http://dx.doi.org/10.1090/gsm/045 An Introductio n to Measur e an d Integratio n This page intentionally left blank An Introductio n to Measur e an d Integratio n SECOND EDITION Inder K. Rana Graduate Studies in Mathematics Volume 45 ;S!HSSJ»0 American Mathematical Society Providence, Rhode Island Editorial Board Steven G. Krantz David Saltman (Chair) David Sattinger Ronald Stern 2000 Mathematics Subject Classification. Primary 28-01; Secondary 28A05, 28A10, 28A12, 28A15, 28A20, 28A25, 28A33, 28A35, 26A30, 26A42, 26A45, 26A46. ABSTRACT. This text presents a motivated introduction to the theory of measure and integration. Starting with an historical introduction to the notion of integral and a preview of the Riemann integral, the reader is motivated for the need to study the Lebesgue measure and Lebesgue integral. The abstract integration theory is developed via measure. Other basic topics discussed in the text are Pubini's Theorem, Lp-spaces, Radon-Nikodym Theorem, change of variables formulas, signed and complex measures. Library of Congress Cataloging-in-Publication Data Rana, Inder K. An introduction to measure and integration / Inder K. Rana.—2nd ed. p. cm. — (Graduate texts in mathematics, ISSN 1065-7339 ; v. 45) Includes bibliographical references and index. ISBN 0-8218-2974-2 (alk. paper) 1. Lebesgue integral. 2. Measure theory. I. Title. II. Graduate texts in mathematics ; 45. QA312 .R28 2002 515/.42—dc21 2002018244 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. Republication, systematic copying, or multiple reproduction of any material in this publica• tion is permitted only under license from Narosa Publishing House. Requests for such permis• sion should be addressed to Narosa Publishing House, 6 Community Centre, Panchscheel Park, New Delhi 110 017, India. First Edition © 1997 by Narosa Publishing House. Second Edition © 2002 by Narosa Publishing House. All rights reserved. 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. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 07 06 05 04 03 02 In memory of my father Shri Omparkash Rana (24th April, 1924-26th May, 2002) This page intentionally left blank Contents Preface xi Preface to the Second Edition xvii Recipe for a one semester course and interdependence of the chapters xix Notations used in the text xxi Prologue. The length function 1 Chapter 1. Riemann integration 5 §1.1. The Riemann integral: A review 5 §1.2, Characterization of Riemann integrable functions 18 §1.3. Historical notes: The integral from antiquity to Riemann 30 §1.4. Drawbacks of the Riemann integral 36 Chapter 2. Recipes for extending the Riemann integral 45 §2.1. A function theoretic view of the Riemann integral 45 §2.2. Lebesgue's recipe 47 §2.3. Riesz-Daniel recipe 49 Chapter 3. General extension theory 51 §3.1. First extension 51 §3.2. Semi-algebra and algebra of sets 54 vm Contents §3.3. Extension from semi-algebra to the generated algebra 58 §3.4. Impossibility of extending the length function to all subsets of the real line 61 §3.5. Countably additive set functions on intervals 62 §3.6. Countably additive set functions on algebras 64 §3.7. The induced outer measure 70 §3.8. Choosing nice sets: Measurable sets 74 §3.9. The cr-algebras and extension from the algebra to the generated cr-algebra 80 §3.10. Uniqueness of the extension 84 §3.11. Completion of a measure space 89 Chapter 4. The Lebesgue measure on R and its properties 95 §4.1. The Lebesgue measure 95 §4.2. Relation of Lebesgue measurable sets with topologically nice subsets of R 99 §4.3. Properties of the Lebesgue measure with respect to the group structure on R 103 §4.4. Uniqueness of the Lebesgue measure 106 §4.5. * Cardinalities of the a-algebras C and B^ HO §4.6. Nonmeasurable subsets of R 113 §4.7. The Lebesgue-Stieltjes measure 114 Chapter 5. Integration 117 §5.1. Integral of nonnegative simple measurable functions 118 §5.2. Integral of nonnegative measurable functions 122 §5.3. Intrinsic characterization of nonnegative measurable functions 130 §5.4. Integrable functions 143 §5.5. The Lebesgue integral and its relation with the Riemann integral 153 §5.6. I/i[a, b] as completion of 1Z[a, b] 158 §5.7. Another dense subspace of L\[a, b] 163 §5.8. Improper Riemann integral and its relation with the Lebesgue integral 168 Contents IX §5.9. Calculation of some improper Riemann integrals 172 Chapter 6. Fundamental theorem of calculus for the Lebesgue integral 175 §6.1. Absolutely continuous functions 175 §6.2. Differentiability of monotone functions 179 §6.3. Fundamental theorem of calculus and its applications 191 Chapter 7. Measure and integration on product spaces 209 §7.1. Introduction 209 §7.2. Product of measure spaces 212 §7.3. Integration on product spaces: Fubini's theorems 221 §7.4. Lebesgue measure on M2 and its properties 229 §7.5. Product of finitely many measure spaces 237 Chapter 8. Modes of convergence and Lp-spaces 243 §8.1. Integration of complex-valued functions 243 §8.2. Convergence: Point wise, almost everywhere, uniform and almost uniform 248 §8.3. Convergence in measure 255 §8.4. Lp-spaces 261 §8.5. *Necessary and sufficient conditions for convergence in Lp 270 §8.6. Dense subspaces of Lp 279 §8.7. Convolution and regularization of functions 281 §8.8. LOO(X,S,IJ,): The space of essentially bounded functions 291 §8.9. 1/2(X, 5,/i): The space of square integrable functions 296 §8.10. Z/2-convergence of Fourier series 306 Chapter 9. The Radon-Nikodym theorem and its applications 311 §9.1. Absolutely continuous measures and the Radon-Nikodym theorem 311 §9.2. Computation of the Radon-Nikodym derivative 322 §9.3. Change of variable formulas 331 Chapter 10. Signed measures and complex measures 345 X Contents §10.1. Signed measures 345 §10.2. Radon-Nikodym theorem for signed measures 353 §10.3. Complex measures 365 r §10.4. Bounded linear functionals on L p(X, 5, /i) 373 Appendix A. Extended real numbers 385 Appendix B. Axiom of choice 389 Appendix C. Continuum hypothesis 391 Appendix D. Urysohn's lemma 393 Appendix E. Singular value decomposition of a matrix 395 Appendix F. Functions of bounded variation 397 Appendix G. Differentiable transformations 401 References 409 Index 413 Index of notations 419 Preface "Mathematics presented as a closed, linearly ordered, system of truths without reference to origin and purpose has its charm and satisfies a philosophical need. But the attitude of introverted science is un• suitable for students who seek intellectual independence rather than indoctrination; disregard for applications and intuition leads to isola• tion and atrophy of mathematics. It seems extremely important that students and instructors should be protected from smug purism." Richard Courant and Fritz John (Introduction to Calculus and Analysis) This text presents a motivated introduction to the subject which goes under various headings such as Real Analysis, Lebesgue Measure and Integration, Measure Theory, Modern Analysis, Advanced Analysis, and so on. The subject originated with the doctoral dissertation of the French mathematician Henri Lebesgue and was published in 1902 under the ti• tle Integrable, Longueur, Aire. The books of C. Caratheodory [8] and [9], S. Saks [35], LP. Natanson [27] and P.R. Halmos [14] presented these ideas in a unified way to make them accessible to mathematicians. Because of its fundamental importance and its applications in diverse branches of mathe• matics, the subject has become a part of the graduate level curriculum. Historically, the theory of Lebesgue integration evolved in an effort to remove some of the drawbacks of the Riemann integral (see Chapter 1). However, most of the time in a course on Lebesgue measure and integra• tion, the connection between the two notions of integrals comes up only XI Xll Preface after about half the course is over (assuming that the course is of one se• mester). In this text, after a review of the Riemann integral, the reader is acquainted with the need to extend it. Possible methods to carry out this extension are sketched before the actual theory is presented. This approach has given satisfying results to the author in teaching this subject over the years and hence the urge to write this text. The nucleus for the text was provided by the lecture notes of the courses I taught at Kurukshetra Uni• versity (India), University of Khartoum (Sudan), South Gujarat University (India) and the Indian Institute of Technology Bombay (India). These notes were slowly augmented with additional material so as to cover topics which have applications in other branches of mathematics. The end product is a text which includes many informal comments and is written in a lecture-note style. Any new concept is introduced only when it is needed in the logical development of the subject and it is discussed informally before the exact definition appears. The subject matter is developed by motivating examples and probing questions, as is normally done while teaching. I have tried to avoid slick proofs. Often a proof is either divided into steps or is presented in such a way that the main ideas of the proof emerge before the details follow. Summary of the text The text opens with a Prologue on the length function and its properties which are basic for the development of the subject.