A superb text on the fundamentals of Lebesgue measure and integration.This book is designed to give the reader a solid understanding of Lebesgue measure and integration. It focuses on only the most fundamental concepts, namely Lebesgue measure for R and Lebesgue integration for extended real-valued functions on R. Starting with a thorough presentation of the preliminary concepts of undergraduate analysis, this book covers all the important topics, including measure theory, measurable functions, and integration. It offers an abundance of support materials, including helpful illustrations, examples, and problems. To further enhance the learning experience, the author provides a historical context that traces the struggle to define "area" and "area under a curve" that led eventually to Lebesgue measure and integration.Lebesgue Measure and Integration is the ideal text for an advanced undergraduate analysis course or for a first-year graduate course in mathematics, statistics, probability, and other applied areas. It will also serve well as a supplement to courses in advanced measure theory and integration and as an invaluable reference long after course work has been completed.
Author(s): Frank Burk
Series: Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts
Edition: 1
Publisher: Wiley-Interscience
Year: 1997
Language: English
Pages: 310
Contents vii......Page f007.djvu
Preface xi......Page f011.djvu
1 Historical Highlights 1......Page p001.djvu
1.1 Rearrangements 2......Page p002.djvu
1.2 Eudoxus (408-355 B.C.E.) and the Method of Exhaustion 3......Page p003.djvu
1.3 The Lune of Hippocrates (430 B.C.E.) 5......Page p005.djvu
1.4 Archimedes (287-212 B.C.E.) 7......Page p007.djvu
1.5 Pierre Fermat (1601-1665) 10......Page p010.djvu
1.6 Gottfried Leibnitz (1646-1716), Issac Newton (1642-1723) 12......Page p012.djvu
1.7 Augustin-Louis Cauchy (1789-1857) 15......Page p015.djvu
1.8 Bernhard Riemann (1826-1866) 17......Page p017.djvu
1.9 Emile Borel (1871-1956), Camille Jordan (1838-1922), Giuseppe Peano (1858-1932) 20......Page p020.djvu
1.10 Henri Lebesgue (1875-1941), William Young (1863-1942) 22......Page p022.djvu
1.11 Historical Summary 25......Page p025.djvu
1.12 Why Lebesgue 26......Page p026.djvu
2.1 Sets 32......Page p032.djvu
2.2 Sequences of Sets 34......Page p034.djvu
2.3 Functions 35......Page p035.djvu
2.4 Real Numbers 42......Page p042.djvu
2.5 Extended Real Numbers 49......Page p049.djvu
2.6 Sequences of Real Numbers 51......Page p051.djvu
2.7 Topological Concepts of R 62......Page p062.djvu
2.8 Continuous Functions 66......Page p066.djvu
2.9 Differentiable Functions 73......Page p073.djvu
2.10 Sequences of Functions 75......Page p075.djvu
3 Lebesgue Measure 87......Page p087.djvu
3.1 Length of Intervals 90......Page p090.djvu
3.2 Lebesgue Outer Measure 93......Page p093.djvu
3.3 Lebesgue Measurable Sets 100......Page p100.djvu
3.4 Borel Sets 112......Page p112.djvu
3.5 "Measuring" 115......Page p115.djvu
3.6 Structure of Lebesgue Measurable Sets 120......Page p120.djvu
4.1 Measurable Functions 126......Page p126.djvu
4.2 Sequences of Measurable Functions 135......Page p135.djvu
4.3 Approximating Measurable Functions 137......Page p137.djvu
4.4 Almost Uniform Convergence 141......Page p141.djvu
5.1 The Riemann Integral 147......Page p147.djvu
5.2 The Lebesgue Integral for Bounded Functions on Sets of Finite Measure 173......Page p173.djvu
5.3 The Lebesgue Integral for Nonnegative Measurable Functions 194......Page p194.djvu
5.4 The Lebesgue Integral and Lebesgue Integrability 224......Page p224.djvu
5.5 Convergence Theorems 237......Page p237.djvu
A Cantor's Set 252......Page p252.djvu
B A Lebesgue Nonmeasurable Set 266......Page p266.djvu
C Lebesgue, Not Borel 273......Page p273.djvu
D A Space-Filling Curve 276......Page p276.djvu
E An Everywhere Continuous, Nowhere Differentiable Function 279......Page p279.djvu
References 285......Page p285.djvu
Index 288......Page p288.djvu
