x ) is bounded on [a, b] and continuous at x, then

lim Bn f (x ) = f (x ).

n

CHAPTER

2

The Riemann Integral

T

with the method of exhaustion of Eudoxus and Archimedes for finding areas of a few important shapes of antiquity. In so doing, they anticipated the key idea of integral calculus over a millennium before its more systematic development. We use the method of exhaustion to find areas related to the power functions xp for p a natural number or zero. This leads to the definite integrals of power functions and polynomials. Next comes our novel definition of the Riemann integral for continuous functions and a collection of its most important properties. Especially noteworthy is the careful treatment of the method of integration by substitution. This method has some subtle aspects that often are swept under the rug in a first course in calculus. The chapter also includes some illuminating examples. Especially striking is the example of Dirichlet from 1837. HIS CHAPTER BEGINS

2.1 METHOD OF EXHAUSTION The method of exhaustion anticipated the key idea of integral calculus nearly a millennium before its systematic development. Archimedes found the area under the arch of a parabola and the area of a circle, the latter by showing how to approximate π as accurately as desired. We use Archimedes’ method to find areas under or above arches of the power functions. Let y = x p for p a natural number. We wish to give a precise definition for the area A between the graph if y = x p and the x-axis for 0 x 1. When a graph lies above the x-axis, we also refer to this area as the area under the graph and above the interval [0, 1]. DOI: 10.1201/9781003389484-3

15

16 ▪ Aspects of Integration

1, n = 1, 2, 3, …, and k /n

For k = 0, 1, …, n

p

k n

k+1 n

xp

x

(k + 1)/n,

p

with strict inequality except at the endpoints because x p is strictly increasing on [0, 1], It follows that the area A should satisfy

1 n

p

1 2 + n n

p

1 + n

n

+ +

p

1

1 1 0. There are two cases to consider, p is even and p is odd. If p is even, then the graph if y = x p is symmetric about the y-axis and

A[a, b] = Ab + A a =

a p +1 b p +1 b p +1 + = p+1 p+1 p+1

a p +1 . p+1

If p is odd, then the graph if y = x p is symmetric with respect to the origin and, hence,

A[a, b] = Ab Second, if a

b

A[a, b] = A[

Aa =

b p +1 p+1

a p +1 b p +1 = p+1 p+1

a p +1 . p+1

0 and p is even, then

b , a]

=

( a) p +1 p+1

( b) p +1 b p +1 = p+1 p+1

a p+1 p+1

The Riemann Integral ▪ 19

while if p is odd,

A[a, b] = A[

( a) p +1 ( b) p +1 b p +1 + = p+1 p+1 p+1

=

b, a]

a p +1 . p+1

The foregoing discussion enables us to define the Riemann integral of the power functions. Shortly, we will use integrals of power functions to define integrals of polynomials and then integrals of continuous functions. Definition 4: For p a nonnegative integer and a and b real numbers with a b, the integral of x p over a x b is b a

n

x pdx = lim n

The integral

a+k

b

a

p

b

n

k =1

a

=

n

b p +1 p+1

a p +1 . p+1

dx becomes an oriented integral by defining b a

x pdx =

a b

x pdx

when b < a. It follows that b a

x pdx =

a b

x pdx

for any real numbers a and b. Observe that b a

x pdx =

c a

x pdx +

b c

x pdx

for any real numbers a, b, and c because c a

x pdx +

b c

x pdx =

c p +1 p+1

a p +1 b p+1 + p+1 p+1

c p +1 . p+1

20 ▪ Aspects of Integration

dx, to the real linear space P = P [a , b] Next, we extend the integral, of all polynomials on the real line with real coefficients in a natural way: if f (x ) = Pp =0 ap x p, then n

f a+

k (b

a) b n

k =1

a

P

=

n

p =0

n

ap

k (b

a+

a)

p

n

k =1

b

a n

.

This shows that there exists n

lim

n

f a+

k (b

a) b n

k =1

n

a

f (x ) dx = lim n

P

=

p =0

b

ap

a

x kdx.

dx to P by defining

and we extend the definition of b

a

n

f a+

k (b

a) b n

k =1

a n

and obtain at once P

b a

p =0

ap x p dx =

P p =0

ap

b a

x kdx.

The integral on P has the following properties, in which f and g are polynomials and c and d are real numbers: b

b

b

1. (Linearity) a (cf (x ) + dg (x )) dx = c a f (x ) dx + d a g (x ) dx . This follows from the representation (cf + dg )(x ) = N p k =0 (ca p + dbp) x for some N and the formula above this item. 2. (Monotonicity) If f

f

P and a

0 on [a , b]

b, then b a

f (x ) dx

0.

The Riemann Integral ▪ 21

Consequently, if m

m (b

f

M on [a, b] for constants m and M, then b

a)

a

f (x ) dx

M (b

a)

and b a

f (x ) dx

(b

a)max( m , M ).

(

k (b

a)

)

b

a

The first assertion follows from nk =0 f a + n n second by applying the first assertion to the polynomials f M f , and the third from the second. 3. (Interval Additivity) If f c a

f (x ) dx +

0, the m and

P and a, b, and c are real numbers, then b c

f (x ) dx =

b a

f (x ) dx.

This follows from linearity and the corresponding property when f (x ) = x p for some nonnegative integer p.

dx to all continuous functions We shall extend the definition of on [a, b] using properties of the Bernstein polynomials. The extended integral will inherit linearity, monotonicity, and interval additivity virtually by its very definition.

2.2 INTEGRAL OF A CONTINUOUS FUNCTION The definition that follows is the basis for our novel approach to the Riemann integral. Let a < b and C = C [a, b] be the linear space of continuous functions on [a, b]. Definition 5: For f

C, b a

f (x ) dx = lim n

b a

(Bn f )(x ) dx ,

where Bn f is the n-th Bernstein polynomial of f on [a, b].

22 ▪ Aspects of Integration

The definition is well-posed. Two things must be established: (a) The indicated limit exists. (b) If f (x ) = x p for p a nonnegative integer, then the limit on the right is (b p +1 a p +1)/(b a). A little later we will confirm that the integral just defined is the (proper) Riemann integral as Riemann defined it. Proof of (a). It follows from Bernstein’s theorem that given > 0 there is an index N = N such that n, m N implies

Bm f (x )

Bn f (x )

(Bm f (x )

f)

[a , b]. Therefore, for m, n

for all x

Bm f (x ) (b

Bn f (x ) b a

a)

(f

Bn f (x ))
0 there is an < pn (x ) Bn f (x ) < for index N = N such that n N implies that all x in [a, b]. Therefore,

(b

a)
0)

26 ▪ Aspects of Integration

provided the indicated limits exist and with a one-side limit understood if c = a or c = b. Now integrals of this type are called improper Riemann integrals, but Riemann just regarded them as integrals. That is, he b

did not distinguish between proper and improper integrals. a f (x ) dx was meaningful for certain functions f (x ) according to Riemann’s definitions, and when the definitions were not satisfied, no meaning was assigned to the integral. The definition displayed above is still used today and is extended in a natural way to embrace the occurrence of a finite number of points c at which the function f (x ) becomes infinite in the interval [a, b]. Similarly, limits of proper Riemann integrals are used to define a f (x ) dx , b

0

f (x ) dx , and f (x ) dx = f (x ) dx + 0 f (x ) dx . Two informative examples are given in Section 2.4, where we discuss integration by substitution. There are bounded functions that are discontinuous and are integrable by Riemann’s original definition. In fact, H. Lebesgue proved that a function defined on a closed bounded interval is (properly) Riemann integrable if and only if it is bounded and continuous almost everywhere. It turns out that if f is bounded and continuous almost everywhere on b

Bn f (x ) dx exists and equals the Riemann integral [a, b], then limn a of f over [a, b]. That is, the limit in Definition 5 determines the Riemann integral of any function that is Riemann integrable according to Riemann’s original definition. These assertions are confirmed in Chapter 6. See Theorem 71.

2.3 FOUNDATIONAL THEOREMS OF INTEGRAL CALCULUS The results presented in this section stand behind most of the myriad applications of calculus. We include among these results the intermediate value theorem for integrals and the basic theorem on the interchange of order of limits and integrals. There are two versions of the fundamental theorem of calculus (FTC). Each shows that differentiation and integration are inverse processes. The first form of the theorem stands behind most standard methods devised to explicitly evaluate definite and indefinite integrals and is used to find the derivative of a function defined by an integral. The second use is especially helpful when the integral in question cannot be expressed in terms of more elementary functions.

The Riemann Integral ▪ 27

The first two foundational results are consequences of the monotonicity of the integral, #2; see page 24. Theorem 7: (Mean Value Theorem for Integrals) If f : [a , b] continuous, then there exists c in [a, b] such that

1

b

b

a

a

R is

f (x ) dx = f (c ).

Proof. Since f is continuous on [a, b] it assumes its minimum value, say m, and its maximum value, say M. By #2 with these choices for m and M,

m

1 b

b

a

a

f (x ) dx

M.

Since, by the intermediate value theorem for continuous functions, a continuous function defined on a closed bounded interval takes on every value between its minimum and maximum, there is a point c in [a, b] with the property asserted in the theorem. ■ Theorem 8: (Interchange of Limits and Integrals) If (fn (x ))n=1 is a sequence of continuous functions on a closed bounded interval [a, b] that converge uniform to a function f (x ) on [a, b], then f (x ) is continuous on [a, b] and

lim

n

b a

fn (x ) dx =

b a

f (x ) dx .

Proof. Since the uniform limit of a sequence of continuous functions is continuous, the function f (x ) is continuous. Given > 0, there is an index N = N such that fn (x ) f (x ) < provided that n N. By #2 b a

for n

fn (x ) dx

b a

f (x ) dx

b a

fn (x )

f (x ) dx

N and the desired conclusion follows. ■

(b

a)

28 ▪ Aspects of Integration

It is critical for many important applications of the mean value theorem of differential calculus that the intermediate point c can be chosen in the interior of the interval [a, b]. The corresponding property is rarely important for applications of the mean value theorem for integrals. Nevertheless, it is true, and a proof using the intermediate value property of continuous functions is not difficult.

Theorem 9: (FTC I) Let f : [a , b] F : [a , b] R by

F (x ) =

x a

R be continuous and define

f (t ) dt .

Then, F is differentiable on [a, b] and

F (x ) = f (x ) for each x in [a, b]. Proof. Let x and x + h be points in [a, b] where x is held fixed and h may vary. The interval additivity property of the integral and the mean value theorem for integration gives

F (x + h) h

F (x )

=

1 h

x+h x

f (t ) dt = f (c )

where c is some point in the interval with endpoints x and x + h. Since c x as h 0 and the function f is continuous at c, it follows that there exists

lim

h

0

F (x + h) h

F (x )

= f (x ).

■

A slight extension of the theorem is worth knowing. If c is any point in x [a, b] and Fc (x ) = c f (t ) dt , then Fc (x ) is differentiable and Fc (x ) = f (x ) for x in [a, b]. This follows from the theorem because c Fc (x ) + a f (t ) dt = F (x ).

The Riemann Integral ▪ 29

R is continuous and g is a real-valued Corollary 10: If f : [a , b] function that is differentiable at a point x in [a, b], then d dx Proof. Let G (x ) =

g (x ) a

g (x ) a

f (t ) dt = f (g (x )) g (x ).

f (t ) dt = F (g (x )) where F is the function in the

theorem. By the chain rule, G (x ) = F (g (x )) g (x ), which is the asserted result. ■ Theorem 11: (FTC II) If f : [a , b] R, then derivative f : [a , b] b a

R has a Riemann integrable

f (x ) dx = f (b)

f (a).

Proof. Let a = x0 < x1 < < xn 1 < xn = b be a partition of [a, b] and set n = max1 k n xk xk 1 . By the mean value theorem of differential calculus there are points ck in (xk 1, xk) such that n

f (b)

f (a) =

k =1

n

(f (xk) b a

f (xk 1)) =

k =1

f (ck)(xk

xk 1)

f (x ) dx

0 because f (x ) is Riemann integrable over [a, b]. ■ As noted earlier, the FTC I stand behind most of the techniques of integration used to evaluate integrals. The most important of these is the use of antiderivatives to evaluate definite integrals. Recall: if I is an interval of any type and g (x ) is a function that is continuous on I, then a function G (x ) is an antiderivative of g (x ) on I if G (x ) is continuous on I and satisfies G (x ) = g (x ) for all x in I except possibly the endpoints of I that belong to I. Since every derivative formula has an equivalent antiderivative formulation, differential calculus provides rich collections of antiderivatives.

as

n

30 ▪ Aspects of Integration

R is a continuous function and G is an Proposition 12: If f : [a , b] antiderivative of f on the interval [a, b], then b a

Proof. Let F (x ) =

x a

f (x ) dx = G (b)

G (a).

f (t ) dt for x in [a, b]. Then, both F and G are

antiderivatives of f, the function H (x ) = F (x ) G (x ) is continuous on [a, b], and H (x ) = 0 on (a , b). It follows from the mean value theorem of differential calculus that H (x ) = c for some constant c and all x in [a, b]. Consequently, F (x ) = G (x ) + c for x in [a, b] and b a

f (x ) dx = F (b)

F ( a) = G ( b )

G (a).

■

In calculus, the notation f (x ) dx , the indefinite integral of f, is used to stand for the collection of all antiderivatives of f (x ). Consequently, if G (x ) is a particular antiderivative of f (x ), then f (x ) dx = G (x ) + c x where c can be any constant. Sometimes the notation f (t ) dt is used to denote a particular antiderivative of f (x ). For example,

cos xdx = sin x + c , x

cos xdx = sin x

and, from the proposition, /2 0

cos xdx = sin /2

sin0 = 1.

One differentiation rule, the product rule, deserves special mention because when it is recast in antiderivative form it becomes an important technique of integration and also has significant theoretical value. If f (x ) and g (x ) are continuously differentiable on an interval, then

(f (x ) g (x )) = f (x ) g (x ) + f (x ) g (x ),

The Riemann Integral ▪ 31

which can be expressed in indefinite integral form as

f (x ) g (x ) dx = f (x ) g (x )

f (x ) g (x ) dx.

in which case it is called integration by parts. This is an effective technique of integration because often a given integrand can be split into factors f (x ) g (x ) for which integration of f (x ) g (x ) is easier. Integration by parts is easiest to remember and apply when expressed in Leibniz notation: if u = f (x ) and v = g (x ) so that du = f (x ) dx and dv = g (x ) dx , then integration by parts can be expressed by

udv = uv

vdu.

The corresponding definite integral formulation is b a

udv = uv ba

b a

vdu.

For example,

x 2 sin xdx =

x 2d ( cos x ) = x 2 ( cos x )

( cos x ) d (x 2)

=

x 2 cos x + 2x cos xdx = x 2 cos x + 2 xd (sin x )

=

x 2 cos x + 2(x sin x

=

x 2 cos x

sin xdx )

+ 2x sin x + 2cos x + c .

2.4 INTEGRATION BY SUBSTITUTION Making a change of variables in an integral, integration by substitution, is such an automatic process using Leibniz notation that the following theorem may seem pedantic. The simple example that follows its proof gives some perspective. Integration by substitution could be included among the foundational results of integration because of its far-reaching practical and theoretical applications. Nevertheless, it is at its core a direct application of the fundamental theorem of calculus. For this reason, we have chosen to put it in a separate section.

32 ▪ Aspects of Integration

To set the stage, it is helpful to review a few properties of continuous one-to-one functions defined on an interval I of any type. An accessible R is continuous and one-to-one, reference is Chapter 2 in [1]. If g : I then g is either strictly increasing or strictly decreasing on I and its range is an interval J . Consequently, g maps I one-to-one onto J and has an inverse I that increases or decreases according as g increases or function h: J decreases and h (g (x )) = x for all x in I . Assuming that g is differentiable at x and h is differentiable at g (x ), the chain rule gives h (g (x )) g (x ) = 1. The needed differentiability of h is provided by the inverse function rule of differential calculus: if g is differentiable at x in I then its inverse function h is differentiable at the point g (x ) in J and h (g (x )) = 1/g (x ) provided g (x ) 0. This is just an analytic way of saying that if the graph u = g (x ) has a tangent line at the point x, then the graph read “backwards” which is the graph of x = h (u) also has a tangent line (the same line) at the point u = g (x ). The proviso that g (x ) 0 is required because a horizontal tangent line to the graph of g becomes a vertical tangent line to the graph of the inverse function h and the derivative is not defined at a point where a graph has a vertical tangent. This result is often written in Leibniz notation as dy /dx = 1/(dx /dy) in introductory calculus. The cases 2 and 3 in the theorem that follows may seem unlikely at first glance but they occur frequently when natural substitutions are used and are easy to apply despite the technical appearance of the statements.

R be conTheorem 13: (Integration by Substitution) Let f : [a , b] tinuous and u = g (x ) be a continuously differentiable function that is either strictly increasing or strictly decreasing on [a, b] with g (x ) 0 on the open interval (a , b). Let x = h (u) be the inverse function of u = g (x ). 1. If g (a) and g (b) are nonzero, then b a

f (x ) dx =

g (b) g (a)

f (h (u)) h (u) du

where the integral on the right arises from the integral on the left by the usual substitution process. Both integrals are proper integrals. 2. If g (a)=0, then

The Riemann Integral ▪ 33 b

f (x ) dx =

a

g (b)

f (h (u)) h (u) du

g (a +)

where the integral on the right is improper and defined by g (b) g (a +)

f (h (u)) h (u) du = lim a

a

g (b) g (a )

f (h (u)) h (u) du.

Similarly if g (b) = 0. 3. If g (a) = 0 in #2, limu g (a) f (h (u)) h (u) exists and is finite, and f (h (u)) h (u) is defined at u = g (a) to be its limiting value as u g (a), then b

g (b)

f (x ) dx =

a

g (a)

f (h (u)) h (u) du

and the integrals on both sides are proper. Similarly if g (b) = 0. Proof. 1: Set G (x ) =

g (x ) g (a)

f (h (u)) h (u) du, and use Corollary 10 to

obtain

G (x ) = f (h (g (x ))) h (g (x )) g (x ) = f (x ) for all x in [a, b] because h (g (x )) = x and h (g (x )) g (x ) = 1 by the chain rule. So G(x) is an antiderivative of f(x) and by Proposition 12 b a

f (x ) dx = G (b)

2: Let a < a

G (a) =

b and set G (x ) = b a

f (x ) dx =

g (x ) g (a )

g (b) g (a )

g (b) g (a)

f (h (u)) h (u) du.

f (h (u)) h (u) du. By #1

f (h (u)) h (u).

Since f (x ) is continuous on [a, b], the left member of the equality converges to

b a

exists as a

a and

f (x ) dx as a

a. Thus the limit of the right member

34 ▪ Aspects of Integration b a

g (b)

f (x ) dx = lim a

g (a )

a

f (h (u)) h (u) =

g (b) g (a +)

f (h (u)) h (u) du.

3: The stated conclusions follow from #2 because under the assumptions in #3 g (b) g (a +)

g (b)

f (h (u)) h (u) du =

g (a)

f (h (u)) h (u) du

by definition of the improper integral on the left. ■ To better appreciate the theorem, consider the substitution u = x 2 to evaluate 1

x 2dx.

1

Since u = x 2, du = 2xdx or dx = du /(±2 u ) and 1 1

x 2dx = ±

1 2

1 1

u du = 0, u

which, of course, is incorrect. The main problem here is that u = g (x ) = x 2 is not one-to-one. Note also that g (0) = 0. However, as x varies from −1 to 0, where g is one-to-one and varies from 1 to 0 and has a nonzero derivative on the open interval from 1 to 0, we have (by part 3 0) of the theorem because the u-integrand has a finite limit as u 0 1

x 2dx =

0 1

u 1 du = . 2 u 3

Likewise, as x varies from 0 to 1, g strictly increases from 0 to 1 and 1 0

1

x 2dx =

1

u

0

2 u

du =

1 . 3

Thus, 1 x 2dx = 2/3 as expected. The point here is that despite its automatic nature, one must be attentive when using integration by substitution.

The Riemann Integral ▪ 35

Integrals with one or both limits of integration infinite have many important applications and can lead to surprising results. We give two examples. The first plays an important role in Fourier series. The second is a surprising example that Whittaker and Watson say Dirichlet put forward in 1837. sin x

Example 14: Show that 0 x dx = 2 . The integral exists as defined by Riemann, but now is called an improper integral. 0. To We interpret sin x/x at x = 0 to be 1, its limiting value as x establish that the integral has a meaning, we must show that

sin x dx = lim b x

0

b 0

sin x dx x

exists and is finite. After that, we give a separate argument to show that its value is /2. It is useful to observe that it suffices to show that the limit exists as b through the values n for n = 1, 2, … . Indeed, given b any there is a greatest integer n such that n b/ < n + 1 and b 0

because limn

limb Now, n 0

sin x dx x

sin x dx = x

0

sin x /x

1/n

n

for n

n sin x dx exists and is 0 x b sin x dx and the two limits 0 x

n

sin x dx = x

k

b n

sin x dx , x

(k +1) k


0 decrease to 0 as k

ck +1 =

n

x (n + 1) . Consequently, if finite, the same is true for

k =1 (k 1) n ( 1)k 1ck k 1

=

b

n k =1

( 1)k

1

k (k 1)

sin x dx x

:

k sin x sin (u + ) dx = du ( k 1) x u+ sin(u + ) du = ck u

36 ▪ Aspects of Integration

and

sin x dx x

k

ck =

(k 1)

1 (k

0

1)

as k

. It follows by the Leibniz alternating series test that the series k +1 c converges and, hence, ( 1) k k =1

sin x dx = x

0

k =1

( 1)k 1ck.

Thus, the integral exists. A more informative expression for the value of the integral can be derived as follows: define

F (t ) = 0. If t > 0 and b > b

for t b 0

e

tx sin x dx

b

x

0

e

0

e

tx sin x dx

x

0,

tx sin x dx

b

x

b

e txdx =

b

e

tb

e

tb

t

0

sin x

e tx x dx exists and is finite. as b , b . Therefore, F (t ) = limb 0 If t = 0, F (0) exists by what we have shown earlier. The estimate above with b = 0 gives b 0

as t

e

tx sin x dx x

1

e t

tb

1 t

, F (t )

0

. We also need the fact that F is continuous at t = 0. To confirm

this, let Fb (t ) =

b 0

F (t )

e

tx sin x dx x

and Rb (t ) =

F (0) = Fb (t )

b

e

tx sin x dx . x

Fb (0) + Rb (t )

Then

Rb (0)

and since both integrals F (t ) and F (0) exist, given > 0 there exist b > 0 such that Rb (t ) Rb (0) < for b b . Consequently,

The Riemann Integral ▪ 37

F (t ) Fb (t ) as t

F (0)

Fb (t ) b 0

Fb (0)

0 and F (t )

F (0) as t

Fb (0) + ,

(1

e

tx ) dx

(1

e

tb ) b

0

0.

Assuming differentiation under the integral sign is permissible for t > 0, we get

F (t ) =

0

e

tx sin xdx .

Reasoning as for F (t ) earlier, it follows that the integral on the right exists for t > 0. Moreover, the differentiation under the integral sign is justified by general results: roughly speaking if the integral that arises by differentiation has an integrand that decreases in magnitude rapidly enough as x and if the original integral exists, then the differentiation is justified. The rapid decay of e tx as x is more than sufficient to apply the general results. Two integration by parts integrating the exponential factor and differentiating the trigonometric factor, gives

F (t ) =

1 1 + t2

for t > 0. Thus, F (t ) = arctan t + c for some constant c. Let t use F (t ) to find that c = /2. Since 0 as t

F (t ) = /2

arctan t

for t > 0 and F (t ) and arctan t are continuous at t = 0, letting t

0

and

0 gives

sin x dx = . x 2

Example 15: (Dirichlet) Show that 0 sin(x 2) dx exists in Riemann’s sense (as an improper integral in current terminology) and is finite.

38 ▪ Aspects of Integration

The conclusion is quite counterintuitive because sin(x 2) does not tend to zero as x . We must show that b 0

sin(x 2) dx

has a finite limit as b . To this end, make the change of variable 2 u = x where 0 x < which is a one-to-one differentiable function on 0 x b with nonzero derivative on 0 < x b. Thus by Theorem 13-(3) b 0

b2

sin(x 2) dx =

0

sin u du , 2 u

0. where we interpret sin u / u at u = 0 to be 0, its limiting value as u Following the line of reasoning in the first part of the solution to the previous example shows that

lim

n

n 0

sin u du = 2 u

( 1)k k =1

1

k (k 1)

sin u du = 2 u

k =1

( 1)k 1dk

where the alternating series converges by the Leibniz alternating series test. Thus, 0 sin(x 2) dx exists and has a finite value. Example 16: (Dirichlet and Du Bois-Reymond) The integral

0

x exp( x 6 (sin x )2 ) dx

exists in Riemann’s sense. Notice that f (x ) = x exp( x 6 (sin x )2 ) is nonnegative has local peaks near x = n where f (x ) = n . The strongly decaying exponential makes the integrand decay rapidly to nearly 0 slightly away from the peaks. This is why the integral exists in Riemann’s sense. Here is a simpler example that exhibits similar behavior. For n = 1, 2, 3, … define f (x ) on the interval [n n 3, n + n 3] to be the piecewise linear function joining n 3, 0) to (n , n ) and finally to (n + n 3, 0). Set the point (n f (x ) = 0 otherwise. Then, f (x ) is unbounded on [0, ) and

The Riemann Integral ▪ 39

0

f (x ) dx = n =1

n2

0 there is a >0 such that f (x , y ) f (x , y ) < when

(x , y )

(x , y ) =

(x

x )2 + (y

y )2 < . Also, f is bounded by a

constant M on R. Let

A1 = A1 (n , x , y ) = (i, j)

{0, 1, …, n}2 : |(x , y)

(xi, yj )|
h2 (x0) or (c) y0 = h2 (x0), If (a) holds, then P0 S1, yn < h2 (x 0) eventually (meaning for all n sufficiently large), Pn S1, and

f (Pn) = f1 (Pn)

f1 (P0) = f (P0)

because f1 is continuous on S1. If (b) holds, then P0 eventually, and

f (Pn) = f2 (Pn)

S2, Pn

S2

f2 (P0) = f (P0).

If (c) holds, P0 S1 S2 and f1 (P0) = f2 (P0) = f (P0). Let Nj be the set of positive integers n such that Pn Sj , for j = 1, 2. If either Nj is infinite, f j (P0) = f (P0) as n through values of n Nj by the then f (Pn) reasoning in cases (a) and (b). It follows that f (Pn) Consequently, h is continuous at every point in S. ■

f (P0) as n

.

Lemma 21: Let h1 (x ) and h2 (x ) be continuous on [a, b] and satisfy h1 (x ) h2 (x ) on [a, b]. Let S be the strip of points (x , y ) in the plane such that a x b and h1 (x ) y h2 (x ) and let k1 (x ) and k2 (x ) be continuous functions on [a, b] that are equal at any points in [a, b] at which h1 and h2 are equal. If (x , y ) S and h1 (x ) < h2 (x ) there is a unique real number t [0,1] such that y = (1 t ) h1 (x ) + th2 (x ), namely t = (y h1 (x ))/(h2 (x ) h1 (x )). Define a function H (x , y ) on S as follows: if h1 (x ) < h2 (x ), then

H (x , y) = (1 =

(1

t ) k1 (x ) + tk2 (x ) y h1 (x ) h2 (x ) h1 (x )

) k (x) + ( 1

y h1 (x ) h2 (x ) h1 (x )

) k (x ) 2

and H (x , y ) = k1 (x ) = k2 (x ) when h1 (x ) = h2 (x ). The function H (x , y ) defined on the strip S in this way is continuous, equals k1 (x ) at the points (x , h1 (x )) of its lower boundary curve, and equals k2 (x ) at the points (x , h2 (x )) of its upper boundary curve. Proof. The function H assumes the stated values on the upper and lower boundary curves by its definition. To confirm continuity of H on S, fix a

Extension to Higher Dimensions ▪ 53

P0 as point P0 = (x 0, y0 ) in the strip S and let points Pn = (xn, yn ) n , where the points Pn lie in the strip S. There are two cases to consider: (a) h1 (x 0) < h2 (x0) and (b) h1 (x 0) = h2 (x 0). In case (a), h1 (xn) < h2 (xn) eventually by continuity of h1 and h2 on [a, b] and

H (Pn) = H (xn, yn ) = (1

tn) k1 (xn) + tn k2 (xn)

(1

t0) k1 (x 0)

+ t0 k2 (x 0) = H (P0) because

tn =

yn h2 (xn)

h1 (xn)

y0

h1 (xn)

h2 (x0)

h1 (x0) h1 (x 0)

= t0

by continuity of h1 and h2. Thus, H is continuous at P0 in case (a). In case (b) h1 (x 0) = h2 (x 0); hence, k1 (x 0) = k2 (x 0). Let Pn = (xn, yn ) P0 = (x0, y0 ), where Pn and P are point of the strip S. Let N1 be the set of indices such that h1 (xn) = h2 (xn) and N2 be the indices such that the h1 (xn) < h2 (xn). If N1 has infinitely many members, then H (Pn) = k1 (xn) k1 (x 0) = H (P0) by continuity of the function k1 on [a, b]. If N2 has infinitely many members, then for such n, h1 (xn) < h2 (xn), and

H (Pn) = H (xn, yn ) = zn = (1

tn) k1 (xn) + tn k2 (xn)

where tn = (yn h1 (xn))/(h2 (xn) h1 (xn)). Since 0 weighted average of k1 (xn) and k2 (xn). Consequently,

min(k1 (xn), k2 (xn))

zn

tn

1, zn is a

max(k1 (xn), k2 (xn)).

k2 (x 0), and k1 (x 0) = k2 (x 0) in case (b), it Since k1 (xn) k1 (x0), k2 (xn) follows that zn k1 (x0) = H (P0) k1 (x 0) = k2 (x0); hence, H (Pn) = zn H (P0) as n as n in N2. It follows that H (Pn) . Thus, H is continuous at P0 in case (b). ■ Now that the lemmas are established, we can define the sequence of continuous extensions fn (x , y ) of f (x , y ) that are used to define f (x , y) dA in (3.1). Enclose the strip S of points (x , y ) defined by D

54 ▪ Aspects of Integration

a x b and h1 (x ) y h2 (x ) in the rectangle R = [a , b] × [c , d], where it is convenient to choose c = 1 + minx [a, b] h1 (x ) and d = 1 + maxx [a, b] h2 (x ). It may be helpful to sketch the decomposition of the rectangle R in the construction that follows. The functions h2 (x ) and + h2 (x ) + 1/n determine the horizontal strip Sn,1 of points (x , y ) with a x b and h2 (x ) y h2 (x ) + 1/n. The function h2 (x ) + 1/n and + the constant function d determine the horizontal strip Sn,2 of points (x , y ) + with a x b and h2 (x ) + 1/n y d . So, the strip Sn,2 sits above the + strip Sn,1 , which sits above the strip S. On the lower boundary of the strip + + (x ) = f (x , h2 (x )) and on Sn,1, the curve of points (x , h2 (x )), define kn,1 + the upper boundary of the strip Sn,1, the curve of points (x , h2 (x ) + 1/n), + (x ) = 0. By Lemma 21 there is a continuous function fn (x , y ) define kn,2 + + + on Sn,1 that agrees with the function kn,1 on the lower boundary of Sn,1 , that is, agrees with f (x , y ) there, and agrees with the function identically + + 0 on the upper boundary of Sn,1 . We extend fn (x , y ) to the strip Sn,2 by defining it to be identically 0 there. We make corresponding constructions in the part of the rectangle R that is below the strip S by introducing the curve (x , h1 (x ) 1/n) for a x b. Since fn (x , y ) is continuous on each of the nonoverlapping strips used in the construction, it follows from Lemma 20 that fn (x , y ) is continuous on the rectangle R. In this way, a sequence of continuous functions fn (x , y ) is defined on R = [a , b] × [c , d] with the properties: 1. fn (x , y ) = f (x , y ) for all (x , y ) in the strip S; 2. fn (x , y ) = 0 at all points in R that are at a distance greater than 1/n from the strip S. 3. fn (x , y )

M = max{ f (x , y) : (x , y )

S}.

The three properties follow directly from the construction. The first and second are evident. The third follows because the values of the extensions + + of f (x , y ) to the strips Sn,1 and Sn,2 (and the corresponding strips below S) are weighted averages of values of f (x , y ) and values of the zero function on the upper and lower boundaries of the strips involved. We define D f (x , y) dA for a horizontal strip D (a y-simple region) by (3.1) using the sequence of continuous functions fn (x , y ) just constructed. We show next that the limit in (3.1) exists and, at the same

Extension to Higher Dimensions ▪ 55

time, show that the double integral can be evaluated as a familiar iterated integral. Iterated Integration Iterated integration of polynomials over rectangles carries over to iterated integration of continuous functions over rectangles and more general domains. Let f be continuous on a rectangle R. Given > 0 there exists an index N = N such that f (x , y ) Bn f (x , y ) < for (x , y ) in R when n > N . Consequently, b a d c

b a

b B f a n

f (x , y) dy d c

f (x , y) dy dx

b a

(x , y) dy
0 there is an index N = N such . v as n that n N implies vn v < , in which case we write vn A Cauchy sequence is a sequence (vn)n=1 whose terms clump so closely together for large n that we suspect it must converge, even if it does not. Definition 23: A sequence (vn)n=1 in a normed space V is a Cauchy sequence if given any > 0 there is an index N = N such that m , n N implies vm vn < .

Extension to the Lebesgue Integral ▪ 61

Definition 24: A normed space is complete if every Cauchy sequence in the space converges to an element in the space. The system of rational numbers with length of the usual absolute value is not complete, as we will soon see. The real number system with length the usual absolute value is complete. Completeness is what makes the system of real numbers what is needed to set calculus on a firm foundation. The completion process that we describe next shows how to “complete” the rational number system by adjoining new numbers to obtain the real number system.

4.2 COMPLETION OF THE RATIONAL NUMBERS In classical times, say around 500 BC, when Euclidean geometry was being developed and the notion and meaning of number systems were evolving, geometers took it for granted that each line segment had a length and that the lengths of various line segments could be compared. The number system in use at the time was essentially what we call the rational number system. Given two line segments, AB and AC , with respective lengths, l = AB and d = AC , it was believed that there were positive integers m and n such that nd = ml . In geometric terms, if m copies of the segment AB were laid end-to-end and n copies of the segment AC were laid end-to-end along a line in the Euclidean plane, then the newly constructed segments would be congruent and have the same length. The relation ml = nd was expressed in proportions

d : l :: m : n read d is to l as m is to n. Equivalently,

d=

m l; n

that is, the length d is a rational multiple of the length l. If the relation ml = nd holds for positive integers m and n, the two segments are called commensurate. So, it was believed that all line segments were commensurate. It is helpful to introduce a standard reference line segment to compare lengths, say the segment AB, and to define its length to be 1 unit

62 ▪ Aspects of Integration

(perhaps, meter, foot, yard, whatever) and measure the lengths of other line segments relative to the standard length. The line segment AC has length

d=

m , n

continuing with the notation above. In summary, at this point in the development of Euclidean geometry and of number systems, every line segment had a length, and its length was a rational number. However, there must have been some doubt about this assertion. Conundrum Around 500 BC the Pythagoreans, followers of the philosopher and mathematician Pythagorus, proved what we now call the Pythagorean theorem. Then a surprise came. They devised a geometric construction that showed that in a square □ABCD whose equal sides had length l = 1 the diagonal AC had length d = AC that was not commensurate with the length of a side. That is, no rational number m/n could be the length of AC. On the other hand, since 12 + 12 = d 2 or d 2 = 2, the geometric construction showed that no rational number could have square 2. The conundrum is this: either the segment AC does not have a length or its length is some kind of new number whose square is 2. There are two ways to look at this situation. View 1 If the only number system used in Euclidean geometry is the system of rational numbers, then we must accept the fact that some line segments cannot be assigned lengths. View 2 Every line segment must have a length, and there must be a number system that extends the rational number system that makes this so. Surely most mathematicians of antiquity held View 2, and the number system they sensed to exist is what we now call the real number system. The numbers adjoined to the rational number system to form the real number system are called irrational numbers. In particular, the length d of the diagonal of a unit square, is the irrational real number that we now denote by 2 . The geometers of antiquity did not have a precise

Extension to the Lebesgue Integral ▪ 63

understanding of what the extended number system was. However, they did anticipate much of the discussion that follows because they developed and used rational approximations to the new irrational numbers. For example, Archimedes found accurate approximations for (which is an irrational number) over 2,000 years before the irrationality of was established and rational approximations to 2 were known as well.

What is

2?

To answer this question, we start with the system of rational numbers and use rational numbers to assign a meaning to the new number 2 . We use decimal notation and recall that any decimal with a finite number of digits, such as N . d1 d2 d3, is a rational number

N . d1 d2 d3 =

103N + 102d1 + 10d2 + d3 , 103

where N is a natural number and d1, d2, and d3 are decimal digits. We (try to) nail down 2 in stages: Stage 1 Routine arithmetic gives

(1.4)2 = 1.96 < 2 < (1.5)2 = 2.25, (1.41)2 = 1.9881 < 2 < (1.42)2 = 2.0164, (1.414)2 = 1.999396 < 2 < (1.415)2 = 2.002225. Continuing in this fashion we obtain two sequences of rational numbers

(rn) = (1.4, 1.41, 1.414, 1.4142, 1.41421, …) and

(sn) = (1.5, 1.42, 1.415, 1.4143, 1.4122, …) with the property that the squares of the successive terms in the first sequence increase but are less than 2 and the squares of the

64 ▪ Aspects of Integration

successive terms in the second sequence decrease but are greater than 2. Both sequences are Cauchy sequences. Indeed, if m n then

0

rn

rm = 0.0

00dm +1

dn

01 = 10 m ;

0.00

0 as m, n hence, rn rm and likewise for the sequence (sn). It is important to notice as well that the two sequences are related; intuitively, they both cluster near the new number we seek. This relation is expressed analytically by 0

sn

rn = 0.00

01 = 10

where 1 is the n-th decimal digit. Hence, sn

n

rn

0 as n

.

Stage 2 Can we think of either sequence (rn) or (sn) as being a new number that we could call 2 ? Well, there is no reason to prefer one of these sequences over the other. Maybe we should think of both of them as being 2 . This is actually what we will do, but with one more twist. Stage 3 There are billions of other sequences of rational numbers that cluster near the new number 2 we seek. For example, the sequence

(1.4, 1.5, 1.41, 1.42, 1.414, 1.415, …) or, in general, any sequence (tn) for which tn rn = zn where (zn) is any sequence of rational numbers that converges to 0. All the sequences (tn) are Cauchy sequences of rational numbers. There is no reason to prefer any of these sequences to (rn) or (sn) to represent the new number 2. Stage 4 So, we take them all and define 2 be the number that stands for the collection of all Cauchy sequences (tn) whose terms . 0 as n are rational numbers such that tn rn The four stages indicate how to define the irrational real number 2 in terms of “equivalent” Cauchy sequences. All other irrational numbers can be defined similarly. There is still much to be done, and we will not

Extension to the Lebesgue Integral ▪ 65

do it. What justifies calling these collections of Cauchy sequences numbers? Are the algebraic and order properties of the rational number system preserved? Do the new numbers contain the rational numbers? Why does the completion process always lead to a complete system, here of numbers? All these questions can be answered satisfactorily. Most are answered in Appendix C. It is important to make a final observation about the completion process that is almost obvious from the construction above. The construction is economical; no new numbers are added that have nothing to do with the numbers we started with. This means: every new irrational real number x can be approximated as closely as desired by rational numbers. Given any positive rational number there is a rational number q such that x q < . Calculus Incredible advancements in mathematics were made using calculus in the 16th, 17th, and 18th centuries (and beyond). However, neither the real number system itself nor the notion of a limit, which are the foundation upon which calculus stands, were fully understood as all the advances took place. In the 19th century, especially its second half, Weierstrass, Dedekind, Cantor, and Heine undertook to put the system of real numbers, and, hence, calculus on a firm foundation. Heine’s 1872 paper [4] is the first published version of the construction of the real numbers from the rational numbers using what we now call Cauchy sequences of rational numbers (called simply sequences of numbers in his paper). In the paper, Heine says that Weierstrass was the principal developer of the ideas presented, but Weierstrass only shared his investigations through his lectures and private correspondence with others. Heine also thanks Cantor for significant contributions to his paper. In the introduction to his paper, Heine says that it is necessary to carefully define the real number system in order to put other analytical results of function theory (calculus) on a firm foundation. Then, he goes on to prove that a continuous function on a closed bounded interval is uniformly continuous. Heine’s reasoning is so close to Borel’s original proof of his covering theorem that they now share credit for that result, the Heine-Borel theorem.

66 ▪ Aspects of Integration

4.3 COMPLETION OF C IN THE 1-NORM We use the completion process described above and fleshed out in Appendix C to extend the Riemann integral (of continuous functions) discussed in Chapter 2 to the Lebesgue integral. We will use the notation from Appendix C, which the reader may wish to consult briefly, and, as is customary, denote the normed linear space of Lebesgue integrable functions on [a, b] by L1 = L1 [a , b]. Equip the space of continuous functions on a closed bounded interval [a, b] with the 1-norm

f

1

=

b a

f (x ) dx and denote the resulting normed linear space by

C1 = C1[a, b] and it completion by C1 = C1 [a , b]. (It is easy to check that the 1-norm is a norm.) The primary conclusion of the section is that C1 “is” L1 in the sense that the two normed linear spaces are the same except for the symbols used to denote their elements. The Completion and its Properties The elements of C1 are collections of 1-norm Cauchy sequences (henceforth, just Cauchy sequences) of functions in C1. By definition two Cauchy sequences (fn ) and (gn) in C1 belong to the same collection in C1 if (and only if)

lim

n

b a

gn (x )

fn (x ) dx = 0.

(4.1)

We write (fn ) (gn) when the Cauchy sequences are related by (4.1), and say that (fn ) and (gn) are equivalent. If (fn ) is a Cauchy sequence in C1, the collection of all Cauchy sequences equivalent to (fn ) will be denoted by fˆ = [(fn )], where the square brackets indicate that fˆ is the collection of Cauchy sequences equivalent to the Cauchy sequence (fn ). Lemma 25: If fˆ = [(fn )] and gˆ = [(gn)] then fˆ = gˆ if and only if (fn ) (gn). Proof.

: If fˆ = gˆ then (fn )

: Suppose (fn )

(gn). If (hn)

gˆ and (fn ) gˆ then (hn)

(gn) by definition of gˆ .

(gn) and

Extension to the Lebesgue Integral ▪ 67 b a

hn (x )

b

fn (x ) dx

a

hn (x )

gn (x ) dx +

b a

gn (x )

fn (x ) dx.

. Hence, the left Both terms in the right member have limit 0 as n member has limit 0, (hn) fˆ , and gˆ fˆ . By symmetry fˆ gˆ and the desired conclusion follows. ■ Definition 26: If fˆ = [(fn )] b a

C1 [a, b], define

fˆ (x ) dx = lim n

b a

fn (x ) dx .

The definition is well posed. First, the indicated limit exists because the real number system is complete and b a

fm (x ) dx

as m , n then (gn) b a

as n Thus,

~

b a

fn (x ) dx

b a

fm (x )

fn (x ) dx = fm

fn

1

0

. Second, if fˆ = [(gn)] where (gn) is a Cauchy sequence in C1, (fn) and

gn (x ) dx

b a

fn (x ) dx

b a

gn (x )

fn (x ) dx = gn

fn

1

0

by definition of the equivalence relation in the completion.

lim

n

b a

gn (x ) dx = lim n

b a

fn (x ) dx ,

and the limit is independent of the representative of fˆ . Just as in the general completion construction, we can regard the space C1 as a subset of C1 by identifying a continuous function f in C1 with fˆ = [(f , f , f , ..)] in C1. With this agreement C1 is dense in C1 and

68 ▪ Aspects of Integration b a

b

fˆ (x ) dx = lim

f (x ) dx =

a

n

b

f (x ) dx ,

a

which means that the definition of the integral in C1 agrees with the integral of a continuous function as defined in Chapter 2. Define the binary relation fˆ gˆ on C1 to mean there are Cauchy sequences (fn ) and (gn) in C1 such that fˆ = [(fn )], gˆ = [(gn)], and fn gn in C1. It is easy to check that fˆ gˆ is a partial order on C1, meaning that

fˆ fˆ

fˆ , gˆ and gˆ

fˆ

gˆ and gˆ

fˆ implies fˆ = gˆ, hˆ implies fˆ hˆ.

The integral in C1 inherits linearity and monotonicity from the corresponding properties of integrals of continuous functions. 1. (Linearity) If fˆ and gˆ belong to C1 and c is a scalar, then b a

(cfˆ (x) + gˆ (x) ) dx = c

b a

fˆ (x ) dx +

b a

gˆ (x ) dx .

If fˆ = [(fn )] and gˆ = [(gn)] then cfˆ + gˆ = [(cfn + gn)] and b a

(cfˆ (x ) + gˆ (x )) dx = lim

b a

(cfn (x ) + gn (x )) dx = c lim b g (x ) dx a n

+ lim 2. (Monotonicity) If fˆ

b a

fˆ (x ) dx +

C1 then

fˆ ˆ Consequently, if m m (b

=c

b

0 fˆ a)

a

fˆ (x ) dx

0.

ˆ then M b a

fˆ (x ) dx

M (b

a)

b f (x ) dx a n b gˆ (x ) dx . a

Extension to the Lebesgue Integral ▪ 69

and b a

fˆ (x ) dx

(b

a)max( m , M )

ˆ = [(M , M , M , …)] and m and M ˆ = [(m, m, m …)], M where m are scalars.

fn 0 fˆ = [(fn )] 0, then for all n and ˆf (x ) dx = lim b f (x ) dx 0, which establishes the first result. a n The second and third results follow from the first because ˆ. ˆ fˆ M m If b a

3. (Additivity) If fˆ = [(fn )] b a

fˆ (x ) dx =

C1 [a, b] and a c a

fˆ (x ) dx +

b c

c

b, then

fˆ (x ) dx ,

where fˆ in the first integral on the right means (just as for ordinary functions) the restriction of fˆ to C1 [a , c], which is [(fn [a , c])]. (Clearly (fn ) Cauchy in C1[a, b] implies (fn [a , c]) is Cauchy in C1[a, c ].) The desired additivity follows from bˆ f (x ) dx a

b c f (x ) dx = lim( a fn (x ) dx a n c ˆ b f (x ) dx + c fˆ (x ) dx. a

= lim =

+

b f (x ) dx ) c n

If fˆ = [(fn )] C1 [a, b] where (fn ) is a Cauchy sequence in C1, then by fn 1 fm fn 1 0 as m, n the triangle inequality fm , ˆ which shows that ( fn ) is a Cauchy sequence in C1. Define f = [( fn )]. The definition is well posed: if fˆ = [(g )] then n

gn

fn

1

gn

fn

1

0

70 ▪ Aspects of Integration

( fn ). Since fˆ ± fˆ

because (gn) (fn ) and, hence, ( gn ) monotonicity result implies that b a

b

fˆ (x ) dx

a

0, the first

fˆ (x ) dx .

For real numbers , , and , +

=

0 and

=(

)

0

by definition. Consequently,

=

+

,

=

+

+ , and

+

+

.

If fˆ = [(fn )] C1 [a, b] where (fn ) is a Cauchy sequence in C1, just as for + the definition of fˆ above, the equivalence classes fˆ = [(fn+ )] and fˆ = [(fn )] are well defined. Consequently, + fˆ = fˆ

+ fˆ and fˆ = fˆ + fˆ .

C1 [a , b] is L1 [a, b] We suppose either that the reader is familiar with the space L1 [a , b] of Lebesgue integrable functions on [a, b] or is willing to take of faith that such a space exists and has the properties used below. (A definition of L1 is given in Part II Section 5.7.) The space L1 is a complete normed linear space, a Banach space, where

f

L1

=

b a

f (x ) dx

and the integral on the right is the Lebesgue integral. (Technically, if is a function that is Lebesgue integrable on [a, b], then in the left member of the equality stands for the equivalence class of all functions that are equal to almost everywhere. This picky point is needed to make L1 a norm on L1.) When we say C1 [a , b] is L1 [a , b] we mean that we can pair off the elements in the two spaces in such a way that the linear structure and

Extension to the Lebesgue Integral ▪ 71

distance structure is preserved under the correspondence. This means we must find a map

L1 [a , b]

: C1 [a, b]

that is a linear isomorphism, a 1-1 onto linear map, when C1 and L1 are regarded as vector spaces, that preserves the distance between corresponding points of the spaces, and assigns the same value to the integrals at corresponding points. That is, we must find a map such that for all ˆ and gˆ in C1 and all scalars c and d it follows that

(c ˆ + dgˆ ) = c ( ˆ ) + d (gˆ ), (ˆ ) (ˆ) g = ˆ gˆ . L1

1

and b a

ˆ (x ) dx =

b a

(x ) dx

where the integral on the right is the Lebesgue integral. The map is defined as follows: if ˆ = [( n)] where ( n) is any Cauchy sequence in C1 that represents ˆ , then since ( n) also is a Cauchy sequence in L1 and L1 is complete, there exists = limn n in L1. By definition ( ˆ ) = . To show this definition is well defined, we must show that the limit in L1 does not depend on the Cauchy sequence in C1 used to represent ˆ . To this end, assume ˆ = [(gn)]. Then there is a function g in L1 such that g = limn gn in L1. Since (gn) ( n), gn 0 by the meaning of equivan L1 = gn n 1 lence and

g

gn =

n

+ (gn

n)

in L1. Thus g = in L1 and the map is well defined. L1 is a linear isomorphism: Let ˆ = [( n)], The map : C1 g in L1, then gˆ = [(gn)], and c and d be scalars. If n and gn c n + dgn c + dg in L1, c + dg is represented by [(c n + dgn)], and by definition

72 ▪ Aspects of Integration

(c

+ dg ) = c + dg = c

(ˆ ) + d

(ˆ). g

and is a linear map. L1 is an isometry, a distance preserving map: If The map : C1 ˆ = [( n)] then

ˆ

1

= lim n

n 1

= lim n

n L1

= lim n

Consequently, if ( ˆ ) = 0 then ˆ = 0 and g = ˆ gˆ, of addition in C1,

ˆ

gˆ

1

=

g

1

=

(

g)

n

L1

=

(ˆ )

L1 .

is one-to-one. By definition

L1

(ˆ )

=

(ˆ) g

L1

and is an isometry. L1 is onto: Given any in L1 there is a sequence of The map : C1 continuous functions n that converge in L1 to . Consequently, ( n) is a Cauchy sequence in C1 and determines an equivalence class ˆ = [( n)] in C1. By definition, ( ˆ ) = and the map is onto. The map with the properties just established is called an isometric isomorphism of C1 onto L1. It remains to show that b a

ˆ (x ) dx =

b a

(x ) dx

where ( ˆ ) = and the integral on the right is the Lebesgue integral. Now, ˆ = [( n)] where ( n) is a Cauchy sequence in C1 and = limn n in L1. Therefore, b a

ˆ (x ) dx = lim

b a

n (x ) dx

=

b a

(x ) dx ,

where the first equals sign holds by the definition of the integral in C1 and . 0 as n the second equals sign holds because n L1

II Lebesgue Integration

P

II AROSE FROM our desire to better understand the presentation given by F. Riesz and Sz-Nagy in the first part of their influential book: F. Riesz & B. Sz-Nagy, Functional Analysis (1952,1953) [8]. The first part of their book is devoted to a Lebesgue-integral-first, Lebesguemeasure-second development of the theory of Lebesgue measure and integration. Riesz and Nagy say that this approach puts the important convergence results of Lebesgue integration front and center and avoids obligating the reader to first learn a good deal of measure theory that plays little role in applying the Lebesgue convergence theorems. Although the presentation here primarily follows that of Riesz and Nagy, it was also influenced by a more recent text: E. Asplund & L. Bungart, A First Course in Integration (1966) [2] which followed the general approach of Riesz and Nagy, added considerably more detail, includes a rich collection of exercises, and has informative historical notes at the end. Both Riesz and Nagy and Asplund and Bungart use integrals of step functions as the starting point for their development of the Lebesgue ART

b

dx integral. Riesz and Nagy almost always express their integrals as a where (a , b) is a bounded or unbounded interval of real numbers, which may be the entire real line. Asplund and Bungart prefer to work with DOI: 10.1201/9781003389484-6

73

74 ▪ Lebesgue Integration

functions defined on the entire real line and denote their integrals by dx. Both treatments were influenced by the Daniell approach to integration. Riesz and Nagy mention that (Riemann) integrals of continuous functions could be used as the starting point in their approach but do not follow up on this observation. It is a matter of taste whether one prefers to start with integrals of step functions or with integrals of continuous functions. We find the latter approach an attractive option and use it here, especially in light of the treatment of the Riemann integral in Part I. A step function, defined on an interval I, is bounded, has one-sided limits at every point, vanishes outside a bounded interval when I is unbounded, and has a (finite) integral defined in the usual way. The class of continuous functions, C0 (I ), that serve as our starting point for developing the Lebesgue integral mimics these step function properties but with the integral of a step function replaced by the Riemann integral of functions in C0 (I ). We use the following definitions and notations throughout Part II. An extended real-valued function f is increasing on I if f (x1) f (x2) for all x1 and x2 in its domain with x1 x2. Likewise, f is decreasing on I if f (x1) f (x2) for all x1 and x2 in its domain with x1 x2. If f is an extended real-valued function on I, the positive part of f is the nonnegative function f + defined by f + (x ) = max(f (x ), 0) and the negative part of f is the nonnegative function f defined by f (x ) = min(f (x ), 0). We also use the notation for max( , ) and for min( , ) so that f + (x ) = f (x ) 0 and f (x ) = (f (x ) 0). Clearly, f = f + f and f = f+ + f . Part II concludes with a chapter that contrasts important differences among the various types of integrals treated in both Part I and Part II.

CHAPTER

5

The Riesz-Nagy Approach to the Lebesgue Integral

T

standing assumptions are in force throughout the chapter. The restrictions placed on the continuous functions in the class C0 (I ) are suggested by corresponding properties of step functions, as was just mentioned in the introductory remarks to Part II. If I is a closed bounded interval of real numbers, then C0 (I ) = C (I ), the set of all continuous functions on I. HE FOLLOWING NOTATION AND

Standing Assumptions 1. All functions are real-valued and defined on an interval I with a b + . The integral of such a endpoints a and b, where b dx. An function, when it exists, will usually be denoted by a dx, and, occaintegral over the real line may be denoted by sionally, the same notation may be used when the interval of integration is clear from the context. When the type of integral b intended may be subject to confusion, we will write (R) a … dx to denote a Riemann integral and use similar notation for other integrals.

DOI: 10.1201/9781003389484-7

75

76 ▪ Aspects of Integration

2. C0 = C0 (I ) denotes the collection of continuous functions f on I that have the following properties: a. the nonzero values of f lie in a bounded subinterval of I, the bounded subinterval may vary from function to function. b. If the endpoint a of I is a real number and a I, f can be extended to a continuous function on the interval {a} I , equivalently limx a f (x ) exists and is a real number. Likewise, if the endpoint b of I is a real number and b I, f can be extended to a continuous function on the interval I {b}, equivalently limx b f (x ) exists and is a real number. If I = [a , b] is a closed bounded interval in , then C0 is simply the class of real-valued continuous functions on I. Regardless of the type of the interval I, 2(b) is equivalent to the assumption that f in C0 means that f has a continuous extension to the closure I in of I. We denote the unique continuous extension of f from I to I by f . When I is unbounded, by 2(a) corresponding to each f in C0 there is a closed bounded interval [A, B] I such that f can assume nonzero values only at points in [A, B] and, hence, f and f both vanish on I \ [A, B]. It follows that each function in C0 is bounded. For example, if I = (0, ) and f (x ) = 1 x for 0 < x < 1 and f (x ) = 0 for 1 x < , then f C0 (I ), I = [0, ), f (x ) = 1 x for 0 x < 1 and f (x ) = 0 for 1 x < , and [A, B] = [0, 1]. Note that A I but A I . Whatever the type of the interval I, the integral b a

f (x ) dx

is defined by Riemann’s original definition where a and b are real numbers and by b a

f (x ) dx =

B A

f (x ) dx

when the interval I is unbounded and f vanishes on I \ [A, B]. With these agreements, we define

The Riesz-Nagy Approach to the Lebesgue Integral ▪ 77 b a

f (x ) dx =

b a

f (x ) dx

for any real-valued continuous function f in C0 (I ).

5.1 NULL SETS AND SETS OF MEASURE ZERO Integrals in this section are Riemann integrals of continuous functions or, in the case of Lemma A and Lemma B, of step functions. The integral of a step function is defined in the customary way. Riesz and Nagy developed a theory of Lebesgue integration based on two lemmas, A and B, which they stated as follows in [8]: Lemma A: For every sequence of step functions { n (x )}, which decrease to 0 almost everywhere, the sequence of values of their integrals tends to zero. Lemma B: If for an increasing sequence of step functions the values of their integrals have a common bound, then the sequence tends almost everywhere to a finite limit. The proofs given are short, and are essentially outlines that suggest how the lemmas can be established; they seem to us to be incomplete. Apparently, Asplund and Bungart agree because they give a much fuller account of Lemmas A and B and their proofs in [2]. Lemma B motivates the following definition, a variation of one used by Riesz and Nagy in their discussion of the Daniell approach to integration and of one given in Asplund and Bungart. Definition 27: Let I be an interval in with endpoints a b. A set E I is a null set if there exists an increasing (meaning nondecreasing) sequence of continuous functions (fn )n in C0 (I ) such that 1. fn (x ) 2.

+

b f (t ) dt a n

for each x

E.

A for some real number A.

Equivalently, we can assume that the functions fn are nonnegative by replacing the sequence fn by the sequence fn

f1 .

78 ▪ Aspects of Integration

When the functions fn are nonnegative we can choose A to be any positive real number by replacing the functions fn by the functions gn = fn /A. There is a useful series formulation of Definition 27. Proposition 28: E is a null set if and only if there is a sequence of functions (fn )n in C0 such that n fn (x ) = + for each x E and b n a fn (t ) dt


0 because, if as not, uN (x ) = u0 (x ) = 0 for all N which contradicts uN (x ) N for x E. Thus, J is a countable collection of intervals whose total length is at most , E is contained in the union of these intervals, and, hence, E has measure zero. : Since E has measure zero, for each n there is a countable n collection of open intervals {In, k}k such that E k In, k and k In, k < 2 . Let the sequence (Im)m be an enumeration of the countable collection {In, k}n, k of open intervals. Then, each x E belongs to an infinite number of the intervals Im and m Im < n k In, k < 1. Each

82 ▪ Aspects of Integration

Im = (am, bm) determines a piecewise linear continuous function fm with

0 x

fm

by fm (x ) = 1 for x

1 defined on

Im, fm (x ) = 0 for

am 2 m and for x bm + 2 m . Since each x E belongs to infinitely many of the intervals Im, m

fm (x ) = +

for x

E

and b m

a

fm (t ) dt =

bm

m

f (t ) dt am m

+2

m

= m

{ Im + 2 m} < 2.

Thus, E is a null set by Proposition 28. ■ A property is said to hold almost everywhere (a.e.) or almost always (a.a.) if it holds at all points not in a set of measure zero.

5.2 LEMMA’S A AND B The standing assumptions stated before Section 5.1 remain in force. Lemmas A and B of Riesz and Nagy, with step functions replaced by continuous functions, are established in this section. It is convenient to start with Lemma B. Lemma 32: (Lemma B) If (fn ) is an increasing sequence of functions in

C0 (I ) and

b f (t ) dt a n

A for all positive integers n and some constant A,

then limn

fn (x )
0, we can fix A and B so that

(x ) dx < /2. Now, Dini’s lemma applies to the

sequence (fn ) on [A , B ] and there is an index N

0

fn (x ) < /2(B 0

b a

+ for n

A ) for x

fn (x ) dx = B B

B A

such that

[A , B ],

fn (x ) dx

A A

f1 (x ) dx +

B A

fn (x ) dx

f1 (x ) dx < ■

N , and there exists lim

n

b a

fn (x ) dx = lim n

b a

fn (x ) dx = 0.

The definition of a null set and Lemma 33 yield a proof of Lemma A in Riesz and Nagy, with continuous functions replacing step functions. Lemma 34: (Lemma A) If (fn ) is a decreasing sequence of functions in C0 (I ) that converges to zero almost everywhere, then

84 ▪ Aspects of Integration b

lim

fn (x ) dx = 0.

a

n

Proof. Let E be a null set (set of measure zero) such that limn fn (x ) = 0 for x E. There is an increasing sequence of nonnegative continuous gn (x ) = + for x E and functions (gn) in C0 (I ) such that limn b a

gn (x ) dx

A for some constant A and all positive integers n.

Let > 0 and define a decreasing sequence (hn) of functions in C0 (I ) gn (x ))+ where ( )+ is the positive part of the by hn (x ) = (fn (x ) indicated function. If x E, then

fn (x ) as n

, while if x

gn (x )

. Consequently, hn

fn (x )

n

fn = fn 0

b

a

fn (x ) dx

g1 (x )

g1 (x )

0

hn (x ) dx = 0.

a

gn + gn b

gn (x )

0 pointwise on I and by Lemma 33

lim

Since 0

f1 (x )

E, then

fn (x ) as n

gn (x )

hn + gn , b a

hn (x ) dx +

b a

gn (x ) dx

The second integral on the right is bounded by A. Let n to obtain

0

lim

n

b a

fn (x ) dx

lim

n

b a

hn (x ) dx = 0.

0 and then

■

5.3 THE CLASS C1(I) OF RIESZ AND NAGY Riesz and Nagy use Lemma’s A and B as an entrée to the Lebesgue integral in which integration comes first and the Lebesgue measure

The Riesz-Nagy Approach to the Lebesgue Integral ▪ 85

comes second. We carry out this approach in the spirit of Riesz and Nagy but with step functions replaced by the class C0 = C0 (I ) of continuous functions. Recall that I is an interval of any type whose endpoints are a a b and b with , I is the closure of I in , and each function f in C0 = C0 (I ) is bounded and vanishes off a closed bounded interval [A, B] I , when I is unbounded. If either endpoint a or b is a real number, then f in C0 has a unique continuous extension f from I to I ; equivalently, limx a f (x ) and limx b f (x ) exist and are finite when a or b are real numbers. (The assumptions on the class C0 were motivated by certain properties of step functions. See the paragraph at the end of the standing assumptions.) Definition 35: Denote by C1 = C1 (I ) the class of extended real-valued functions f on I such that there is an increasing sequence (fn ) of continuous functions in C0 (I ) such that fn and

b f (x ) dx a n

f for almost all x

I,

A for some real number A and all natural numbers n.

1. If f C1, then f is finite almost everywhere. If (fn ) is an increasing sequence in C0 as in the definition, then fn g = limn fn (x ) finite or infinite for all x in I. By Lemma B g f almost everywhere, f = g is finite almost everywhere. Since fn almost everywhere and, hence, f is finite almost everywhere. 2. If f C1 and g = f almost everywhere, then g C1. A sequence (fn ) that implies that f C1 also implies that g

C1.

3. If f C1 and (fn ) is an increasing sequence as in the definition, then (fn ) converges almost everywhere to f on I. By #1 f is finite almost everywhere. Definition 36: If f for f if fn

C1, a sequence of functions (fn ) in C0 is approximate

f almost everywhere, fn

fn+1 , and

b f (x ) dx a n

A for some

real number A and all positive integers n.

C0

C1 because given f

C0 the sequence fn = f for all n is

approximate for f . It is natural to try to extend the (Riemann) integral defined on C0 to the class of functions C1 by defining

86 ▪ Aspects of Integration b a

f (x ) dx = lim n

b a

fn (x ) dx ,

where the limit of the integrals on the right exists and is finite because the sequence of integrals is increasing and bounded above. We must show that the definition is well posed. That is, if the sequences fn and gn are both approximate for f , then b

lim

a

n

b

fn (x ) dx = lim

a

n

gn (x ) dx .

This equality is a direct consequence of Lemma 37: If the sequences fn and gn are approximate for f and g respectively and f

limn

b a

g almost everywhere, then limn

b f (x ) dx a n

gn (x ) dx.

Proof. Fix a positive integer m. The sequence of functions (fm

gn)n is

decreasing and converges pointwise almost everywhere to fm

g

Therefore, (fm By Lemma A

0.

)+

gn decreases pointwise almost everywhere to zero on I.

lim

n

b a

gn)+dx = 0

(fm

for each m. Since fm = gn + (fm gn) gn + (fm gn)+ , integration from a to b, letting n and then letting m completes the proof. ■ The lemma justifies the following definition. Definition 38: If f

C1 and the sequence (fn ) is approximate for f , then b a

f (x ) dx = lim

We have already observed that C0 is approximate for f and, hence,

n

b a

C1. If f

fn (x ) dx.

C0, then the sequence fn = f

The Riesz-Nagy Approach to the Lebesgue Integral ▪ 87

(C1)

b a

f (x ) dx = (C0)

b a

f (x ) dx .

A direct consequence of the definition of the class C1 and the preceding lemma is:

C1 and g = f almost everywhere, then g

If f b a

g (x ) dx =

b a

C1 and

f (x ) dx .

Functions in the class C1 have the following properties: 1. If r and s are nonnegative real numbers and f and g are in C1, then rf + sg C1 and b a

(rf (x ) + sg (x )) dx = r

b a

f (x ) dx + s

b a

g (x ) dx .

If (fn ) and (gn) are approximate for f and g, then (rfn + sgn) is approximate for rf + sg . Thus, rf + sg C1 and the additivity for nonnegative scalars of the integral follows directly from the definition of the integral in C1 and the corresponding property in C0. 2. If f and g are in C1 and f g almost everywhere, then b b f (x ) dx g (x ) dx . a a This property was established in the previous lemma. 3. C1 is a lattice; that is, if f and g are in C1, then f g and f g are in C1. First, consider the special case in which f and g are nonnegative. Since f and g are nonnegative there are nonnegative sequences (fn ) and (gn) that are approximate for f and g respectively. Let h = f g . Then hn = fn gn is approximate for h. Indeed, since (fn ) and (gn) are increasing sequences in C0, hn C0 and

hn = fn

gn

fn +1

gn +1 = hn +1.

So, (hn) is an increasing sequence in C0. Since there are constants A b b and B such that a fn (x ) dx A , a gn (x ) dx B , and fn and gn are nonnegative,

88 ▪ Aspects of Integration b a

b

hn (x ) dx

a

(fn (x ) + gn (x )) dx

A+B

for all n. Finally, limn hn (x ) = limn fn (x ) gn (x ) = f (x ) g (x ) almost everywhere. Therefore, h = f g C1. The proof is almost the same for k = f g except that a little more care is needed to see that the sequence k n = fn gn is increasing. Fix x and assume (without loss in generality by symmetry) that fn (x ) gn (x ). Then, k n (x ) = fn (x ) gn (x ) and since (fn ) and (gn) are increasing sequences,

k n (x ) = fn (x )

fn +1 (x )

gn +1 (x ) = k n +1 (x ).

Second, consider the general case where f and g are in C1. Let h = f g and let (fn ) and (gn) be approximate for f and g respectively and with A and B as above. Set h0 = f1 g1 C0 so that (fn h0) and (gn h0) are increasing sequences of nonnegative functions in C0 that tend almost everywhere to the (nonnegative) limits f h0 and g h0 respectively. Moreover, b a

( fn

h0) dx

A

b a

h0 dx and

b a

( gn

h0) dx

B

b a

h0 dx

so the sequences (fn h0) and (gn h0) are approximate for f h0 and g h0 respectively, these nonnegative functions are in C1, and (f h0) (g h0) C1 by what was already established. Finally,

f

g = (f

h 0)

(g

h 0) + h 0

by #1. Similar reasoning establishes that f

g

C1 C1.

4. If f is in C1, c is a point in I that is strictly between a and b, and I = I1 I2 where I1 and I2 are disjoint intervals with endpoints a and c and c and b respectively, then f is in C1 (Ij ) for j = 1, 2 and b a

f (x ) dx =

c a

f (x ) dx +

b c

f (x ) dx.

The Riesz-Nagy Approach to the Lebesgue Integral ▪ 89

If (fn ) is approximate for f on I, then clearly fn I j is approximate for f Ij on I j for j = 1, 2 and the additivity of the integrals follows from the definition of the integral and the corresponding result for functions in C0. The following example will be used later when we compare the Riesz and Nagy approach to the Lebesgue integral that starts with step functions with the current approach that starts with the class of continuous functions C0 (I ). Example 39: A step function defined on an interval I is in the class C1 (I ) and its C1-integral is the same as its Riemann integral. Any step function s1 can be expressed as a finite linear combination of characteristic functions of disjoint intervals. Some of the intervals in the representation of s1 may be degenerate, that is consist of a single point. In that case, deleting the terms that correspond to degenerate intervals yields a step function s expressed in terms of non-degenerate intervals. Since s1 = s almost everywhere, if s is in C1 (I ) so is s1. So, it suffices to establish that s =

K k =1 ck Ik

is in C1 and that (C1) s (x ) dx =

K k =1 ck Ik

where Ik are

pairwise disjoint bounded subintervals of I, ck are real numbers, each Ik has positive length, and Ik is the length of the interval Ik . By item #1 above, s will be in C1 and its C1-integral will be its Riemann integral if we confirm that ck Ik belongs to C1 for each k = 1, 2, …, K and that its C1-integral is

ck Ik . Thus, it suffices to show that the function f = c

J

C1 where c a real

number and J a bounded subinterval of I with endpoints u < v and that its C1-integral is c J . The verification that f C1 is slightly different for the two cases c 0 and c < 0. It is helpful to sketch the approximations that follow in each case. Case: c 0. For each positive integer n 2, f = c J can be approximated by the piecewise linear function fn defined: fn is 0 for x for x

v and fn is piecewise linear and joins the point (u, 0) to

(u + (v

u)/n, c ) to (v

functions fn

fn

u and

(v

u)/n, c ) to (v , 0). The sequence of

C0 and satisfies fn

fn+1 , fn dx fdx for all n, and f pointwise except at the points u and v. Thus, c J C1 when c 0

and its C1-integral is

90 ▪ Aspects of Integration

(C1)

f (x ) dx = lim n

fn (x ) dx = lim c (v

u)

c

2, f = c

J

n

Case: c < 0. For each positive integer n

v

u n

=cJ.

can be approxi-

mated by the piecewise linear function fn defined: fn is 0 for x and for x

(u

u

1/n

v + 1/n and fn is piecewise linear and joins the point

1/n, 0) to (u, c ) to (v , c ) to (v + 1/n, 0). The sequence of functions

fn C0 and satisfies fn fn+1 , fn dx f fdx for all n, and fn pointwise except at the points u and v. Thus, c J C1 when c < 0 and (C1)

f (x ) dx = lim n

fn (x ) dx = lim c (v n

u) + c

v

u n

=cJ.

Example 40: The function f (x ) = x 1/2 cos x C1 ((0, 1]) and its C1 -integral equals its Riemann integral, as understood by Riemann but now called an improper Riemann integral. The Riemann integral

1 0

x

1/2 cos xdx

exists and is finite by the basic

comparison test for improper integrals, comparing with x 1/2. The sequence of continuous functions fn (x ) = x 1/2 cos x for 1/n x 1 and fn (x ) = n1/2 cos(1/n) for 0 < x because fn

f pointwise and 1 0

for all n. Thus, f 1 f 0

1/n is approximate for f (x )

fn (x ) dx

n

1/2

+

1 1/ n

x

1/2dx

2

C1 ((0, 1]) and

(x ) dx = lim n

1 f (x ) dx 0 n

= lim n

1 x 1/2 cos xdx 1/ n

= lim

1 x 1/2 cos xdx 1/ n

n

+ =

1/ n 1/2 n cos(1/n) dx 0 1 f 0

(x ) dx ;

The Riesz-Nagy Approach to the Lebesgue Integral ▪ 91

that is, the C1-integral of f exists and is equal to the (improper) Riemann integral of f.

5.4 THE CLASS C2 OF RIESZ AND NAGY It will turn out that the class of functions C2 = C2 (I ) defined next is the class of Lebesgue integrable functions on I. If f1 and f2 are functions in C1 = C1 (I ) the function f1 f2 is defined almost everywhere because functions in C1 are finite almost everywhere by Lemma B. Consequently, the following definition makes sense. Definition 41: Denote by C2 the class of extended real-valued functions f on I for which there are functions f1 and f2 in C1 such that f (x ) =

f1 (x )

f2 (x ) almost everywhere.

By a slight abuse of notation, if f C2 we write f = f1 f2 although the equality may not hold on the set of measure zero. Sometimes it is convenient given two functions f1 and f2 in C1 to define a function f in C2 by setting f (x ) = f1 (x ) f2 (x ) for all x where the right member is defined or for which the right member is finite and for other x (which lie in a set of measure zero) to define f (x ) in any convenient way. For example, define f (x ) = 0 at such points. C1 C2 because f = f 0 for each f C1 and the 0 function belongs to C1. As for functions in C1, it is natural to try and extend the integral to functions in C2 by setting b a

f (x ) dx =

b a

f1 (x ) dx

b a

f2 (x ) dx.

For this proposed definition to be well posed we must show that if f = f1 f2 and f = f3 f4 where f1 ,..., f4 C1 then b a

f1 (x ) dx

b a

f2 (x ) dx =

b a

f3 (x ) dx

b a

f4 (x ) dx .

This equality follows at once because f1 + f4 = f2 + f3 almost everywhere, both members are functions in C1, and hence

92 ▪ Aspects of Integration b a

f1 (x ) dx +

b a

f4 (x ) dx =

b a

f2 (x ) dx +

b a

f3 (x ) dx .

Thus, the following definition is well posed. Definition 42: If f b a

C2 and f = f1 f (x ) dx =

b a

f2 for functions f1 and f2 in C1, b

f1 (x ) dx

a

f2 (x ) dx.

A direct consequence of the two preceding definitions is: If f b a

C2 and g = f almost everywhere, then g

g (x ) dx =

b a

f (x ) dx .

We just observed that C1

(C2)

C2 and

b a

C2 because f = f f (x ) dx = (C1)

b a

0 and it now follows that

f (x ) dx.

Functions in the classes C1 and C2 are called integrable because their integrals are finite. Riesz and Nagy say that C2 is the class of functions that Lebesgue called summable (Lebesgue integrable) on I but defer the proof, which seems to be given piecemeal later in their discussion of various other types of integrals. After proving the Lebesgue dominated convergence theorem for C2-integrals, Riesz and Nagy write on page 38 in [8] “Of course, Lebesgue proved this theorem starting with his definition, which we still have to prove is equivalent to ours.” We were unable to find their proof of equivalence later in Part I of [8], although we may just have missed it. Functions in the class C2 have the following properties: 1. If f C2, then f is finite almost everywhere. f = f1 f2 where f1 C1 and f2 C1 are finite almost everywhere. 2. If r and s are real numbers and f and g are in C2, then rf + sg C2 and

The Riesz-Nagy Approach to the Lebesgue Integral ▪ 93 b a

(rf (x ) + sg (x )) dx = r

Let f = f1

b a

f + g = (f1 + g1)

tf = tf1 tf2 C2 when t t 0 and f and g in C2, tf (x ) dx = = b a

( f (x ) + g (x )) dx = = b g (x ) dx a 2

=

g (x ) dx .

a

g2 where f1 , f2 , g1, and g2 are in C1. Then

f2 and g = g1

b a

b

f (x ) dx + s

b a

(f2 + g2)

0, and

(tf1 (x )

C 2,

f = f2

f1

C2. Hence, for

tf2 (x )) dx

b b b t a f1 (x ) dx t a f2 (x ) dx = t a f (x ) dx , b (( f1 (x ) + g1 (x )) ( f2 (x ) + g2 (x ))) dx a b b b f (x ) dx f (x ) dx + a g1 (x ) dx a 1 a 2 b b f (x ) dx + a g (x ) dx , a

and b a

f (x ) dx = =

b a

( f2 (x )

f1 (x )) dx =

b f (x ) dx a 1

b f (x ) dx a 2

b f (x ) dx a 2

=

b f (x ) dx a 1 b a

f (x ) dx.

If sigma subscript z equal to +1 if z ≥ 0 and equal to −1 if z < 0, then rf + sg = r ( r f ) + s ( s g ) C2 and b a

(rf (x ) + sg (x )) dx = r = r

3. If f and g are in C2 and f b a

b a b f a

( r f (x )) dx + s (x ) dx + s

b g a

b a

( s g (x )) dx

(x ) dx.

g almost everywhere, then

f (x ) dx

b a

g (x ) dx.

If f = f1 f2 and g = g1 g2 with f1 , f2 , g1, and g2 in C1, then f g almost everywhere means that f1 + g2 g1 + f2 almost everywhere, and, since both members of the inequality are in C1,

94 ▪ Aspects of Integration b a

b

(f1 (x ) + g2 (x )) dx

a

(g1 (x ) + f2 (x )) dx

and the stated result follows by rearranging the terms in the inequality and the definition of the integral in C2. 4. If f C2, then f , f + , and f are functions in C2. Let f = f1 f2 where f1 and f2 are functions in C1. The stated properties follow from the fact that C1 is a lattice and the identities

f = f1 f + = f1 f = f1 5. If f , g

f2 f2 f2

f1 f2 , f2 , f1 .

C2, then b a b a

f (x ) dx

b a

f (x ) dx ,

f (x ) + g (x ) dx

b a

f (x ) dx +

The first inequality follows from f inequality follows from f (x ) + g (x )

b a

g (x ) dx .

f f and #3. The second f (x ) + g (x ) and #2, #3.

6. If f is in C2, c is a point in I that is strictly between a and b, and I = I1 I2 where I1 and I2 are disjoint intervals with endpoints a and c and c and b respectively, then f is in C2 (Ij ) for j = 1, 2 and b a

f (x ) dx =

c a

f (x ) dx +

b c

f (x ) dx.

This additivity property follows at once from the corresponding property for functions in C1. 7. C2 is a lattice. Use #2, #4 and the relations f (g f )+ + g .

g = (f

g )+ + g and f

g=

The Riesz-Nagy Approach to the Lebesgue Integral ▪ 95

Proposition 43: If f in C0 (I ) such that 1. fn

C2 there is a sequence (fn ) of continuous functions

f pointwise almost everywhere on I.

2. limn

b a

fn (x )

f (x ) dx = 0.

Proof. The function f = f1

f2 where f1 and f2 are in C1. Consequently,

there are sequences (f1,n ) and (f2,n ) in C0 that are approximate for f1 and

f2 . The function fn = f1, n everywhere because f1,n and f1 b a

f (x )

f2, n is in C0 and fn

f1 and f2,n

f pointwise almost

f2 pointwise almost everywhere,

f2 is defined almost everywhere. (See Definition 41.) Finally, fn (x ) dx =

b a b a

f1 (x )

f1, n (x ) dx +

( f1 (x )

f1, n (x )) dx +

b f (x ) a 2 b ( f2 (x ) a

f2, n (x ) dx f2, n (x )) dx

0

as n because the sequences f1,n and f2,n are approximate for f1 and f2. ■ Step Functions vs Continuous Functions Riesz and Nagy used the collection C0 of step functions on the real line as the starting point for their development of the Lebesgue integral. As we noted earlier, they mention that the class of continuous functions, each of which vanishes off a closed bounded interval, could serve as the starting point. That is the point of view used here. The remarks that follow provide some perspective on the two approaches. First, following the current approach, each step function belongs to the class C1 as was shown in Example 39 and not to the class C0 of Riesz and Nagy. Following the approach of Riesz and Nagy, every continuous function f in the class C0 (I ) is in their class C1 because f vanishes off a closed bounded interval, say [A, B] I , and there is a sequence of step functions (sn) such that sn sn+1, sn f almost everywhere, and sn dx fdx for all n. One such sequence of step functions can be constructed as follows: Partition [A, B] by 2n + 1 equally spaced points that determine 2n pairwise disjoint intervals In, k . Define sn to be the step

96 ▪ Aspects of Integration

function that takes the value mn, k = inf{ f (x ) : x In, k} on In, k and is 0 on I \ [A, B]. Theorem 44 in the next section implies that applying the construction that leads from the class C0 to the class C1 to the class C1 itself does not lead to a new class of integrable functions. Therefore, the remarks above show that the Riesz and Nagy approach to integration starting either with step functions or with continuous functions in C0 (I ) lead to the same class C2 (I ) of integrable functions.

5.5 CONVERGENCE THEOREMS Can we extend the class of functions that have integrals still further? A natural next step would be to apply the same procedure to functions in the class C1 that was applied to functions in C0 to determine the class C1 and its integrals. The two theorems that follow show that applying the prior procedure to functions in C1 or C2 does not lead to a new classes of functions or integrals. These two theorems and especially Theorem 47, which follows them are versions of a result known as Beppo Levi’s lemma or theorem. Beppo Levi Theorems Theorem 44: Let (fn ) be an increasing sequence of functions in C1 such that

b f (x ) dx a n

C1 such that fn

function f

limn

A for some constant A and all n. Then there is a

b f (x ) dx a n

=

b a

f pointwise almost everywhere and

f (x ) dx , the integral defined for C1 functions.

Consequently, b a

f (x ) dx =

b

b

lim fn (x ) dx = lim

a n

a

n

fn (x ) dx

and the sequence ( fn) converges (has a finite limit) almost everywhere. Proof. Let (gn, k)k be a sequence that is approximate for fn ; that is,

(gn, k )k is an increasing sequence of functions in C 0, gn, k b a

fn almost everywhere as k

gn, k (x ) dx

An, k