Lebesgue Measure and Integration

An Introduction
1. Auflage 1997
ISBN: 978-0-471-17978-8
Verlag: Wiley

Buch, Englisch, 312 Seiten, Format (B × H): 161 mm x 240 mm, Gewicht: 636 g

Reihe: Wiley Series in Pure and Applied Mathematics

Ein lebendig und präzise geschriebenes Lehrbuch zu Lebesgue-Maß und Lebesgue-Integration, das sehr ins Detail geht und viele historische Aspekte anspricht. Reich ausgestattet mit Illustrationen, Übungsaufgaben und zahlreichen Beispielen, legt dieses Buch die Grundlage für eine Vorlesung über abstrakte Maße und Integrationstheorie. (11/97)

Autoren/Hrsg.

Burk, Frank

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Reelle Analysis

Inhaltsverzeichnis

Historical Highlights;
Preliminaries;
Lebesgue Measure;
Lebesgue Measurable Functions;
Lebesgue Integration;
Appendices;
References.

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

Über Autor(innen)

FRANK BURK teaches in the Department of Mathematics at California State University.