Lebesgue Measure and Integration

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

• Preface xiChapter 1. Historical Highlights 11.1 Rearrangements 21.2 Eudoxus (408-355 B.C.E.) and the Method of Exhaustion 31.3 The Lune of Hippocrates (430 B.C.E.) 51.4 Archimedes (287-212 B.C.E.) 71.5 Pierre Fermat (1601-1665)1.6 Gottfried Leibnitz (1646-1716), Issac Newton (1642-1723) 121.7 Augustin-Louis Cauchy (1789-1857) 151.8 Bernhard Riemann (1826-1866) 171.9 Emile Borel (1871 -1956), Camille Jordan (1838-1922), Giuseppe Peano (1858-1932) 201.10 Henri Lebesgue (1875-1941), William Young (1863-1942) 221.11 Historical Summary 251.12 Why Lebesgue 26Chapter 2. Preliminaries 322.1 Sets 322.2 Sequences of Sets 342.3 Functions 352.4 Real Numbers 422.5 Extended Real Numbers 492.6 Sequences of Real Numbers 512.7 Topological Concepts of R 622.8 Continuous Functions 662.9 Differentiable Functions 732.10 Sequences of Functions 75Chapter 3. Lebesgue Measure 873.1 Length of Intervals 903.2 Lebesgue Outer Measure 933.3 Lebesgue Measurable Sets 1003.4 BorelSets 1123.5 "Measuring" 1153.6 Structure of Lebesgue Measurable Sets 120Chapter 4. Lebesgue Measurable Functions 1264.1 Measurable Functions 1264.2 Sequences of Measurable Functions 1354.3 Approximating Measurable Functions 1374.4 Almost Uniform Convergence 141Chapter 5. Lebesgue Integration 1475.1 The Riemann Integral 1475.2 The Lebesgue Integral for Bounded Functions on Sets of Finite Measure 1735.3 The Lebesgue Integral for Nonnegative Measurable Functions 1945.4 The Lebesgue Integral and Lebesgue Integrability 2245.5 Convergence Theorems 237Appendix A. Cantor's Set 252Appendix B. A Lebesgue Nonmeasurable Set 266Appendix C. Lebesgue, Not Borel 273Appendix D. A Space-Filling Curve 276Appendix E. An Everywhere Continuous, Nowhere Differentiable,Function 279