Lebesgue Measure and Integration: An Introduction (Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts)

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. Contents......Page 7 Preface......Page 11 1 Historical Highlights......Page 17 1.1 REARRANGEMENTS......Page 18 1.2 EUDOXUS (408-355 B.C.E.) AND THE METHOD OF EXHAUSTION......Page 19 1.3 THE LUNE OF HIPPOCRATES (430 B.C.E.)......Page 21 1.4 ARCHIMEDES (287 -212 B.C.E.)......Page 23 1.5 PIERRE FERMAT (1601-1665)......Page 26 1.6 GOTTFRIED LEIBNITZ {1646-1716), ISSAC NEWTON (1642-1723)......Page 28 1.7 AUGUSTIN-LOUIS CAUCHY (1789-1857)......Page 31 1.8 BERNHARD RIEMANN (1826-1866)......Page 33 1.9 EMILE BOREL (1871-1956), CAMILLE JORDAN (1838-1922), GIUSEPPE PEANO (1858-1932)......Page 36 1.10 HENRI LEBESGUE (1875-1941), WILLIAM YOUNG (1863-1942}......Page 38 1.11 HISTORICAL SUMMARY......Page 41 1.12 WHY LEBESGUE?......Page 42 2.1 SETS......Page 48 2.2 SEQUENCES OF SETS......Page 50 2.3 FUNCTIONS......Page 51 2.4 REAL NUMBERS......Page 58 2.5 EXTENDED REAL NUMBERS......Page 65 2.6 SEQUENCES OF REAL NUMBERS......Page 67 2.7 TOPOLOGICAL CONCEPTS OF R......Page 78 2.8 CONTINUOUS FUNCTIONS......Page 82 2.9 DIFFERENTIABLE FUNCTIONS......Page 89 2.10 SEQUENCES OF FUNCTIONS......Page 91 3 Lebesgue Measure......Page 103 3.1 LENGTH OF INTERVALS......Page 106 3.2 LEBESGUE OUTER MEASURE......Page 109 3.3 LEBESGUE MEASURABLE SETS......Page 116 3.4 BOREL SETS......Page 128 3.5 "MEASURING"......Page 131 3.6 STRUCTURE OF LEBESGUE MEASURABLE SETS......Page 136 4.1 MEASURABLE FUNCTIONS......Page 142 4.2 SEQUENCES OF MEASURABLE FUNCTIONS......Page 151 4.3 APPROXIMATING MEASURABLE FUNCTIONS......Page 153 4.4 ALMOST UNIFORM CONVERGENCE......Page 157 5.1 THE RIEMANN INTEGRAL......Page 163 5.2 THE LEBESGUE INTEGRAL FOR BOUNDED FUNCTIONS ON SETS OF FINITE MEASURE......Page 189 5.3 THE LEBESGUE INTEGRAL FOR NONNEGATIVE MEASURABLE FUNCTIONS......Page 210 5.4 THE LEBESGUE INTEGRAL AND LEBESGUE INTEGRABILITY......Page 240 5.5 CONVERGENCE THEOREMS......Page 253 A.1 CANTOR'S SET......Page 268 B.1 A LEBESGUE NONMEASURABLE SET......Page 282 C.1 LEBESGUE, NOT BOREL......Page 289 D.1 A SPACE-FILLING CURVE......Page 292 E.1 AN EVERYWHERE CONTINUOUS, NOWHERE DIFFERENTIABLE, FUNCTION......Page 295 References......Page 301 Index......Page 304 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 Struggles 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. Front Matter -- Historical Highlights -- Preliminaries -- Lebesgue Measure -- Lebesgue Measurable Functions -- Lebesgue Integration -- Appendix A: Cantor's Set -- Appendix B: A Lebesgue Nonmeasurable Set -- Appendix C: Lebesgue, Not Borel -- Appendix D: A Space-filling Curve -- Appendix E: An Everywhere Continuous, Nowhere Differentiable, Function -- References -- Index -- Pure And Applied Mathematics -- Historical Highlights -- Preliminaries -- Lebesgue Measure -- Lebesgue Measurable Functions -- Lebesgue Integration -- Appendices -- References -- Index. Frank Burk. A Wiley-interscience Publication. Includes Bibliographical References And Index. The figures below demonstrate the general idea of "rearranging"; in the first example, a circle rearranged into a parallelogram.