Recent Titles in This Series 4 Russell A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, 1994 3 William W. Adams and Philippe Loustaunau, An introduction to Grobner bases, 1994 2 Jack Graver, Brigitte Servatius, and Herman Servatios, Combinatorial rigidity, 1993 1 Ethan Akin, The general topology of dynamical systems, 1993

http://dx.doi.org/10.1090/gsm/004

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The Integrals of Lebesgue, Denjoy, Perron, and Henstock

Russell A. Gordon

Graduate Studies

in Mathematics

Volume 4

±>2S**\

))i American Mathematical Society

Editorial Board James E. Humphreys

Robion C. Kirby Lance W. Small

1991 Mathematics Subject Classification. Primary 26Axx, 28Axx.

A B S T R A C T . Suppose that / : [a, 6] - • R is differentiable at each point of [a, 6]. Is / ' integrable on [a, 6]? The answer to this question depends on the integral that is used. For example, the answer is no for the Riemann and Lebesgue integrals. In this century, three integration processes have been developed that provide an affirmative answer to this question. The principal investigators of these integrals were Denjoy, Perron, and Henstock. Each of these integrals generalizes a different property of the Lebesgue integral, but it turns out that all three integrals are equivalent.

In this book, the properties of the Lebesgue, Denjoy, Perron, and Henstock integrals are devel­oped fully from their definitions. The equivalence of the last three integrals is then established. Discussions of the integration by parts formula and convergence theorems are included. In the last part of the book, we consider approximate derivatives and attempts to develop an integration process for which every approximate derivative is integrable.

Library of Congress Cataloging-in-Publication D a t a

Gordon, Russell A., 1955-The integrals of Lebesgue, Denjoy, Perron, and Henstock/Russell A. Gordon.

p. cm. —(Graduate studies in mathematics, ISSN 1065-7339; 4) Includes bibliographical references and index. ISBN 0-8218-3805-9 1. Measure theory. 2. Functions of real variables. I. Title. II. Series.

QA312.G63 1994 515'.42-dc20 94-19080

CIP

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

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© Copyright 1994 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights

except those granted to the United States Government. Printed in the United States of America.

© The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

This publication was typeset by the author, with assistance from the American Mathematical Society, using AMS-T&L, the American Mathematical Society's IfcjX macro system.

10 9 8 7 6 5 4 3 10

To Brenda and Charles

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Contents

Preface ix Chapter 1. Lebesgue Measure 1 Chapter 2. Measurable Functions 17 Chapter 3. The Lebesgue Integral 29 Chapter 4. Bounded Variation and Absolute Continuity 49 Chapter 5. Darboux and Baire Class One Functions 69 Chapter 6. Functions of Generalized Bounded Variation 89 Chapter 7. The Denjoy Integral 107 Chapter 8. The Perron Integral 121 Chapter 9. The Henstock Integral 137 Chapter 10. The McShane Integral 157 Chapter 11. Equivalence of Integrals 169 Chapter 12. Integration by Parts 181 Chapter 13. Convergence Theorems 201 Chapter 14. Approximate Derivatives 223 Chapter 15. The Khintchine Integral 237 Chapter 16. The Approximately Continuous Henstock Integral 245 Chapter 17. The Approximately Continuous Perron Integral 259 Solutions to Exercises 271 References 389 Notation Index 391 Subject Index 393

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Preface

The Fundamental Theorem of Calculus is one of the highlights of a first year calculus course. Its importance lies in the fact that it reveals a significant rela­tionship between integration and differentiation. As usually stated in calculus books, the functions involved are continuous. However, the following version of the Fundamental Theorem of Calculus is valid.

If F is differentiate on [a,b] and if F' is Riemann integrable on [a, b], then / * F' = F(x) - F(a) for each x € [a, 6].

At first glance, it might appear that there is an extra hypothesis. Aren't all derivatives Riemann integrable? A brief search leads one to derivatives that are not bounded and, as a result, not Riemann integrable. However, there are even bounded derivatives (existing at all points) that are not Riemann integrable. In other words, the extra hypothesis is essential.

What about the Lebesgue integral? This integral was designed to overcome the deficiencies of the Riemann integral. Are all derivatives Lebesgue integrable? The answer once again is no. However, all bounded derivatives are Lebesgue integrable so the following version of the Fundamental Theorem of Calculus is valid.

If F is differentiate on [a, b] and if F' is bounded on [a, 6], then F' is Lebesgue integrable on [a, b] and fx F' = F{x) — F(a) for each x G [a, 6].

This discussion leads naturally to the following question. Is it possible to define an integration process for which the theorem

If F is differentiate on [a, 6], then the function F' is integrable on [a, b] and / * F' = F(x) - F(a) for each x G [a, &].

is valid? The answer is yes. In this century, three integration processes have been developed for which this ideal version of the Fundamental Theorem of Calculus is valid. These integrals, named after their principal investigators Denjoy, Perron, and Henstock, each generalize some aspect of the Lebesgue integral. Since each of these new integrals focuses on a different property of the Lebesgue integral, the definitions of the integrals are radically different. However, it turns out that all three integrals are equivalent. The purpose of this book is to present, in an elementary fashion, these integration processes.

ix

X PREFACE

The material for this book has been drawn from a number of different sources. The books by Saks and Natanson are the standard references for the Denjoy and Perron integrals. However, Natanson presents little more than a brief introduc­tion to these integrals and the classic work by Saks is not easily accessible to a novice in the field. The theory of the Henstock integral and recent developments related to all three integrals exist primarily in research articles. For this reason, it is difficult to follow these new developments. In addition, there have been some dead ends, the notation is inconsistent and often confusing, and some of the proofs contain errors. A detailed treatment of all three integrals from the ground up does not exist. As these integrals have found several applications and since they are interesting in their own right, it seems fitting that such a devel­opment should be available. The hope here is that a reader with a thorough understanding of basic real analysis, such as that found in Rudin's Principles of Mathematical Analysis, will be able to read this book.

The book essentially has three parts. Chapters 1 through 4 provide an intro­duction to the Lebesgue integral. No familiarity with the Lebesgue integral or the concept of measure is assumed. However, it is not the purpose of this book to provide an in-depth study of the Lebesgue integral. Only the bare essentials of Lebesgue integration are discussed here, but these essentials are discussed in full detail. The driving force is to present those aspects of the Lebesgue integral that are necessary to develop the Denjoy, Perron, and Henstock integrals. These first four chapters have been used successfully as an independent study course for advanced undergraduates.

The next section of the book, Chapters 7 through 13, presents the integrals of Denjoy, Perron, and Henstock. The definition and basic properties of each integral is considered in a separate chapter. Included in each of these chapters is, of course, a proof that the integral satisfies the ideal form of the Fundamental Theorem of Calculus. Chapter 11 establishes the equivalence of these three integrals. Chapter 10 is an aside to consider the McShane integral, a Riemann type integral that is equivalent to the Lebesgue integral. The integration by parts formula for each of these integrals is the primary focus of Chapter 12. An equally important part of this chapter is a brief introduction to the Riemann-Stieltjes integral. Convergence theorems are the topic of Chapter 13. A general convergence theorem for each integral is stated and proved, then it is shown which of these theorems is the most general. Some essential prerequisites for this section of the book are discussed in Chapters 5 and 6. These chapters form a bridge between the Lebesgue integral and its generalizations. The level of sophistication is increased at this point. The concepts of Darboux function, Baire class one function, and functions of generalized bounded variation are defined and studied.

Another transition occurs in Chapter 14. At this point, the notions of ap­proximate continuity and approximate derivative are introduced. It is shown how these concepts fit naturally in the theory of the Lebesgue integral. A num-

PREFACE XI

ber of interesting facts about the approximate derivative are stated and proved. The last part of the book, Chapters 15 through 17, considers integration pro­cesses that attempt to recover a function from its approximate derivative. In other words, the focus changes to the following theorem.

If F is approximately differentiable on [a, 6], then F^p is integrable on [a, b] and f* F^p = F(x) — F(a) for each x G [a, b].

This section looks at the Khintchine integral as well as some recent attempts to generalize the Henstock and Perron integrals to the case in which the indefi­nite integral is only approximately continuous. The book ends with some open questions related to these integrals.

Each of the chapters contains a set of exercises. Some of the exercises are sim­ple consequences of definitions and theorems. These are intended to provide some practice with the concepts. Other exercises introduce important /interesting re­sults that are not discussed in the text. It should be pointed out that some of the exercises are rather difficult. The order of the exercises corresponds to the order of the text. It may happen that a problem is trivial as a result of a later theorem. In such a case, it is usually intended that the problem be solved directly from the definition or earlier results. The reader who desires to develop a working knowledge of this subject should try all of the exercises. Complete solutions to all of the exercises are provided.

Most, but not all, of the notation is standard. A notation index can be found at the back of the book. A short list of references is also given. This list is by no means exhaustive. Several of these works (especially the book by Bruckner and the paper by Bullen on nonabsolute integrals) have extended lists of articles in them and should be consulted by the reader who wants to dig deeper. Another good source is the journal Real Analysis Exchange. There has been no attempt in this work to trace the history of the subject and, when simpler, newer proofs are given rather than those found in the original sources. As mentioned above, this work is drawn from a number of different sources and the author is deeply indebted to them. However, my own style and methods are evident throughout. Some of the proofs and ideas in this book are entirely new. I have worked independently on this book and would appreciate any comments, suggestions, and /or questions. Correspondence should be sent to the author at Whitman College, Walla Walla, WA 99362. A response will be sent as quickly as possible.

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References

1. R. G. Bartle, A convergence theorem for generalized Riemann integrals, Real Analysis Exchange 20 (1994-95), no. 2, 119-124.

2. R. G. Bartle and D. R. Sherbert, Introduction to real analysis, John Wiley & Sons, Inc., New York, 1992.

3. A. M. Bruckner, Differentiation of real functions, Lect. Notes in Math, Vol. 659, Springer-Verlag, Heidelberg, 1978.

4. P. S. Bullen, The Burkill approximately continuous integral, J. Austral. Math. Soc. (Series A) 35 (1983), 236-253.

5. , A survey of integration by parts for Perron integrals, J. Austral. Math. Soc. (Series A) 40 (1986), 343-363.

6. , A simple proof of integration by parts for the Perron integral, Canad. Math. Bull. 28 (2) (1985), 195-199.

7. , Nonabsolute integrals, Real Analysis Exchange 5 (1979-80), no. 2, 195-259.

8. J. C. Burkill, The approximately continuous Perron integral, Math. Z. 34 (1931), 270-278.

9. R. O. Davies and Z. Schuss, A proof that Henstoch's integral includes Lebesgue's, J. London Math. Soc. (2) 2 (1970), 561-562.

10. J. D. DePree and C. W. Swartz, Introduction to real analysis, John Wiley & Sons, Inc., New York, 1988.

11. A. G. Dzvarseisvili, On a sequence of integrals in the sense of Denjoy, Akad. Nauk Gruzin. SSR. Trudy Tbiliss. Mat. Inst. Razmadze 18 (1951), 221-236. (Russian)

12. C. Goffman, Real functions, Holt, Rinehart and Winston, New York, 1953. 13. C. Goffman and C. Neugebauer, On approximate derivatives, Proc. Amer.

Math. Soc. 11 (1960), 962-966. 14. R. A. Gordon, Some comments on an approximately continuous Khintchine

integral, Real Analysis Exchange 20 (1994-95), no. 2, 831-841. 15 , An iterated limits theorem applied to the Henstock integral, Real

Analysis Exchange 21 (1995-96), no. 2, 774-781. 16. R. Henstock, A Riemann-type integral of Lebesgue power, Canad. Jour.

Math. 20 (1968), 79-87. 17 , Theory of integration, Butterworths, London, 1963.

389

390 REFERENCES

18. E. W. Hobson, The theory of functions of a real variable and the theory of Fourier series, Vol. 1, 3rd ed., Cambridge, 1927.

19. H. Kestelman, Modern theories of integration, Oxford, 1937. 20. A. Khintchine, Recherches sur la structure des fonctions mesurables, Fund.

Math. 9 (1927), 212-279. 21. J. Kurzweil, Generalized ordinary differential equations and continuous de­

pendence on a parameter, Czech. Math. J. 7 (1957), 418-446. 22 , Nichtabsolut konvergente integrate, Leipzig, 1988. 23. P. Y. Lee, Lanzou lectures on Henstock integration, World Scientific, Singa­

pore, 1989. 24. R. M. McLeod, The generalized Riemann integral, Carus Mathematical Mono­

graphs, No. 20, Mathematical Association of America, 1980. 25. E. J. McShane, A unified theory of integration, Amer. Math. Monthly 80

(1973), 349-359. 26.1. P. Natanson, Theory of functions of a real variable, Vols. 1 and 2, Frederick

Ungar Publishing Co., New York, 1964. 27. W. F. Pfeffer, A note on the generalized Riemann integral, Proc. Amer.

Math. Soc. 103 (1988), 1161-1166. 28 , The Riemann approach to integration, Cambridge Univ. Press, Lon­

don, 1993. 29. P. Romanovski, Essai d'une exposition de I'integrate de Denjoy sans nombres

transfini, Fund. Math. 19 (1932), 38-44. 30. H. L. Royden, Real analysis, Third edition, Macmillan, New York, 1988. 31. W. Rudin, Principles of mathematical analysis, Third edition, McGraw-Hill,

New York, 1976. 32. S. Saks, Theory of the integral, 2nd. ed. revised, Hafner, New York, 1937. 33. G. Tolstov, Sur Vintegrale de Perron, Mat. Sb. 5 (1939), 647-659.

Notation Index

Z + set of positive integers

Q set of rational numbers

R set of real numbers

Re extended real numbers, R U {±00}

E closure of the set E

E° interior of the set E

CE complement of the set E

dE boundary of the set E

Ed set of points of density of E that belong to E

XE characteristic function of the set E

£(I) length of interval /

[i{E) measure of the set E

li* (E) outer measure of the set E

H*(E) inner measure of the set E

dxE density of the set E at the point x

p(x, E) distance from the point x to the set E

d(A, B) distance between the sets A and B

A C B A is a subset of B or A = B

A-B AnCB

A&B symmetric difference of the sets A and B

Gs set any set that is a countable intersection of open sets

Fa set any set that is a countable union of closed sets

f(x+) lim f(t)

f(x-) lim f{t) t-*X~

/ + / + ( x ) = m a x { / ( x ) , 0 }

/ " / - ( x ) = max{-/( :r) ,0}

rrif rrif(x) = lim inf{/(£) : t G [x - r, x + r) R [a, b]} r—*0 +

Mf Mj{x)— lim sup{/(£) : t e {x - r,x + r) n [a, 6]} r—>0+

391

392 NOTATION INDEX

4>f distribution function of /

F'(x) derivative of F at x

FLp(x) approximate derivative of F at x

-̂ st (x) strong derivative of F at x

DF(x) upper derivate of F at x

ADF(x) lower approximate derivate of F at x

D+F(x) lower right derivate of F at x

AD~F(x) upper left approximate derivate of F at x

SDF(x) upper strong derivate of F at x

UJ(F, [a, 6]) oscillation of the function F on the interval [a, b]

V(F, [a, b]) variation of the function F on the interval [a, b]

V(F, E) weak variation of the function F on the set E

V* (F, E) strong variation of the function F on the set E

sgn x 1 if x > 0, 0 if x = 0, and — 1 if x < 0

$ collection of Borel sets

V a collection of non-overlapping tagged intervals

V a collection of non-overlapping free tagged intervals

A an approximate full cover

A# approximate full cover with tags in the set E

Uba U(b) - U(a)

f{V) Riemann sum associated with V

f(V^) Riemann-Stieltjes sum associated with <f> and V

F(V) sum of the increments of F over V

Subject Index

absolute continuity, 49 AC function, 90 ACs function, 146 AC A function, 252 AC* function, 90 ACG function, 90 ACG6 function, 146 ACGA function, 252 ACG* function, 90 algebra, 7 almost everywhere, 18 almost uniform convergence, 22 ap-ACG*, 265 ap-Henstock integral, 247 ap-major function, 259 ap-minor function, 259 ap-Perron integral, 261 APC integral, 267 approximately continuous, 225 approximate derivates, 249, 255 approximate derivate

properties, 249 approximately differentiable,

228 approximate full cover, 245 approximate lower limit, 242 approximate upper limit, 242 Axiom of Choice, 6, 230

Baire Category Theorem, 69 Baire classes, 73 Baire class one function, 73 Borel measurable function, 21 Borel set, 7, 20, 21 Bounded Convergence

Theorem, 34

bounded variation, 49 BV function, 89 BV6 function, 178 BV* function, 89 BVG function, 89 BVG8 function, 178 BVG* function, 89

Cantor function, 14, 59 Cantor set, 13 Cauchy criterion, 32, 142, 189 characteristic function, 17 common major function, 208 component of a set, 232 condensation point, 16 condition (AT), 96 contiguous intervals, 92 continuous interval function,

175 convergence theorem, 42, 201 convergent interval function, 80

Darboux function, 71 (5-Cauchy sequence, 221 Denjoy Convergence Theorem,

205 Denjoy integral, 108 dense set, 69 density, 223 derivates of a function, 54 derivate properties, 122 distance between sets, 16 distribution function, 197 Dominated Convergence

Theorem, 45 doubly indexed sequence, 204

394 SUBJECT INDEX

Egoroff's Theorem, 22 equi-absolutely continuous, 67 equi AC, 202 equi AC with respect to

Lebesgue measure, 201 equi ACG, 241 equi ACG*, 204 equi W * , 212 equicontinuous, 221 ex-major function, 132 ex-minor function, 132

fAC6, 168 Fatou's Lemma, 43 first category, 69 free tagged interval, 158 Fa set, 8, 88 Fubini's Theorem, 229 Fundamental Theorem of

Calculus, 57, 59

G6 set, 8, 15, 87 generalized P-Cauchy

sequence, 213

Henstock Convergence Theorem, 210, 219

Henstock integral, 140

improper Riemann integral, 152 infinite-valued functions, 62 inner measure, 14 integrable set, 154 integration by parts, 182 interchange of limit operations,

204

Khintchine Convergence Theorem, 241

Khintchine integral, 237

Lebesgue integral, 30, 32, 40, 41 left-hand density, 255 limsup of sets, 15 linear extension, 25 Lipschitz function, 105 lower Lebesgue integral, 32 lower Riemann integral, 30 lower semicontinuous, 81 Lusin's Theorem, 26

major function, 121 McShane integral, 158 Mean Value Theorem, 97, 234 measurable set, 3 measurable function, 17 metric, 68 minor function, 121 Monotone Convergence

Theorem, 43, 161

nearly everywhere, 101 nonmeasurable set, 7 nowhere dense, 12, 69 nowhere differentiate

function, 52

oscillation of a function, 89 outer measure, 1

'P-Cauchy sequence, 213 Pc integral, 127 perfect portion, 70 perfect set, 12 Perron Convergence Theorem,

208 Perron integral, 123 point of density, 223 point of dispersion, 223 point of divergence, 115 point of nonsummability, 114 portion, 69 Px integral, 132

restriction of a function, 18 Riemann integral, 30, 137 Riemann-Stieltjes integral, 189 Riemann-Stieltjes sum, 188 right-hand density, 255 Romanovski's Lemma, 78

Saks-Henstock Lemma, 144, 159, 249

second category, 69 cr-algebra, 7 simple function, 23 singular function, 59 step function, 29 strong derivative, 165 strong derivates, 167

SUBJECT INDEX 395

strong variation. 89 subadditive, 2 subordinate to <5, 138 summable, 41, 114 superadditive, 175 symmetric difference, 9

tagged interval, 138 tagged partition, 138 Tietze Extension Theorem, 24 totalizable, 117 totalization, 108 transfinite numbers, 78, 116

uniformly Henstock integrable, 209

uniformly Lebesgue integrable, 220

upper Lebesgue integral, 32 upper semicontinuous, 81 upper Riemann integral, 30

variational integral, 175 variation of a function, 49 Vitali Convergence Theorem,

203, 219 Vitali cover, 52 Vitali Covering Lemma, 52

weak variation, 89

Zorn's Lemma, 230