Lebesgue integration on Euclidean space

Frank Jones0763717088, 9780763717087

Lebesgue Integration on Euclidean Space contains a concrete, intuitive, and patient derivation of Lebesgue measure and integration on Rn. Throughout the text, many exercises are incorporated, enabling students to apply new ideas immediately. Jones strives to present a slow introduction to Lebesgue integration by dealing with n-dimensional spaces from the outset. In addition, the text provides students a through treatment of Fourier analysis, while holistically preparing students to become “workers” in real analysis.

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Table of contents :
Front cover……Page 1
Various Integration Formulas……Page 3
Various Special Integrals and Sums……Page 4
Title……Page 5
Series……Page 6
Title page……Page 7
Date-line……Page 8
Contents……Page 9
Preface……Page 13
Bibliography……Page 15
Acknowledgments……Page 17
Dedication……Page 19
A Sets……Page 21
B Countable Sets……Page 24
C Topology……Page 25
D Compact Sets……Page 30
E Continuity……Page 35
F The Distance Function……Page 40
A Construction……Page 45
B Properties of Lebesgue Measure……Page 69
C Appendix: Proof of P1 and P2……Page 80
3 Invariance of Lebesgue Measure……Page 85
A Some Linear Algebra……Page 86
B Translation and Dilation……Page 91
C Orthogonal Matrices……Page 93
D The General Matrix……Page 95
A A Nonmeasurable Set……Page 101
B A Bevy of Cantor Sets……Page 103
C The Lebesgue Function……Page 106
D Appendix: The Modulus of Continuity of the Lebesgue Functions……Page 115
A Algebras and $sigma$-Algebras……Page 123
B Borel Sets……Page 127
C A Measurable Set which Is Not a Borel Set……Page 130
D Measurable Functions……Page 132
E Simple Functions……Page 137
A Nonnegative Functions……Page 141
B General Measurable Functions……Page 150
C Almost Everywhere……Page 155
D Integration Over Subsets of $mathbb{R}^n$……Page 159
E Generalization: Measure Spaces……Page 162
F Some Calculations……Page 167
G Miscellany……Page 172
A Riemann Integral……Page 177
B Linear Change of Variables……Page 190
C Approximation of Functions in $L^1$……Page 191
D Continuity of Translation in $L^1$……Page 200
8 Fubini’s Theorem for $mathbb{R}^n$……Page 201
A Definition and Simple Properties……Page 219
B Generalization……Page 222
C The Measure of Balls……Page 225
D Further Properties of the Gamma Function……Page 229
E Stirling’s Formula……Page 232
F The Gamma Function on $mathbb{R}$……Page 236
A Definition and Basic Inequalities……Page 241
B Metric Spaces and Normed Spaces……Page 247
C Completeness of $L^p$……Page 251
D The Case $p=infty$……Page 255
E Relations between $L^p$ Spaces……Page 258
F Approximation by $C_c^infty(mathbb{R}^n)$……Page 264
G Miscellaneous Problems……Page 266
H The Case $0