Lebesgue Measure and Integration: An Introduction

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John Wiley & Sons, 1998 - Mathematics - 292 pages
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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
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About the author (1998)

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

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