Lebesgue Measure and Integration: An Introduction

Preface xi

Chapter 1. Historical Highlights 1

1.1 Rearrangements 2

1.2 Eudoxus (408-355 B.C.E.) and the Method of Exhaustion 3

1.3 The Lune of Hippocrates (430 B.C.E.) 5

1.4 Archimedes (287-212 B.C.E.) 7

1.5 Pierre Fermat (1601-1665)

1.6 Gottfried Leibnitz (1646-1716), Issac Newton (1642-1723) 12

1.7 Augustin-Louis Cauchy (1789-1857) 15

1.8 Bernhard Riemann (1826-1866) 17

1.9 Emile Borel (1871 -1956), Camille Jordan (1838-1922), Giuseppe Peano (1858-1932) 20

1.10 Henri Lebesgue (1875-1941), William Young (1863-1942) 22

1.11 Historical Summary 25

1.12 Why Lebesgue 26

Chapter 2. Preliminaries 32

2.1 Sets 32

2.2 Sequences of Sets 34

2.3 Functions 35

2.4 Real Numbers 42

2.5 Extended Real Numbers 49

2.6 Sequences of Real Numbers 51

2.7 Topological Concepts of R 62

2.8 Continuous Functions 66

2.9 Differentiable Functions 73

2.10 Sequences of Functions 75

Chapter 3. Lebesgue Measure 87

3.1 Length of Intervals 90

3.2 Lebesgue Outer Measure 93

3.3 Lebesgue Measurable Sets 100

3.4 BorelSets 112

3.5 "Measuring" 115

3.6 Structure of Lebesgue Measurable Sets 120

Chapter 4. Lebesgue Measurable Functions 126

4.1 Measurable Functions 126

4.2 Sequences of Measurable Functions 135

4.3 Approximating Measurable Functions 137

4.4 Almost Uniform Convergence 141

Chapter 5. Lebesgue Integration 147

5.1 The Riemann Integral 147

5.2 The Lebesgue Integral for Bounded Functions on Sets of Finite Measure 173

5.3 The Lebesgue Integral for Nonnegative Measurable Functions 194

5.4 The Lebesgue Integral and Lebesgue Integrability 224

5.5 Convergence Theorems 237

Appendix A. Cantor's Set 252

Appendix B. A Lebesgue Nonmeasurable Set 266

Appendix C. Lebesgue, Not Borel 273

Appendix D. A Space-Filling Curve 276

Appendix E. An Everywhere Continuous, Nowhere Differentiable,

Function 279