Measure and Integration A Concise Introduction to Real Analysis

by Richardson, Leonard F.

• ISBN13: 9780470259542
• ISBN10: 047025954X
• Edition: 1st
• Format: Hardcover
• Copyright: 2009-07-07
• Publisher: Wiley

Summary

Measure and Integration A concise Introduction to Real Analysis Leonard F. Richardson

Author Biography

Leonard F. Richardson, PhD, is Herbert Huey McElveen Professor and Director of Graduate Studies in Mathematics at Louisiana State University, where he is also Assistant Chair of the Department of Mathematics. Dr. Richardson's research interests include harmonic analysis, homogeneous spaces, and representation theory. He is the author of Advanced Calculus: An Introduction to Linear Analysis, also published by Wiley.

Table of Contents

Preface	p. xi
Acknowledgments	p. xiii
Introduction	p. xv
History of the Subject	p. 1
History of the Idea	p. 1
Deficiencies of the Riemann Integral	p. 3
Motivation for the Lebesgue Integral	p. 6
Fields, Borel Fields, and Measures	p. 11
Fields, Monotone Classes, and Borel Fields	p. 11
Additive Measures	p. 18
Carathéodory Outer Measure	p. 20
E. Hopf's Extension Theorem	p. 24
Fields, ¿-Fields, and Measures Inherited by a Subset	p. 29
Lebesgue Measure	p. 31
The Finite Interval [-N, N)	p. 31
Measurable Sets, Borel Sets, and the Real Line	p. 34
Lebesgue Measure on R	p. 36
Measure Spaces and Completions	p. 38
Minimal Completion of a Measure Space	p. 41
A Nonmeasurable Set	p. 41
Semimetric Space of Measurable Sets	p. 43
Lebesgue Measure in Rn	p. 50
Jordan Measure in Rn	p. 52
Measurable Functions	p. 55
Measurable Functions	p. 55
Baire Functions of Measurable Functions	p. 56
Limits of Measurable Functions	p. 58
Simple Functions and Egoroff's Theorem	p. 61
Double Sequences	p. 63
Convergence in Measure	p. 65
Lusin's Theorem	p. 66
The Integral	p. 69
Special Simple Functions	p. 69
Extending the Domain of the Integral	p. 72
The Class L+ of Nonnegative Measurable Functions	p. 74
The Class L of Lebesgue Integrable Functions	p. 78
Convex Functions and Jensen's Inequality	p. 81
Lebesgue Dominated Convergence Theorem	p. 83
Monotone Convergence and Fatou's Theorem	p. 89
Completeness of L1 (X, $$, ¿) and the Pointwise Convergence Lemma	p. 92
Complex-Valued Functions	p. 100
Product Measures and Fubini's Theorem	p. 103
Product Measures	p. 103
Fubini's Theorem	p. 108
Comparison of Lebesgue and Riemann Integrals	p. 117
Functions of a Real Variable	p. 123
Functions of Bounded Variation	p. 123
A Fundamental Theorem for the Lebesgue Integral	p. 128
Lebesgue's Theorem and Vitali's Covering Theorem	p. 131
Absolutely Continuous and Singular Functions	p. 139
General Countably Additive Set Functions	p. 151
Hahn Decomposition Theorem	p. 152
Radon-Nikodym Theorem	p. 156
Lebesgue Decomposition Theorem	p. 161
Examples of Dual Space from Measure Theory	p. 165
The Banach Space Lp (X, $$, ¿)	p. 165
The Dual of a Banach Space	p. 170
The Dual Space of Lp (X, $$, ¿)	p. 174
Hilbert Space, Its Dual, and L2 (X, $$, ¿)	p. 178
Riesz-Markov-Saks-Kakutani Theorem	p. 185
Translation Invariance in Real Analysis	p. 195
An Orthonormal Basis for L2(T)	p. 196
Closed, Invariant Subspaces of L2(T)	p. 203
Integration of Hilbert Space Valued Functions	p. 204
Spectrum of a Subset of L2(T)	p. 206
Schwartz Functions: Fourier Transform and Inversion	p. 208
Closed, Invariant Subspaces of L2(T)	p. 213
The Fourier Transform in L2(R)	p. 213
Translation-Invariant Subspaces of L2(R)	p. 216
The Fourier Transform and Direct Integrals	p. 218
Irreducibility of L2(R) Under Translations and Rotations	p. 219
Position and Momentum Operators	p. 221
The Heisenberg Group	p. 222
The Banach-Tarski Theorem	p. 225
The Limits to Countable Additivity	p. 225
References	p. 229
Index	p. 231
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