Principles of Real Analysis: Measure, Integration, Functional Analysis, and Applications

541 Pages·2018·5.56 MB·English

by  Hugo D Junghenn

Principles of Analysis

Principles of Analysis
Measure, Integration, Functional Analysis,
and Applications
Hugo D. Junghenn
CRC Press
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Library of Congress Cataloging-in-Publication Data
Names: Junghenn, Hugo D. (Hugo Dietrich), 1939- author.
Title: Principles of real analysis : measure, integration, functional
analysis, and applications / Hugo D. Junghenn.
Description: Boca Raton : CRC Press, Taylor & Francis Group, 2018. | Includes
bibliographical references and index.
Identifiers: LCCN 2017061660 | ISBN 9781498773287
Subjects: LCSH: Functions of real variables--Textbooks. | Mathematical
analysis--Textbooks.
Classification: LCC QA331.5 .J86 2018 | DDC 515/.8--dc23
LC record available at https://lccn.loc.gov/2017061660
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TO MY CHILDREN
Katie and Patrick
AND TO MY WIFE
Mary
AS ALWAYS

Contents
Preface xix
0 Preliminaries 1
0.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Number Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
0.2 Algebraic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Semigroups and Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Quotient Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
0.3 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Open and Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Interior, Closure, and Boundary . . . . . . . . . . . . . . . . . . . . . . . 12
Sequential Convergence. Completeness . . . . . . . . . . . . . . . . . . . 12
Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
0.4 Normed Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Norms and Seminorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Completion of a Normed Space . . . . . . . . . . . . . . . . . . . . . . . 16
Infinite Series in Normed Spaces . . . . . . . . . . . . . . . . . . . . . . 16
Unordered Sums in Normed Spaces . . . . . . . . . . . . . . . . . . . . . 17
Bounded Linear Transformations . . . . . . . . . . . . . . . . . . . . . . 18
Banach Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
0.5 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Open and Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Neighborhood Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Neighborhood Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Relative Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
0.6 Continuity in Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . 23
Definition and General Properties . . . . . . . . . . . . . . . . . . . . . . 23
Initial Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Product Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Final Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Quotient Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
vii
viii Contents
The Space of Continuous Functions . . . . . . . . . . . . . . . . . . . . . 25
F-sigma and G-delta Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 25
0.7 Normal Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Urysohn’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Tietze Extension Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 27
0.8 Compact Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 27
Convergence in Compact Spaces . . . . . . . . . . . . . . . . . . . . . . . 28
Compactness of Cartesian Products . . . . . . . . . . . . . . . . . . . . . 29
Continuity and Compactness . . . . . . . . . . . . . . . . . . . . . . . . . 29
0.9 Totally Bounded Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . 30
0.10 Equicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
0.11 The Stone-Weierstrass Theorem . . . . . . . . . . . . . . . . . . . . . . . 32
0.12 Locally Compact Topological Spaces . . . . . . . . . . . . . . . . . . . . . 33
General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Functions with Compact Support . . . . . . . . . . . . . . . . . . . . . . 34
Functions That Vanish at Infinity . . . . . . . . . . . . . . . . . . . . . . 35
The One-Point Compactification . . . . . . . . . . . . . . . . . . . . . . . 35
0.13 Spaces of Differentiable Functions . . . . . . . . . . . . . . . . . . . . . . 36
0.14 Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
0.15 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
I Measure and Integration 41
1 Measurable Sets 43
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.2 Measurable Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Fields and Sigma Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Generated Sigma Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Borel Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Extended Borel Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Product Sigma Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Pi-Systems and Lambda-Systems . . . . . . . . . . . . . . . . . . . . . . 47
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
1.3 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Set Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Properties and Examples of Measures . . . . . . . . . . . . . . . . . . . . 51
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
1.4 Complete Measure Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Completion Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Null Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.5 Outer Measure and Measurability . . . . . . . . . . . . . . . . . . . . . . 55
Construction of an Outer Measure . . . . . . . . . . . . . . . . . . . . . 56
Carath´eodory’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
1.6 Extension of a Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
The Measure Extension Theorem . . . . . . . . . . . . . . . . . . . . . . 59
Approximation Property of the Extension . . . . . . . . . . . . . . . . . . 61
Completeness of the Extension . . . . . . . . . . . . . . . . . . . . . . . . 61
Uniqueness of the Extension . . . . . . . . . . . . . . . . . . . . . . . . . 62
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Contents ix
1.7 Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
The Volume Set Function . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Construction of the Measure . . . . . . . . . . . . . . . . . . . . . . . . . 65
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
1.8 Lebesgue-Stieltjes Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
One-Dimensional Distribution Functions . . . . . . . . . . . . . . . . . . 67
∗Higher Dimensional Distribution Functions . . . . . . . . . . . . . . . . . 69
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
*1.9 Some Special Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
An Uncountable Set with Lebesgue Measure Zero . . . . . . . . . . . . . . 71
Non-Lebesgue-Measurable Sets . . . . . . . . . . . . . . . . . . . . . . . . 71
A Lebesgue Measurable, Non-Borel Set . . . . . . . . . . . . . . . . . . . 72
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2 Measurable Functions 75
2.1 Measurable Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 75
General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.2 Measurable Numerical Functions . . . . . . . . . . . . . . . . . . . . . . . 78
Criteria for Measurability . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Almost Everywhere Properties . . . . . . . . . . . . . . . . . . . . . . . . 79
Combinatorial and Limit Properties of Measurable Functions . . . . . . . 79
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.3 Simple Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
A Fundamental Convergence Theorem . . . . . . . . . . . . . . . . . . . . 82
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.4 Convergence of Measurable Functions . . . . . . . . . . . . . . . . . . . . 85
Modes of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Relationships Among the Modes of Convergence . . . . . . . . . . . . . . 86
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3 Integration 89
3.1 Construction of the Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Integral of a Nonnegative Simple Function . . . . . . . . . . . . . . . . . 89
Integral of a Real-Valued Function . . . . . . . . . . . . . . . . . . . . . . 90
Integral of a Complex-Valued Function . . . . . . . . . . . . . . . . . . . 91
Integral over a Measurable Set . . . . . . . . . . . . . . . . . . . . . . . . 91
3.2 Basic Properties of the Integral . . . . . . . . . . . . . . . . . . . . . . . . 92
Almost Everywhere Properties . . . . . . . . . . . . . . . . . . . . . . . . 92
Monotone Convergence Theorem . . . . . . . . . . . . . . . . . . . . . . 93
Linearity of the Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Integration Against an Image Measure . . . . . . . . . . . . . . . . . . . 96
Integration Against a Measure with Density . . . . . . . . . . . . . . . . . 96
Change of Variables Theorem . . . . . . . . . . . . . . . . . . . . . . . . 97
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.3 Connections with the Riemann Integral on Rd . . . . . . . . . . . . . . . 100
The Darboux Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
The Riemann Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Measure Zero Criterion for Riemann Integrability . . . . . . . . . . . . . 104