A (terse) introduction to Lebesgue integration

219 Pages·2009·1.914 MB·English

by  John Franks

A (Terse)
Introduction
to Lebesgue
Integration

STUDENT MATHEMATICAL LIBRARY
Volume 48
A (Terse)
Introduction
to Lebesgue
Integration
John Franks
Providence, Rhode Island
Editorial Board
Gerald B. Folland Brad G. Osgood (Chair)
Robin Forman Michael Starbird
2000 Mathematics Subject Classification. Primary28A20, 28A25, 42B05.
Theimagesonthecoverarerepresentationsoftheergodictransformationsin
Chapter 7. The figure with the implied cardioid traces iterates of the squaring
map on the unit circle. The “spirograph” figures trace iterates of an irrational
rotation. The arc of + signs consists of iterates of an irrational rotation. I am
gratefultoEdwardDunneforprovidingthefigures.
For additional informationand updates on this book, visit
www.ams.org/bookpages/stml-48
Library of Congress Cataloging-in-Publication Data
Franks,JohnM.,1943–
A(terse)introductiontoLebesgueintegration/JohnFranks.
p.cm. –(Studentmathematicallibrary;v. 48)
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-4862-3(alk.paper)
1. Lebesgue integral. I. Title. II. Title: Introduction to Lebesgue integra-
tion.
QA312.F698 2009
515(cid:1).43–dc22 2009005870
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(cid:1)c 2009byJohnFranks.Allrightsreserved.
PrintedintheUnitedStatesofAmerica.
(cid:1)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
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VisittheAMShomepageathttp://www.ams.org/
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To my family: Judy, Josh, Mark and Alex

Contents
Preface xi
Chapter 1. The Regulated and Riemann Integrals 1
§1.1. Introduction 1
§1.2. Basic Properties of an Integral 2
§1.3. Step Functions 4
§1.4. Uniform and Pointwise Convergence 7
§1.5. Regulated Integral 8
§1.6. The Fundamental Theorem of Calculus 13
§1.7. The Riemann Integral 16
Chapter 2. Lebesgue Measure 25
§2.1. Introduction 25
§2.2. Null Sets 27
§2.3. Sigma Algebras 29
§2.4. Lebesgue Measure 31
§2.5. The Lebesgue Density Theorem 35
§2.6. Lebesgue Measurable Sets – Summary 37
Chapter 3. The Lebesgue Integral 41
§3.1. Measurable Functions 41
vii
viii Contents
§3.2. The Lebesgue Integral of Bounded Functions 48
§3.3. The Bounded Convergence Theorem 56
Chapter 4. The Integral of Unbounded Functions 63
§4.1. Non-negative Functions 63
§4.2. Convergence Theorems 67
§4.3. Other Measures 72
§4.4. General Measurable Functions 77
Chapter 5. The Hilbert Space L2 83
§5.1. Square Integrable Functions 83
§5.2. Convergence in L2 89
§5.3. Hilbert Space 95
§5.4. Fourier Series 99
§5.5. Complex Hilbert Space 104
Chapter 6. Classical Fourier Series 111
§6.1. Real Fourier Series 111
§6.2. Integrable Complex-Valued Functions 119
§6.3. The Complex Hilbert Space L2[−π,π] 122
C
§6.4. The Hilbert Space L2[T] 125
C
Chapter 7. Two Ergodic Transformations 129
§7.1. Measure Preserving Transformations 130
§7.2. Ergodicity 134
§7.3. The Birkhoff Ergodic Theorem 137
Appendix A. Background and Foundations 141
§A.1. The Completeness of R 141
§A.2. Functions and Sequences 143
§A.3. Limits 145
§A.4. Complex Limits 148
§A.5. Set Theory and Countability 151
§A.6. Open and Closed Sets 156