Basic Analysis IV: Measure Theory and Integration [1 ed.] 1138055115, 9781138055117

Table of contents :
Cover
Half Title
Title Page
Copyright Page
Dedication
Abstract
Acknowledgments
Table of Contents
I Introductory Matter
1 Introduction
1.1 The Analysis Courses
1.1.1 Senior Level Analysis
1.1.2 The Graduate Analysis Courses
1.1.3 More Advanced Courses
1.2 Teaching the Measure and Integration Course
1.3 Table of Contents
1.4 Acknowledgments
II Classical Riemann Integration
2 An Overview of Riemann Integration
2.1 Integration
2.1.1 The Riemann Integral as a Limit
2.1.2 The Fundamental Theorem of Calculus
2.1.3 The Cauchy Fundamental Theorem of Calculus
2.2 Handling Jumps
2.2.1 Removable Discontinuity
2.2.2 Jump Discontinuity
3 Bounded Variation
3.1 Partitions
3.2 Monotone
3.2.1 The Saltus Function
3.2.2 The Continuous Part of a Monotone Function
3.3 Bounded Variation
3.4 The Total Variation Function
3.5 Continuous Functions of Bounded Variation
4 Riemann Integration
4.1 Definition
4.2 Existence
4.3 Properties
4.4 Riemann Integrable?
4.5 More Properties
4.6 Fundamental Theorem
4.7 Substitution
4.8 Same Integral?
5 Further Riemann Results
5.1 Limit Interchange
5.2 Riemann Integrable?
5.3 Content Zero
III Riemann - Stieltjes Integration
6 The Riemann - Stieltjes Integral
6.1 Properties
6.2 Step Integrators
6.3 Monotone Integrators
6.4 Equivalence Theorem
6.5 Further Properties
6.6 Bounded Variation Integrators
7 Further Riemann - Stieltjes Results
7.1 Fundamental Theorem
7.2 Existence
7.3 Computations
IV Abstract Measure Theory One
8 Measurability
8.1 Borel Sigma-Algebra
8.2 The Extended Borel Sigma-Algebra
8.3 Measurable Functions
8.3.1 Examples
8.4 Properties
8.5 Extended Real-Valued
8.6 Extended Properties
8.7 Continuous Compositions
8.7.1 The Composition with Finite Measurable Functions
8.7.2 The Approximation of Non-Negative Measurable Functions
8.7.3 Continuous Functions of Extended Real-Valued Measurable Functions
9 Abstract Integration
9.1 Measures
9.2 Properties
9.3 Sequences of Sets
9.4 Integration
9.5 Integration Properties
9.6 Equality a.e. Problems
9.7 Complete Measures
9.8 Convergence Theorems
9.8.1 Monotone Convergence Theorems
9.8.2 Fatou's Lemma
9.9 The Absolute Continuity of a Measure
9.10 Summable Functions
9.11 Extended Integrands
9.12 Levi's Theorem
9.13 Constructing Charges
9.14 Properties of Summable Functions
9.15 The Dominated Convergence Theorem
9.16 Alternative Abstract Integration Schemes
9.16.1 Properties of the Darboux Integral
10 The Lp Spaces
10.1 The General Lp Spaces
10.2 The World of Counting Measure
10.3 Essentially Bounded Functions
10.4 The Hilbert Space L2
V Constructing Measures
11 Building Measures
11.1 Measures from Outer Measure
11.2 The Properties of the Outer Measure
11.3 Measures Induced by Outer Measures
11.4 Measures from Metric Outer Measures
11.5 Constructing Outer Measure
12 Lebesgue Measure
12.1 Outer Measure
12.2 Lebesgue Outer Measure is a Metric Outer Measure
12.3 Lebesgue Measure is Regular
12.4 Approximation Results
12.4.1 Approximating Measurable Sets
12.4.2 Approximating Measurable Functions
12.5 The Summable Functions are Separable
12.6 The Existence of Non-Lebesgue Measurable Sets
12.7 Metric Spaces
13 Cantor Sets
13.1 Generalized
13.2 Representation
13.3 The Cantor Functions
13.4 Consequences
14 Lebesgue - Stieltjes Measure
14.1 Lebesgue - Stieltjes Outer Measure and Measure
14.2 Approximation Results
14.3 Properties
VI Abstract Measure Theory Two
15 Convergence Modes
15.1 Extracting Subsequences
15.2 Egoroff's Theorem
15.3 Vitali's Theorem
15.4 Summary
16 Decomposing Measures
16.1 Jordan Decomposition
16.2 Hahn Decomposition
16.3 Variation
16.4 Absolute Continuity
16.5 Radon-Nikodym
16.6 Lebesgue Decomposition
17 Connections to Riemann Integration
18 Fubini Type Results
18.1 The Riemann Setting
18.1.1 Fubini on a Rectangle
18.2 The Lebesgue Setting
19 Differentiation
19.1 Absolutely Continuous Functions
19.2 LS and AC
19.3 Bounded Variation Derivatives
19.4 Measure Estimates
19.5 Extending the Fundamental Theorem of Calculus
19.6 Charges Induced by Absolutely Continuous Functions
VII Summing It All Up
20 Summing It All Up
VIII References
IX Detailed Index
X Appendix: Undergraduate Analysis Background Check
A Undergraduate Analysis Part One
A-1 Sample Exams
A-1.1 Exam 1
A-1.2 Exam 2
A-1.3 Exam 3
A-1.4 Final
B Undergraduate Analysis Part Two
B-1 Sample Exams
B-1.1 Exam 1
B-1.2 Exam 2
B-1.3 Exam 3
B-1.4 Final
XI Appendix: Linear Analysis Background Check
C Linear Analysis
C-1 Sample Exams
C-1.1 Exam 1
C-1.2 Exam 2
C-1.3 Exam 3
C-1.4 Final
XII Appendix: Preliminary Examination Check
D The Preliminary Examination in Analysis
D-1 Sample Exams
D-1.1 Exam 1
D-1.2 Exam 2
D-1.3 Exam 3
D-1.4 Exam 4
D-1.5 Exam 5
D-1.6 Exam 6
D-1.7 Exam 7
D-1.8 Exam 8

Citation preview

Basic Analysis IV

Basic Analysis IV: Measure Theory and Integration

The giant squids are interested in advanced mathematical training and Jim is happy to help!

James K. Peterson Department of Mathematical Sciences Clemson University

First edition published 2021 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2020 Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot as­ sume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including pho­ tocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for iden­ tification and explanation without intent to infringe. ISBN: 978-1-138-05511-7 (hbk) ISBN: 978-1-315-16618-6 (ebk) LCCN: 2019059882

Dedication We dedicate this work to all of our students who have been learning these ideas of analysis through our courses. We have learned as much from them as we hope they have from us. We are a firm believer that all our students are capable of excellence and that the only path to excellence is through discipline and study. We have always been proud of our students for doing so well on this journey. We hope these notes in turn make you proud of our efforts. Abstract This book introduces graduate students in mathematics concepts from measure theory and also, continues their training in the abstract way of looking at the world. We feel that is a most important skill to have when your life’s work will involve quantitative modeling to gain insight into the real world. Acknowledgments I want to acknowledge the great debt I have to my wife, Pauli, for her pa­ tience in dealing with the long hours spent in typing and thinking. You are the love of my life.

The cover for this book is an original painting by me done in July 2017. It shows the moment when the giant squids reached out to me to learn advanced mathematics.

Table of Contents I

Introductory Matter

1

Introduction 1.1 The Analysis Courses . . . . . . . . . . . . . 1.1.1 Senior Level Analysis . . . . . . . . 1.1.2 The Graduate Analysis Courses . . . 1.1.3 More Advanced Courses . . . . . . . 1.2 Teaching the Measure and Integration Course 1.3 Table of Contents . . . . . . . . . . . . . . . 1.4 Acknowledgments . . . . . . . . . . . . . . .

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Classical Riemann Integration

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An Overview of Riemann Integration 2.1 Integration . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 The Riemann Integral as a Limit . . . . . . . . 2.1.2 The Fundamental Theorem of Calculus . . . . 2.1.3 The Cauchy Fundamental Theorem of Calculus 2.2 Handling Jumps . . . . . . . . . . . . . . . . . . . . . 2.2.1 Removable Discontinuity . . . . . . . . . . . 2.2.2 Jump Discontinuity . . . . . . . . . . . . . . .

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Bounded Variation 3.1 Partitions . . . . . . . . . . . . . . . . . . . . . . . 3.2 Monotone . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The Saltus Function . . . . . . . . . . . . . 3.2.2 The Continuous Part of a Monotone Function 3.3 Bounded Variation . . . . . . . . . . . . . . . . . . 3.4 The Total Variation Function . . . . . . . . . . . . . 3.5 Continuous Functions of Bounded Variation . . . . .

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Riemann Integration 4.1 Definition . . . . . . . 4.2 Existence . . . . . . . 4.3 Properties . . . . . . . 4.4 Riemann Integrable? . 4.5 More Properties . . . . 4.6 Fundamental Theorem 4.7 Substitution . . . . . . 4.8 Same Integral? . . . .

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TABLE OF CONTENTS Further Riemann Results 93

5.1 Limit Interchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2 Riemann Integrable? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.3 Content Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

III Riemann - Stieltjes Integration 6

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The Riemann - Stieltjes Integral 6.1 Properties . . . . . . . . . . . 6.2 Step Integrators . . . . . . . . 6.3 Monotone Integrators . . . . . 6.4 Equivalence Theorem . . . . . 6.5 Further Properties . . . . . . . 6.6 Bounded Variation Integrators

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Further Riemann - Stieltjes Results 133

7.1 Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.2 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.3 Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

IV Abstract Measure Theory One 8

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Measurability 8.1 Borel Sigma-Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Extended Borel Sigma-Algebra . . . . . . . . . . . . . . . . . . . . . . . 8.3 Measurable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Extended Real-Valued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Extended Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Continuous Compositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1 The Composition with Finite Measurable Functions . . . . . . . . . . . 8.7.2 The Approximation of Non-Negative Measurable Functions . . . . . . 8.7.3 Continuous Functions of Extended Real-Valued Measurable Functions

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Abstract Integration 9.1 Measures . . . . . . . . . . . . . . . . . 9.2 Properties . . . . . . . . . . . . . . . . . 9.3 Sequences of Sets . . . . . . . . . . . . . 9.4 Integration . . . . . . . . . . . . . . . . . 9.5 Integration Properties . . . . . . . . . . . 9.6 Equality a.e. Problems . . . . . . . . . . 9.7 Complete Measures . . . . . . . . . . . . 9.8 Convergence Theorems . . . . . . . . . . 9.8.1 Monotone Convergence Theorems 9.8.2 Fatou’s Lemma . . . . . . . . . . 9.9 The Absolute Continuity of a Measure . . 9.10 Summable Functions . . . . . . . . . . . 9.11 Extended Integrands . . . . . . . . . . . . 9.12 Levi’s Theorem . . . . . . . . . . . . . .

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TABLE OF CONTENTS 9.13 9.14 9.15 9.16

Constructing Charges . . . . . . . . . . . Properties of Summable Functions . . . . The Dominated Convergence Theorem . . Alternative Abstract Integration Schemes 9.16.1 Properties of the Darboux Integral

10 The Lp Spaces 10.1 The General Lp Spaces . . . . . 10.2 The World of Counting Measure 10.3 Essentially Bounded Functions . 10.4 The Hilbert Space L2 . . . . . .

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V Constructing Measures

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11 Building Measures 11.1 Measures from Outer Measure . . . . 11.2 The Properties of the Outer Measure . 11.3 Measures Induced by Outer Measures 11.4 Measures from Metric Outer Measures 11.5 Constructing Outer Measure . . . . .

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12 Lebesgue Measure 12.1 Outer Measure . . . . . . . . . . . . . . . . . . . . 12.2 Lebesgue Outer Measure is a Metric Outer Measure 12.3 Lebesgue Measure is Regular . . . . . . . . . . . . 12.4 Approximation Results . . . . . . . . . . . . . . . 12.4.1 Approximating Measurable Sets . . . . . . 12.4.2 Approximating Measurable Functions . . . 12.5 The Summable Functions are Separable . . . . . . 12.6 The Existence of Non-Lebesgue Measurable Sets . 12.7 Metric Spaces . . . . . . . . . . . . . . . . . . . .

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13 Cantor Sets 13.1 Generalized . . . . . 13.2 Representation . . . . 13.3 The Cantor Functions 13.4 Consequences . . . .

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14 Lebesgue - Stieltjes Measure 14.1 Lebesgue - Stieltjes Outer Measure and Measure . . . . . . . . . . . . . . . . . . . 14.2 Approximation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract Measure Theory Two

15 Convergence Modes 15.1 Extracting Subsequences 15.2 Egoroff’s Theorem . . . 15.3 Vitali’s Theorem . . . . . 15.4 Summary . . . . . . . .

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TABLE OF CONTENTS

16 Decomposing Measures 16.1 Jordan Decomposition . 16.2 Hahn Decomposition . . 16.3 Variation . . . . . . . . . 16.4 Absolute Continuity . . . 16.5 Radon-Nikodym . . . . . 16.6 Lebesgue Decomposition

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17 Connections to Riemann Integration

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18 Fubini Type Results 379

18.1 The Riemann Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

18.1.1 Fubini on a Rectangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

18.2 The Lebesgue Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

19 Differentiation 19.1 Absolutely Continuous Functions . . . . . . . . . . . 19.2 LS and AC . . . . . . . . . . . . . . . . . . . . . . . 19.3 Bounded Variation Derivatives . . . . . . . . . . . . 19.4 Measure Estimates . . . . . . . . . . . . . . . . . . 19.5 Extending the Fundamental Theorem of Calculus . . 19.6 Charges Induced by Absolutely Continuous Functions

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Summing It All Up

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20 Summing It All Up

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VIII References

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IX Detailed Index

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X

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Appendix: Undergraduate Analysis Background Check

A Undergraduate Analysis Part One A-1 Sample Exams . . . . . . . . A-1.1 Exam 1 . . . . . . . A-1.2 Exam 2 . . . . . . . A-1.3 Exam 3 . . . . . . . A-1.4 Final . . . . . . . .

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B Undergraduate Analysis Part Two B-1 Sample Exams . . . . . . . . B-1.1 Exam 1 . . . . . . . B-1.2 Exam 2 . . . . . . . B-1.3 Exam 3 . . . . . . . B-1.4 Final . . . . . . . .

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TABLE OF CONTENTS

XI

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Appendix: Linear Analysis Background Check

C Linear Analysis C-1 Sample Exams . C-1.1 Exam 1 C-1.2 Exam 2 C-1.3 Exam 3 C-1.4 Final .

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XII Appendix: Preliminary Examination Check D The Preliminary Examination in Analysis D-1 Sample Exams . . . . . . . . . . . . D-1.1 Exam 1 . . . . . . . . . . . D-1.2 Exam 2 . . . . . . . . . . . D-1.3 Exam 3 . . . . . . . . . . . D-1.4 Exam 4 . . . . . . . . . . . D-1.5 Exam 5 . . . . . . . . . . . D-1.6 Exam 6 . . . . . . . . . . . D-1.7 Exam 7 . . . . . . . . . . . D-1.8 Exam 8 . . . . . . . . . . .

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Part I

Introductory Matter

1

Chapter 1

Introduction We believe that all students who are seriously interested in mathematics at the master’s and doctoral level should have a passion for analysis even if it is not the primary focus of their own research interests. So you should all understand that my own passion for the subject will shine though in the notes that follow! And, it goes without saying that we assume that you are all mature mathematically and eager and interested in the material! Now, the present text focuses on the topics of Measure and Integration from a very abstract point of view, but it is very helpful to place this course into its proper context. Also, for those of you who are preparing to take the qualifying examination in analysis, the overview below will help you see why all this material fits together into a very interesting web of ideas.

1.1

The Analysis Courses

In outline form, the classical material on analysis would cover the material using textbooks equivalent to the ones listed below: (A): Undergraduate Analysis, text Advanced Calculus: An Introduction to Analysis, by Watson Fulks. Our take on this material can be seen in (Peterson (8) 2020). Additional material, not covered in courses now but very useful is discussed in (Peterson (10) 2020). (B): Introduction to Abstract Spaces, text Introduction to Functional Analysis and Applications, by Ervin Kreyszig. Our take on this material is seen in (Peterson (9) 2020). (C): Measure Theory and Abstract Integration, texts General Theory of Functions and Integra­ tion, by Angus Taylor and Real Analysis, by Royden are classics but actually very hard to read and absorb. Our take on this material is the text you are currently reading. (D:) Topology and Functional Analysis. A classical text is Topology and Analysis by Simmons. Our take on this material, and other topics, is seen in (Peterson (6) 2020). In addition, a nice book that organizes the many interesting examples and counterexamples in this area is good to have on your shelf. We recommend the text Counterexamples in Analysis by Gel­ baum and Olmstead. There are thus essentially five courses required to teach you enough of the concepts of mathematical analysis to enable you to read technical literature (such as engineering, control, physics, mathematics, statistics and so forth) at the beginning research level. Here are some more details about these courses. 3

4

BASIC ANALYSIS IV: MEASURE THEORY AND INTEGRATION

1.1.1 Senior Level Analysis Typically, this is a full two semester sequence that discusses thoroughly what we would call the analysis of functions of a real variable. This two semester sequence covers the following: Advanced Calculus I: This course studies sequences and functions whose domain is simply the real line. There are, of course, many complicated ideas, but everything we do here involves things that act on real numbers to produce real numbers. If we call these things that act on other things, OPERATORS, we see that this course is really about real-valued operators on real numbers. This course invests a lot of time in learning how to be precise with the notion of convergence of sequences of objects, that happen to be real numbers, to other numbers. 1. Basic Logic, Inequalities for Real Numbers, Functions 2. Sequences of Real Numbers, Convergence of Sequences 3. Subsequences and the Bolzano - Weierstrass Theorem 4. Cauchy sequences 5. Continuity of Functions 6. Consequences of Continuity 7. Uniform Continuity 8. Differentiability of Functions 9. Consequences of Differentiability 10. Taylor Series Approximations Advanced Calculus II: In this course, we rapidly become more abstract. First, we develop carefully the concept of the Riemann Integral. We show that although differentiation is intellectually quite a different type of limit process, it is intimately connected with the Riemann integral. Also, for the first time, we begin to explore the idea that we could have sequences of objects other than real numbers. We study carefully their convergence properties. We learn about two fundamental concepts: pointwise and uniform convergence of sequences of objects called functions. We are beginning to see the need to think about sets of objects, such as functions, and how to define the notions of convergence and so forth in this setting. 1. The Riemann Integral 2. Sequences of Functions 3. Uniform Convergence of Sequence of Functions 4. Series of Functions Our version of this course, (Peterson (8) 2020) adds two variable calculus, convex analysis in � and a nice discussion of Fourier Series. It turns out many of our mathematics majors do not take an advanced engineering mathematics class, so they are not necessarily exposed to Fourier Series tools. And there is a lot of nice mathematics in that!

1.1.2 The Graduate Analysis Courses There are three basic courses at most universities. First, linear analysis, then measure and integration and finally, functional analysis. Linear analysis is a core analysis course and all master’s students usually take it. Also, linear analysis and measure theory and integration form the two courses which we test prospective Ph.D. students on as part of the analysis preliminary examination. The content of these courses, also must fit within a web of other responsibilities. Many students are typically weak

INTRODUCTION

5

in abstraction coming in, so if we teach the material too fast, we lose them. Now if 20 students take linear analysis, usually 15 or 75% are already committed to an master’s program which emphasizes Operations Research, Statistics, Algebra/ Combinatorics or Computation in addition to applied Anal­ ysis. Hence, currently, there are only about 5 students in linear analysis who might be interested in an master’s specialization in analysis. The other students typically either don’t like analysis at all and are only there because they have to be or they like analysis but it is part of their studies in number theory, partial differential equations for the computation area, and so forth. Either way, the students will not continue to study analysis for a degree specialization. However, we think it is important for all students to both know and appreciate this material. Traditionally, there are several ways to go. The Cynical Approach: Nothing you can do will make students who don’t like analysis change their mind. So teach the material hard and fast and target the 2 - 3 students who can benefit. The rest will come along for the ride and leave the course convinced that analysis is just like they thought – too hard and too complicated. If you do this approach, you can pick about any book you like. Most books for our students are too abstract and so are very hard for them to read. But the 2 -3 students who can benefit from material at this level, will be happy with the book. We admit this is not our style although some think it is a good way to find the really bright analysis students. We prefer the alternate Enthusiastic “maybe I can get them interested anyway” Approach: The instructor scours the available literature in order to make up notes to lead the students “gently” into the required abstract way of thinking. We haven’t had much luck finding a published book for this, so since this is our preferred plan of action: we have worked on notes such as the ones you have in your hand for a long time. These notes start out handwritten and slowly mature into the typed versions. Many students have suffered with the earlier versions and we thank them for that! We believe it is important to actively try to get all the students interested but, of course, this is never completely successful. However, we still think there is great value in this approach and it is the one we have been trying for many years. Introductory Linear Analysis: Our constraints here are to choose content so the students are ade­ quately exposed to a more abstract way of thinking about the world. We generally cover • Metric Spaces. • Vector Spaces with a Norm. • Vector Spaces with an Inner Product. • Linear Operators. • Basic Linear Functional Analysis such as Hahn - Banach Theorems and so forth. • The Baire Category, the Open Mapping and Closed Graph Theorems. It doesn’t sound like much but there is a lot of material in here the students haven’t seen. For example, we typically focus a lot on how we are really talking about sets of objects with some additional structure. A set plus a way to measure distance between objects gives a metric space; if we can add and scale objects, we get a vector space; if we have a vector space and add the structure that allows us to project a vector to a subspace, we get an inner product space. We also mention we could have a set of objects and define one operation to obtain a group or if we define a special collection of sets we call open, we get a topological space and so forth. If we work hard, we can help open their minds to the fact that each of the many sub-disciplines in the Mathematical Sciences focuses on special structure we add to a set to help us solve problems in that arena. There are lots of ways to cover the important material in these topic areas and even many ways to decide on exactly what is important from metric, normed and inner product spaces. So there is that kind of freedom, but not so much freedom that you can decide to drop say, inner product

6

BASIC ANALYSIS IV: MEASURE THEORY AND INTEGRATION ¨ - Liouville systems as an example when the discus­ spaces. For example, we could use Sturm sion turns to eigenvalues of operators. It is nice to use projection theorems in an inner product setting as a big finishing application, but remember the students are weak in background, e.g. their knowledge of ordinary differential equations and Calculus in �n is normally weak. So we are limited in our coverage of the completeness of an orthonormal sequence in an inner product space in many respects. In fact, since the students don’t know measure theory (which is in this text) the discussion of Hilbert spaces of functions is inherently weak. However, we decided to address this by constructing the reals from scratch and then showing how to build a set of functions L2 ([a, b]) which is a Hilbert space without having to resort to measure theory. Of course, this is all equivalent to what you would get with a measure theory approach, but it has the advantage of allowing the students to see how all the details work without brushing stuff under the rug, so to speak! However, we can now develop this material in another way using the ideas from measure theory which is the content of the present text. Also, if you look carefully at that material, you need to cover some elementary versions of the Hahn - Banach theorem to do it right which is why we have placed it the linear analysis course. In general, in the class situation, we run out of time to cover such advanced topics. The trade-off seems to be between thorough coverage of a small number of topics or rapid coverage of many topics superficially. However, we can do it all in a book treatment and we have designed all of the basic analysis books for self-study too. We believe this course is a very critical one in the evolution of the student’s ability to think abstractly and so teaching there is great value in teaching very, very carefully the basics of this material. This course takes a huge amount of time for lecture preparation and student interaction in the office, so when we teach this material, we slow down in our research output! In more detail, in this text, we begin to rephrase all of our knowledge about convergence of sequence of objects in a much more general setting. 1. Metric Spaces: A set of objects and a way of measuring distance between objects which satisfies certain special properties. This function is called a metric and its properties were chosen to mimic the properties that the absolute value function has on the real line. We learn to understand convergence of objects in a general metric space. It is really important to note that there is NO additional structure imposed on this set of objects; no linear structure (i.e. vector space structure), no notion of a special set of elements called a basis which we can use to represent arbitrary elements of the set. The metric in a sense generalizes the notion of distance between numbers. We can’t really measure the size of an object by itself, so we do not yet have a way of generalizing the idea of size or length. A fundamentally important concept now emerges: the notion of completeness and how it is related to our choice of metric on a set of objects. We learn a clever way of constructing an abstract representation of the completion of any metric space, but at this time, we have no practical way of seeing this representation. 2. Normed Spaces: We add linear structure to the set of objects and a way of measuring the magnitude of an object; that is, there is now an operation we think of as addition and another operation which allows us to scale objects and a special function called a norm whose value for a given object can be thought of as the object’s magnitude. We then develop what we mean by convergence in this setting. Since we have a vector space structure, we can now begin to talk about a special subset of objects called a basis which can be used to find a useful way of representing an arbitrary object in the space. Another most important concept now emerges: the cardinality of this basis may be finite or infinite. We begin to explore the consequences of a space being finite versus infinite dimensional.

INTRODUCTION

7

3. Inner Product Spaces: To a set of objects with vector space structure, we add a function called an inner product which generalizes the notion of dot product of vectors. This has the extremely important consequence of allowing the inner product of two objects to be zero even though the objects are not the same. Hence, we can develop an abstract notion of the orthogonality of two objects. This leads to the idea of a basis for the set of objects in which all the elements are mutually orthogonal. We then finally can learn how to build representations of arbitrary objects efficiently. 4. Completions: We learn how to complete an arbitrary metric, normed or inner product space in an abstract way, but we know very little about the practical representations of such completions. 5. Linear Operators: We study a little about functions whose domain is one set of objects and whose range is another. These functions are typically called operators. We learn a little about them here. 6. Linear Functionals: We begin to learn the special role that real-valued functions acting on objects play in analysis. These types of functions are called linear functionals and learning how to characterize them is the first step in learning how to use them. We just barely begin to learn about this here. We have implemented this approach in (Peterson (9) 2020). Measure Theory: This course generalizes the notion of integration to a very abstract setting. The set of notes you are reading is a textbook for this material. Roughly speaking, we first realize that the Riemann integral is a linear mapping from the space of bounded real-valued functions on a compact interval into the reals which has a number of interesting properties. We then study how we can generalize such mappings so that they can be applied to arbitrary sets X, a special collection of subsets of X called a sigma-algebra and a new type of mapping called a measure which on � generalizes our usual notion of the length of an interval. In this class, we discuss the following: 1. The Riemann Integral. 2. Measures on a sigma-algebra S in the set X and integration with respect to the measure. 3. Measures specialized to sigma-algebras on the set �n and integrations with respect to these measures. The canonical example of this is Lebesgue measure on �n . 4. Differentiation and integration in these abstract settings and their connections. This is what we cover in the notes you are reading now and some other things!

1.1.3

More Advanced Courses

It is also recommended that students consider taking a course in what is called Functional Analy­ sis. Our version is (Peterson (6) 2020) which tries hard to put topology, analysis and algebra ideas together into one pot. It is carefully designed for self-study as this course is just not being offered as much as it should be. While not part of the qualifying examination, in this course, we can finally develop in a careful way the necessary tools to work with linear operators, weak convergence and so forth. This is a huge area of mathematics, so there are many possible ways to design an introductory course. A typical such course would cover: 1. An Introduction to General Operator Theory. 2. Topological Vector Spaces and Distributions.

8

BASIC ANALYSIS IV: MEASURE THEORY AND INTEGRATION 3. An Introduction to the Spectral Theory of Linear Operators; this is the study of the eigenvalues and eigenobjects for a given linear operator; there are lots of applications here! 4. Differential Geometry ideas. 5. Degree Theory ideas. 6. Some advanced topics using these ideas: possibilities include (a) Existence Theory of Boundary Value Problems. (b) Existence Theory for Integral Equations. (c) Existence Theory in Control.

1.2

Teaching the Measure and Integration Course

So now that you have seen how the analysis courses all fit together, it is time for the main course. So roll up your sleeves and prepare to work! Let’s start with a few more details on what this course on Measure and Integration will cover. In this course, we assume mathematical maturity and we tend to follow The Enthusiastic “maybe I can get them interested anyway” Approach in lecturing (so, be warned)! It is difficult to decide where to start in this course. There is usually a reasonable fraction of you who have never seen an adequate treatment of Riemann Integration. For example, not everyone may have seen the equivalent of the second part of undergraduate analysis where Riemann integration is carefully discussed. We therefore have taught several versions of this course and we have divide the material into blocks as follows: We believe there are a lot of advantages in treating integration abstractly. So, if we covered the Lebesgue integral on � right away, we can take advantage of a lot of the special structure � has which we don’t have in general. It is better for long-term intellectual development to see measure and integration approached without using such special structure. Also, all of the standard theorems we want to do are just as easy to prove in the abstract setting, so why specialize to �? So we tend to do abstract measure stuff first. The core material for Block 1 is as follows: 1. Abstract measure ν on a sigma-algebra S of subsets of a universe X. 2. Measurable functions with respect to a measure ν; these are also called random variables when ν is a probability measure. 3. Integration

f dν.

4. Convergence results: monotone convergence theorem, dominated convergence theorem, etc. Then we develop the Lebesgue Integral in �n via outer measures as the great example of a nontrivial measure. So Block 2 of material is thus: 1. Outer measures in �n . 2. Caratheodory conditions for measurable sets. 3. Construction of the Lebesgue sigma-algebra. 4. Connections to Borel sets. To fill out the course, we pick topics from the following:

INTRODUCTION

9

1. Riemann and Riemann - Stieltjes integration. This would go before Block 1 if we do it. Call it block Riemann. 2. Decomposition of measures – I love this material so this is after Block 2. Call it block Decom­ position. 3. Connection to Riemann integration via absolute continuity of functions. This is actually hard stuff and takes about 3 weeks to cover nicely. Call it Block Riemann and Lebesgue. If this is done without Block Riemann, you have to do a quick review of Riemann stuff so they can follow the proofs. 4. Fubini type theorems. This would go after Block 2. Call this Block Fubini. 5. Differentiation via the Vitali approach. This is pretty hard too. Call this Differentiation. 6. Treatment of the usual Lp spaces. Call this Block Lp . 7. More convergence stuff like convergence in measure, Lp convergence implies convergence of a subsequence pointwise, etc. These are hard theorems and to do them right requires a lot of time. Call this More Convergence. We have taught this in at least the following ways. And always, lots of homework and projects, as we believe only hands-on work really makes this stuff sink in. Way 1: Block Riemann, Block 1, Block 2 and Block Decomposition. Way 2: Block 1, Block 2, Block Decomposition and Block Riemann and Lebesgue. Way 3: Block 1, Block 2, Block Decomposition and Differentiation. Way 4: Block 1, Block 2, Block Lp , Block More Convergence and Block Decomposition. Way 5: Block 1, Block 2, Block Fubini, Block More Convergence and Block Decomposition. In book form, we can cover all of these topics in some detail; not as much as we like, but enough to get you started on your lifelong journey into these areas. We use Octave (Eaton et al. (3) 2020), which is an open source GPL licensed (Free Software Foun­ R , as a computational engine and we are not afraid to use it as dation (4) 2020) clone of MATLAB� R (MATLAB (5) 2018 ­ an adjunct to all the theory we go over. Of course, you can use MATLAB� 2020) also if your university or workplace has a site license or if you have a personal license. Get used to it: theory, computation and science go hand in hand! Well, buckle up and let’s get started!

1.3

Table of Contents

This text is based on quite a few years of teaching graduate courses in analysis and began from hand­ written notes starting roughly in the late 1990’s but with material being added all the time. Along the way, these notes have been helped by the students we have taught and also by our own research interests as research informs teaching and teaching informs research in return. In this text, we go over the following blocks of material. Part One: These are our beginning remarks you are reading now which are in Chapter 1. Part Two: Classical Riemann Integration Here we are concerned with classic Riemann integra­ tion and the new class of functions called functions of bounded variation.

10

BASIC ANALYSIS IV: MEASURE THEORY AND INTEGRATION • In Chapter 2, we provide a quick overview of Riemann Integration just to set the stage. • Chapter 3 discusses a new topic: functions of bounded variation. • In Chapter 4, we carefully go over the basics of the theory of Riemann integration. We want you to see how these proofs are done, so that is easy to see how they are modified for the case of Riemann - Stieltjes integration. • Chapter 5 goes over further important details of classical Riemann integration theory.

Part Three: Riemann - Stieltjes Integration We extend Riemann integration to Riemann - Stielt­ jes integration. • Chapter 6 provides the basics of Riemann - Stieltjes integration theory. • In Chapter 7, we add further details about Riemann - Stieltjes integrals. Part Four: Abstract Measure Theory One We define what a measure is and then develop a corre­ sponding theory of integration based on this measure. • Chapter 8 defines measures and their associated spaces. We do not know how to construct measures yet, but we can work out many of the consequences of a measure if we assume it is there to use. • Chapter 9 develops the theory of integration based on a measure. • In Chapter 10, we generalize the discussion of the gp sequence spaces to the more general setting of equivalence classes of functions whose pth power is integrable in this more general sense. Part Five: Constructing Measures We learn how to construct measures in various ways. • Chapter 11 discusses how to build or construct measures in very general ways. • In Chapter 12, we carefully construct what is called Lebesgue measure. • Chapter 13 shows you how to build specialized subsets of the real line called Cantor sets which are the source of many examples and ideas. This chapter is a series of exercises which as you finish them, you see how these sets are constructed via limiting processes and what their properties are. • In Chapter 14, we extend the construction of Lebesgue measures to Lebesgue - Stieltjes measures. Part Six: Abstract Measure Theory Two We want to finish with general comments on a variety of topics: the different kinds of convergence of sequences of functions we now have, the way measures can be decomposed into fundamental modes, how the Lebesgue integral connects to the Riemann Integral, generalized Fubini results and finally some thoughts on differentiation in this context. This last part allows us to state and prove an extension of the Fundamental Theorem of Calculus. • In Chapter 15, we discuss carefully the many types of convergence of sequences of func­ tions we have in this context. • Chapter 16 explains how we can write measures as sums of simpler parts giving us a variety of decomposition theorems. • In Chapter 17, we develop connections between Lebesgue and Riemann Integration. • Chapter 18 looks at the idea of multidimensional abstract integration in terms of iterated integrals: i.e. extensions of the classical Fubini Theorems. • Chapter 19 develops the idea of absolutely continuous functions and measures which we use to extend the Fundamental Theorem of Calculus.

INTRODUCTION

11

Part Seven: Summing It All Up In Chapter 20, we talk about the things you have learned here and where you should go next to continue learning more analysis.

There is much more we could have done, but these topics are a nice introduction into the further use of abstraction in your way of thinking about models and should prepare you well for the next step, which is functional analysis and topology.

1.4

Acknowledgments

I have been steadily shrinking since 1996 and currently I can only be seen when standing on a large box. This has forced me to alter my lecture style somewhat so that my students can see me and I am considering the use of platform shoes. However, if I can ever remember the code to my orbiting spaceship, I will be able to access the shape changing facilities on board and solve the problem per­ manently. However, at the moment, I do what I can with this handicap. There were many problems with the handwritten version of these notes, but with the help of my many students in this class over the years, they have gotten better. However, being written in my handwrit­ ing was not a good thing. In fact, one student some years ago was so unhappy with the handwritten notes, that he sent me the names of four engineering professors who actually typed their class notes nicely. He told me I could learn from them how to be a better teacher. Well, it has taken me awhile, but at least the notes are now typed. The teaching part, though, is another matter. I will leave you with an easy proposition to prove; consider it your first mathematical test! Proposition 1.4.1 My Proposition My proposition is this: chocolate makes one happier! Proof 1.4.1 Eat a piece of chocolate and the proof is there! And, of course, there is a corollary involving donuts which I will leave to you!

Jim Peterson School of Mathematical and Statistical Sciences Clemson University

�

Part II

Classical Riemann Integration

13

Chapter 2

An Overview of Riemann Integration

In this chapter, we will give you a quick overview of Riemann integration. There are few real proofs but it is useful to have a quick tour before we get on with the job of extending this material to a more abstract setting.

2.1

Integration

There are two intellectually separate ideas here: 1. The idea of a Primitive or Antiderivative of a function f . This is any function F which is differentiable and satisfies F ' (t) = f (t) at all points in the domain of f . Normally, the domain of f is a finite interval of the form [a, b], although it could also be an infinite interval like all of � or [1, ∞) and so on. Note that an antiderivative does not require any understanding of the process of Riemann integration at all – only what differentiation is! 2. The idea of the Riemann integral of a function. You should have been exposed to this in your first Calculus course and perhaps a bit more rigorously in your undergraduate second semester analysis course. Let’s review what Riemann Integration involves. First, we start with a bounded function f on a finite interval [a, b]. This kind of function f need not be continuous! Then select a finite number of points from the interval [a, b], {x0 , x1 , , . . . , xn−1 , xn }. We don’t know how many points there are, so a different selection from the interval would possibly give us more or less points. But for convenience, we will just call the last point xn and the first point x0 . These points are not arbitrary – x0 is always a, xn is always b and they are ordered like this: x0 = a < x1 < x2
0 be given. Then, using the Supremum Tolerance Lemma, Lemma 3.2.2, there is a y ∗ ∈ [a, x) such that sup Tx − E < f (y ∗ ) ≤ sup Tx . For any y ∈ (y ∗ , x), we have f (y ∗ ) ≤ f (y) since f is increasing. Thus, 0 ≤ (sup Tx − f (y)) ≤ (sup Tx −f (y ∗ )) < E for y ∈ (y ∗ , x). Let δ = (x−y ∗ )/2. Then, if 0 < x−y < δ, sup Tx −f (y) < E. Since E was arbitrary, this shows that limy→x− f (y) = supTx . The proof for f (x+ ) is similar, using the Infimum Tolerance Lemma, Lemma 3.2.1. You should be able to see that f (x− ) is less than or equal to f (x+ ) for all x. We will define f (a− ) = f (a) and f (b+ ) = f (b) since f is not defined prior to a or after b.

To prove the stated result holds, first choose an arbitrary yj ∈ (xj , xj+1 ) for each j = 0, . . . , p − 1. + Then, since f is increasing, for each j = 1, . . . , p, we have f (yj−1 ) ≤ f (x− j ) ≤ f (xj ) ≤ f (yj ). Thus, − f (x+ j ) − f (xj )

≤

f (yj ) − f (yj−1 )

(3.1)

BOUNDED VARIATION

33

We also have f (a) ≤ f (a+ ) ≤ f (y0 ) and f (yp−1 ) ≤ f (b− ) ≤ f (b). Thus, it follows that p �

− f (x+ j ) − f (xj )

− f (x+ 0 ) − f (x0 ) +

=

j=0

p−1 � − + − [f (x+ j ) − f (xj )] + f (xp ) − f (xp ) j=1

f (a+ ) − f (a− ) +

≤

p−1 �

[f (yj ) − f (yj−1 )] + f (b+ ) − f (b− )

j=1

Using Equation 3.1 and replacing x0 by a and xp with b, we then note the sum on the right-hand side collapses to f (yp−1 ) − f (y0 ). Finally, since f (a− ) = f (a) and f (b+ ) = f (b), we obtain p �

− f (x+ j ) − f (xj )

≤

f (a+ ) − f (a) + f (yp−1 ) − f (y0 ) + f (b) − f (b− )

≤ ≤

f (y0 ) − f (a) + f (yp−1 ) − f (b− ) + f (b) − f (y0 ) f (b) − f (a) + f (yp−1 ) − f (b− )

j=0

But f (yp−1 ) − f (b− ) ≤ 0, so p �

− f (x+ j ) − f (xj )

≤

f (b) − f (a)

j=0

� Theorem 3.2.4 A Monotone Function Has a Countable Number of Discontinuities If f is monotone on [a, b], the set of discontinuities of f is countable. Proof 3.2.4 For concreteness, we assume f is monotone increasing. The decreasing case is shown similarly. Since f is monotone increasing, the only types of discontinuities it can have are jump discontinuities. If x ∈ [a, b] is a point of discontinuity, then the size of the jump is given by f (x+ ) − f (x− ). Define Dk = {x ∈ (a, b) : f (x+ ) − f (x− ) > 1/k}, for each integer k ≥ 1. We want to show that Dk is finite. Select any finite subset S of Dk and label the points in S by x1 , . . . , xp with x1 < x2 < · · · < xp . If we add the point x0 = a and xp+1 = b, these points determine a partition π. Hence, by Theorem 3.2.3, we know that p � − [f (x+ j ) − f (xj )]

≤

� − [f (x+ j ) − f (xj )] ≤ f (b) − f (a) π

j=1

But each jump satisfies f (xj+ ) − f (xj− ) > 1/k and there are a total of p such points in S. Thus, we must have p/k


0. Hence, there is a positive � integer k0 so that f (x+ ) − f (x− ) > 1/k0 . This means x is in Dk0 and so is in D. Definition 3.2.2 The Discontinuity Set of a Monotone Function Let f be monotone increasing on [a, b]. We will let S denote the set of discontinuities of f on [a, b]. We know this set is countable by Theorem 3.2.4 so we can label it as S = {xj }. Define functions u and v on [a, b] by u(x)

=

0, x=a f (x) − f (x− ), x ∈ (a, b]

v(x)

=

f (x+ ) − f (x), x ∈ [a, b) 0, x=b

In Figure 3.1, we show a monotone increasing function with several jumps. You should be able to compute u and v easily at these jumps. There are several very important points to make about these functions u and v which are listed below. Comment 3.2.1

1. Note that u(x) is the left-hand jump of f at x ∈ (a, b] and v(x) is the right-hand jump of f at x ∈ [a, b). 2. Both u and v are non-negative functions and u(x) + v(x) = f (x+ ) − f (x− ) is the total jump in f at x, for x ∈ (a, b). 3. Moreover, f is continuous at x from the left if and only if u(x) = 0, and f is continuous from the right at x if and only if v(x) = 0. 4. Finally, f is continuous on [a, b] if and only if u(x) = v(x) = 0 on [a, b]. Homework Exercise 3.2.1 Let f be defined by f (x)

=

⎧ 0≤x 0 such that y ∈ [a, b] and |x − y| < δ ⇒ |f (x) − f (y)| < E. But by the second part of this proof, we

38

BASIC ANALYSIS IV: MEASURE THEORY AND INTEGRATION

have |Sf (x) − Sf (y)| ≤ |f (y) − f (x)| < E. Thus, Sf is continuous at x.

�

Homework Exercise 3.2.5 Let f be defined by f (x)

=

⎧ 0≤x