Table Of Content

BARTIE The Elements of Integration and Lebesgue Measure The Elements of Integration and Lebesgue Measure The Elements of Integration and Lebesgue Measure ROBERT G. BARTLE Eastern Michigan University and University of Illinois Wiley Classics Library Edition Published 1995 A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Brisbane Toronto Singapore This text is punted on acid-free paper. Copyright © 1966 by John Wiley & Sons, Inc Wiley Classics Library Edition Published 1995 All rights reserved Published simultaneously in Canada Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, Inc , 605 Third Avenue, New York, NY 10158-0012 Library of Congress Cataloging in Publication Data Library of Congress Catalog Card Number 66-21048 ISBN 0-471-04222-6 Punted in the United States of America 10 9 8 7 6 Preface This book consists of two separate, but closely related, parts. The first part (Chapters 1-10) is subtitled The Elements of Integra- tion; the second part (Chapters 11-17) is subtitled The Elements of Lebesgue measure. It is possible to read these two parts in either order, with only a bit of repetition. The Elements of Integration is essentially a corrected reprint of a book with that title, originally published in 1966, designed to present the chief results of the Lebesgue theory of integration to a reader hav- ing only a modest mathematical background. This book developed from my lectures at the University of Illinois, Urbana Champaign, and it was subsequently used there and elsewhere with considerable success. Its only prerequisites are a understanding of elementary real analysis and the ability to comprehend "- S arguments". We suppose that the reader has some familarity with the Riemann integral so that it is not necessary to provide motivation and detailed discussion, but we do not assume that the reader has a mastery of the subtleties of that theory. A solid course in "advanced calculus", an understanding of the first third of my book The Elements of Real Analysis, or of most of my book Introduction to Real Analysis with D. R. Sherbert provides an adequate background. In preparing this new edition, I have seized the opportunity to correct certain errors, but I have re- sisted the temptation to insert additional material, since I believe v vl Preface that one of the features of this book that is most appreciated is its brevity. The Elements of Lebesgue Measure is descended from class notes written to acquaint the reader with the theory of Lebesgue measure in the space RP. While it is easy to find good treatments of the case p = 1, the case p > 1 is not quite as simple and is much less frequently discussed. The main ideas of Lebesgue measure are presented in detail in Chapters 10-15, although some relatively easy remarks are left to the reader as exercises. The final two chapters venture into the topic of nonmeasurabl a sets and round out the subject. There are many expositions of the Lebesgue integral from various points of view, but I believe that the abstract measure space approach used here strikes directly towards the most important results: the convergence theorems. Further, this approach is particularly well- suited for students of probability and statistics, as well as students of analysis. Since the book is intended as an introduction, I do not follow all of the avenues that are encountered. However, I take pains not to attain brevity by leaving out important details, or assigning them to the reader. Readers who complete this book are certainly not through, but if this book helps to speed them on their way, it has accomplished its purpose. In the References, I give some books that I believe readers can profitably explore, as well as works cited in the body of the text. I am indebted to a number of colleagues, past and present, for their comments and suggestions; I particularly wish to mention N. T. Hamilton, G. H. Orland, C. W. Mullins, A. L. Peressini, and J. J. Uhl, Jr. I also wish to thank Professor Roy G. Davies of Leicester University for pointing out a number of errors and possible improve- ments. ROBERT G. BARTLE Ypsilanti and Urbana November 20, 1994 Contents The Elements of Integration 1. Introduction 1 Reasons for the development of the Lebesgue integral, comparison with the Riemann integral, the extended real number system 2. Measurable I Cnctious 6 Measurable sets and functions, combinations, complex- valued functions, functions between measurable spaces 3. Measures 19 Measures, measure spaces, almost everywhere, charges 4. The Integral 27 Simple functions and their integrals, the integral of a non- negative extended real-valued measurable function, the Monotone Convergence Theorem, Fatou's Lemma, properties of the integral vii Contents 5. Integrable Functions 41 Integrable real-valued functions, positivity and linearity of the integral, the Lebesgue Dominated Convergence Theorem, integrands that depend on a parameter 6. The Lebesgue Spaces Lp 52 Normed linear spaces, the Lp spaces, Holder's Inequality, Minkowski's Inequality, the Completeness Theorem, the space L,,. 7. Modes of Convergence 65 Relations between convergence in mean, uniform convergence, almost everywhere convergence, convergence in measure, and almost uniform convergence, Egoroff's The- orem, the Vit al i Convergence Theorem 8. Decomposition of Measures 80 Hahn and Jordan Decomposition Theorems, the Radon- Nikodym Theorem, Lebesgue Decomposition Theorem, the Riesz Representation Theorem for Lp 9. Generation of Measures 96 Measures on algebras of sets, the extension of measures, Caratheodory and Hahn Extension Theorems, Lebesgue and Lebesgue-Stieltjes measure, the Riesz Representation Theorem for C 10. Product Measures 113 Rectangles, the Product Measure Theorem, sections, the Monotone Class Lemma, Tonelli and Fubini Theorems Contents The Elements of Lebesgue Measure 11. Volumes of Cells and Intervals 127 Cells, intervals, length, cells in RP, p--dimensional volume, translation invariance 12. The Outer Measure 131 The outer measure in RP, properties of m*, translation invariance 13. Measurable Sets 139 o-Algebras, measure on a o-algebra, the Caratheodory condition, Caratheodory's theorem, Lebesgue sets and Lebesgue measure, uniqueness of Lebesgue measure, some useful properties 14. Examples of Measurable Sets 149 Borel sets, null sets, translation invariance, existence of non-Borel sets 15. Approximation of Measurable Sets 155 Approximation by open sets, approximation by closed sets, approximation by compact sets, approximation by cells 16. Additivity and Nonadditivity 161 Additivity, Caratheodory revisited, inner measure