# Lebesgue Measure and Integration: An Introduction

John Wiley & Sons, 1998 - Mathematics - 292 pages
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.

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This book is a good companion for mathematics students especially the undergraduates.

### Contents

 HISTORICAL HIGHLIGHTS 1 11 REARRANGEMENTS 2 12 EUDOXUS 408355 BCE AND THE METHOD OF EXHAUSTION 3 13 THE LUNE OF HIPPOCRATES 430 BCE 5 14 ARCHIMEDES 287212 BCE 7 15 PIERRE FERMAT 16011665 10 16 GOTTFRIED LEIBNITZ 16461716 ISSAC NEWTON 16421723 12 17 AUGUSTINLOUIS CAUCHY 17891857 15
 28 CONTINUOUS FUNCTIONS 66 29 DIFFERENTIABLE FUNCTIONS 73 210 SEQUENCES OF FUNCTIONS 75 LEBESGUE MEASURE 87 31 LENGTH OF INTERVALS 90 32 LEBESGUE OUTER MEASURE 93 33 LEBESGUE MEASURABLE SETS 100 34 BOREL SETS 112

 18 BERNHARD RIEMANN 18261866 17 19 EMILE BOREL 18711956 CAMILLE JORDAN 18381922 GIUSEPPE PEANO 18581932 20 110 HENRI LEBESGUE 18751941 WILLIAM YOUNG 18631942 22 111 HISTORICAL SUMMARY 25 112 WHY LEBESGUE 26 PRELIMINARIES 32 22 SEQUENCES OF SETS 34 23 FUNCTIONS 35 24 REAL NUMBERS 42 25 EXTENDED REAL NUMBERS 49 26 SEQUENCES OF REAL NUMBERS 51 27 TOPOLOGICAL CONCEPTS OF R 62
 35 MEASURING 115 36 STRUCTURE OF LEBESGUE MEASURABLE SETS 120 LEBESGUE MEASURABLE FUNCTIONS 126 42 SEQUENCES OF MEASURABLE FUNCTIONS 135 43 APPROXIMATING MEASURABLE FUNCTIONS 137 44 ALMOST UNIFORM CONVERGENCE 141 LEBESGUE INTEGRATION 147 52 THE LEBESGUE INTEGRAL FOR BOUNDED FUNCTIONS ON SETS OF FINITE MEASURE 173 53 THE LEBESGUE INTEGRAL FOR NONNEGATIVE MEASURABLE FUNCTIONS 194 54 THE LEBESGUE INTEGRAL AND LEBESGUE INTEGRABILITY 224 55 CONVERGENCE THEOREMS 237 Copyright