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Theory of the Integral THEORY OF THE INTEGRAL Brian S. Thomson Simon Fraser University CLASSICALREALANALYSIS.COM This text is intended as a treatise for a rigorous course introducing the ele- ments of integration theory on the real line. All of the important features of the Riemann integral, the Lebesgue integral, and the Henstock-Kurzweil integral are covered. The text can be considered a sequel to the four chapters of the more elementary text THE CALCULUS INTEGRAL which can be downloaded from our web site. For advanced readers, however, the text is self-contained. For further information on this title and others in the series visit our website. www.classicalrealanalysis.com There are free PDF files of all of our texts available for download as well as in- structions on how to order trade paperback copies. We also allow access to the content of our books on GOOGLE BOOKS and on the AMAZON Search Inside the Book feature. COVER IMAGE: This mosaic of M31 merges 330 individual images taken by the Ultraviolet/Optical Telescope aboard NASA’s Swift spacecraft. It is the highest-resolution image of the galaxy ever recorded in the ultraviolet. The image shows a region 200,000 light-years wide and 100,000 light- years high (100 arcminutes by 50 arcminutes). Credit: NASA/Swift/Stefan Immler (GSFC) and Erin Grand (UMCP) —http://www.nasa.gov/mission_pages/swift/bursts/uv_andromeda.html Citation: Theory of the Integral, Brian S. Thomson, ClassicalRealAnalysis.com (2012), [ISBN 1467998168] Date PDF file compiled: December 27, 2011 ISBN-13: 978-1467998161 ISBN-10: 1467998168 CLASSICALREALANALYSIS.COM Preface Work in Progress: This is a preliminary version of a text planned for an April 2012 publication date. This file is dated December 27, 2011. Return to our website occa- sionally for updates. Please send comments to the author. The text is a self-contained account of integration theory on the real line. The usual curricula in real analysis courses do not allow for much time to be spent on the Henstock-Kurzweil integral. Instead extensive accounts of Riemann’s integral and the Lebesgue integral are presented. Accordingly the version here would be mostly recommended for supplementary reading. Even so it would be a reasonable course design to teach this material prior to a course in abstract measure and integration. The student should end up as well-prepared as in more traditional courses. Certainly every professional math- ematician should be aware of more than Lebesgue’s theory; while nonabsolutely convergent integrals do not play an extensive role in applications, they are part of our history and of our culture. The reader might want to view first the prequel to this text: B. S. Thomson, The Calculus Integral, ClassicalRealAnalysis.com (2008). ISBN-13: 978-1442180956, ISBN-10: 1442180951 That text is an (experimental) outline of an elementary real analysis course in which the Newton integral plays the key role. Since the presentation in the present textbook also uses the Newton integral (in its various versions) as a motivating tool, the reader may wish also to consult the prequel to see how this would work at an introductory level. There are innumerable books published on the Lebesgue integral. The reader needing instruction in that theory is faced with too many choices, although many of them are truly excellent. For the more general integral (called here the gen- eral Newton integral or simply “the integral”) that is best known classically as the Denjoy-Perron integral and, more recently, as the Henstock-Kurzweil integral, there are far fewer choices and not all of them are excellent. I have resisted for many years writing a lengthy account of this integral, partly because the topic is not widely thought of as being of much significance. ii As an further experiment, however, I offer this account of the theory of that integral. The challenge as I see it is to present a coherent narrative leading the readers to a deep understanding of the nature of integration on the real line and, moreover, fully preparing them to study abstract measure and integration. But that is just a goal, not necessarily realized here. Most teachers will, doubtless, remain with the usual sequence of instruction: basic calculus (the Riemann and improper Riemann integrals vaguely presented), elementary analysis (the Riemann integral treated in depth), then abstract measure and integration in graduate school. My guess is that few graduate students, freshly taught this sequence, could survive an oral examination on the statement b F′(x)dx = F(b) F(a), Za − giving all conditions on how this might or might not hold. I hope my readers do better. BST Contents Preface i Table of Contents ii 1 By way of an introduction 1 1.1 The classical Newton integral ................... 1 1.2 Continuity and integrability ..................... 3 1.2.1 Upper functions ....................... 3 1.2.2 Continuous functions are Newton integrable ........ 4 1.2.3 Proof of Lemma 1.2 ..................... 5 1.3 Riemann sums ........................... 7 1.3.1 Mean-value theorem and Riemann sums ......... 9 1.3.2 Uniform Approximation by Riemann sums ......... 12 1.3.3 Cauchy’s theorem ...................... 12 1.3.4 Robbins’s theorem ..................... 13 1.3.5 Proof of Theorem 1.6 .................... 14 1.4 Characterization of Newton’s integral ............... 18 1.4.1 Proof of Theorem 1.8 .................... 20 1.5 How to generalize the integral? .................. 24 1.6 Exceptional sets .......................... 25 1.6.1 Sets of measure zero .................... 26 1.6.2 Proof of Lemma 1.10 .................... 27 1.7 Zero variation ............................ 28 1.8 Generalized Newton integral .................... 29 1.8.1 Justifications ........................ 30 1.8.2 Continuous linear functionals ................ 32 1.8.3 Which Newton variant should we teach .......... 33 1.9 Constructive aspects: the regulated integral ............ 34 1.9.1 Step functions and regulated functions ........... 35 1.10 Riemann’s integral ......................... 37 1.10.1 Integrability criteria ..................... 38 1.10.2 Proof of Theorem 1.20 ................... 39 iii iv CONTENTS 1.10.3 Volterra’s Example ..................... 43 1.11 Integral of Henstock and Kurzweil ................. 44 1.11.1 A Cauchy criterion ..................... 45 1.11.2 The Henstock-Saks Lemma ................ 45 1.11.3 Proof of Theorem 1.23 ................... 46 1.11.4 The Henstock-Kurzweil integral includes all Newton integrals 49 1.11.5 Proof of Theorem 1.24 ................... 50 1.12 Integral of Lebesgue ........................ 51 1.13 Lebesgue differentiation theorem ................. 52 1.13.1 Bounded variation ..................... 53 1.13.2 The Dini derivatives ..................... 54 1.13.3 Two easy lemmas ...................... 54 1.13.4 Proof of the Lebesgue differentiation theorem ....... 56 1.14 Where are we? ........................... 59 2 Covering Theorems 61 2.1 Covering Relations ......................... 62 2.1.1 Partitions and subpartitions ................. 62 2.1.2 Covering relations ..................... 62 2.1.3 Prunings .......................... 62 2.1.4 Full covers ......................... 63 2.1.5 Fine covers ......................... 63 2.1.6 Uniformly full covers .................... 64 2.1.7 Cousin covering lemma ................... 67 2.1.8 Decomposition of full covers ................ 67 2.1.9 Riemann sums ....................... 68 2.2 Sets of Lebesgue measure zero .................. 71 2.2.1 Lebesgue measure of open sets .............. 71 2.2.2 Sets of Lebesgue measure zero .............. 73 2.2.3 Sequences of Lebesgue measure zero sets ........ 73 2.2.4 Almost everywhere language ................ 76 2.3 Full null sets ............................ 77 2.4 Fine null sets ............................ 79 2.5 The Mini-Vitali Covering Theorem ................. 80 2.5.1 Covering lemmas for families of compact intervals ..... 81 2.5.2 Proof of the Mini-Vitali covering theorem .......... 82 2.6 Functions having zero variation .................. 84 2.6.1 Zero variation and zero derivatives ............. 86 2.6.2 Generalization of the zero derivative/variation ....... 87 2.6.3 Zero variation and mapping properties ........... 88 2.7 Absolutely continuous functions .................. 89 2.7.1 Absolute continuity in the sense of Vitali .......... 90 CONTENTS v 2.7.2 Proof of Lemma 2.29 .................... 91 2.7.3 Absolute continuity in the variational sense ........ 92 2.7.4 Absolute continuity and derivatives ............. 93 2.8 An application to the Henstock-Kurzweil integral .......... 95 2.8.1 Proof of Theorem 2.34 ................... 95 2.9 Lebesgue differentiation theorem ................. 96 2.9.1 Upper and lower derivates ................. 96 2.9.2 Geometrical lemmas .................... 97 2.9.3 Proof of the Lebesgue differentiation theorem ....... 98 2.9.4 Fubini differentiation theorem ................ 100 2.10 An application to the Riemann integral ............... 101 2.10.1 Riesz’s problem ....................... 105 2.10.2 Other variants ........................ 105 3 The Integral 107 3.1 Upper and lower integrals ..................... 108 3.1.1 The integral and integrable functions ............ 109 3.1.2 HK criterion ......................... 110 3.1.3 Cauchy criterion ...................... 111 3.1.4 McShane’s criterion ..................... 112 3.2 Elementary properties of the integral ................ 113 3.2.1 Integration and order .................... 113 3.2.2 Integration of linear combinations
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