Elementary introduction to the Lebesgue integral

197 Pages·2018·0.92 MB·English

by  Krantz, Steven George

Elementary
Introduction to the
Lebesgue Integral
Textbooks in Mathematics
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ELEMENTARY INTRODUCTION TO THE LEBESGUE INTEGRAL
Steven G. Krantz
Elementary
Introduction to the
Lebesgue Integral
Steven G. Krantz
CRC Press
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Library of Congress Cataloging-in-Publication Data
Names: Krantz, Steven G. (Steven George), 1951- author.
Title: An introduction to the Lebesgue integral / Steven G. Krantz.
Description: Boca Raton : CRC Press, Taylor & Francis Group, 2018. | Includes
bibliographical references and index.
Identifiers: LCCN 2017061661 | ISBN 9781138482760
Subjects: LCSH: Lebesgue integral--Textbooks. | Integrals,
Generalized--Textbooks. | Measure theory--Textbooks.
Classification: LCC QA312 .K7325 2018 | DDC 515/.43--dc23
LC record available at https://lccn.loc.gov/2017061661
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To the memory of Phil Curtis, a great mentor and friend.

Contents
Preface xi
1 Introductory Thoughts 1
1.1 Review of the Riemann Integral . . . . . . . . . . . . . . . . 1
1.2 The Idea of the Lebesgue Integral . . . . . . . . . . . . . . 4
1.3 Measurable Sets . . . . . . . . . . . . . . . . . . . . . . . . 8
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 The Purpose of Measures 19
2.1 What Is a Measure? . . . . . . . . . . . . . . . . . . . . . . 19
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 The Lebesgue Integral 25
3.1 The Integration Theory of Lebesgue . . . . . . . . . . . . . 25
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4 Integrable Functions 35
4.1 Functions with Finite Integral . . . . . . . . . . . . . . . . . 35
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5 The Lebesgue Spaces 43
5.1 Definition of the Spaces . . . . . . . . . . . . . . . . . . . . 43
5.2 The Case p= . . . . . . . . . . . . . . . . . . . . . . . . 49
∞
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6 The Concept of Outer Measure 53
6.1 Outer Lebesgue Measure . . . . . . . . . . . . . . . . . . . . 53
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7 What Is a Measurable Set? 59
7.1 Identifying Measurable Sets . . . . . . . . . . . . . . . . . . 59
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
8 Decomposition Theorems 69
8.1 Signed Measures And the Hahn Decomposition . . . . . . . 69
8.2 The Radon–Nikody´m Theorem . . . . . . . . . . . . . . . . 73
vii
viii Contents
8.3 The Riesz Representation Theorem . . . . . . . . . . . . . . 76
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
9 Creation of Measures 83
9.1 Outer Measure . . . . . . . . . . . . . . . . . . . . . . . . . 83
9.2 Construction of Lebesgue Measure . . . . . . . . . . . . . . 91
9.3 Borel-Stieltjes Measure . . . . . . . . . . . . . . . . . . . . . 92
9.4 Linear Functionals on C(X) . . . . . . . . . . . . . . . . . . 92
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
10 Instances of Measurable Sets 97
10.1 Particular Sets . . . . . . . . . . . . . . . . . . . . . . . . . 97
10.2 Lebesgue Null Sets . . . . . . . . . . . . . . . . . . . . . . . 98
10.3 Invariance under Translation . . . . . . . . . . . . . . . . . 99
10.4 A Lebesgue Measurable Set That Is Not Borel . . . . . . . 100
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
11 Approximation by Open And Closed Sets 103
11.1 Interior Approximation by Closed Sets . . . . . . . . . . . . 105
11.2 Approximation from Inside by Compact Sets . . . . . . . . 106
11.3 Approximation by Intervals . . . . . . . . . . . . . . . . . . 107
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
12 Different Methods of Convergence 109
12.1 Review of Convergence Techniques . . . . . . . . . . . . . . 109
12.2 Convergence in Measure . . . . . . . . . . . . . . . . . . . . 111
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
13 Measure on a Product Space 117
13.1 Product Measures . . . . . . . . . . . . . . . . . . . . . . . 117
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
14 Additivity for Outer Measure 125
14.1 A New Look at Carath´eodory . . . . . . . . . . . . . . . . . 126
14.2 A Few Words about Inner Measure . . . . . . . . . . . . . . 127
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
15 Nonmeasurable Sets And Non-Borel Sets 129
15.1 Nonmeasurable Sets . . . . . . . . . . . . . . . . . . . . . . 129
15.2 Existence of a Measurable Set That Is Not Borel . . . . . . 131
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
16 Applications 135
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Table of Notation 141
Contents ix
Glossary 145
Solutions to Selected Exercises 159
References 179
Index 181