Measure, Integration and Function Spaces

✍ Scribed by Charles W. Swartz

• Publisher: World Scientific Publishing Company
• Year: 1994
• Tongue: English
• Leaves: 288
• Category: Library

✦ Synopsis

This text contains a basic introduction to the abstract measure theory and the Lebesgue integral. Most of the standard topics in the measure and integration theory are discussed. In addition, topics on the Hewitt-Yosida decomposition, the Nikodym and Vitali-Hahn-Saks theorems and material on finitely additive set functions not contained in standard texts are explored. There is an introductory section on functional analysis, including the three basic principles, which is used to discuss many of the classic Banach spaces of functions and their duals. There is also a chapter on Hilbert space and the Fourier transform.

✦ Table of Contents

TABLE OF CONTENTS......Page 8
Preface......Page 6
1.1 Preliminaries ......Page 12
1.2 Extended Real Numbers (R^*) and R^n ......Page 15
1.3 Lebesgue's Definition of the Integral ......Page 20
2.1 Semi-rings and Algebras of Sets ......Page 26
2.2 Additive Set Functions ......Page 31
2.2.1 Jordan Decomposition ......Page 39
2.2.2 Hahn Decomposition ......Page 44
2.2.3 Drewnowski's Lemma ......Page 46
2.3 Outer Measures ......Page 47
2.3.1 Metric Outer Measures ......Page 50
2.4 Extensions of Premeasures ......Page 52
2.4.1 Hewitt-Yosida Decomposition ......Page 60
2.5 Lebesgue Measure ......Page 61
2.6 Lebesgue-Stieltjes Measure ......Page 68
2.6.1 Hewitt-Yosida Decomposition for Lebesgue-Stieltjes Measures ......Page 71
2.7 Regular Measures ......Page 73
2.8 The Nikodym Convergence and Boundedness Theorems ......Page 77
3.1 Measurable Functions ......Page 82
3.1.1 Approximation of Measurable Functions ......Page 89
3.2 The Lebesgue Integral ......Page 92
3.3 The Riemann and Lebesgue Integrals ......Page 107
3.4 Integrals Depending on a Parameter ......Page 111
3.5 Convergence in Mean ......Page 114
3.6 Convergence in Measure ......Page 118
3.7 Comparison of Modes of Convergence ......Page 123
3.8 Mikusinski's Characterization of the Lebesgue Measure ......Page 126
3.9 Product Measures and Fubini's Theorem ......Page 129
3.10 A Geometric Interpretation of the Integral ......Page 137
3.11 Convolution Product ......Page 138
3.12 The Radon-Nikodym Theorem ......Page 143
3.12.1 The Radon-Nikodym Theorem for Finitely Additive Set Functions ......Page 148
3.13 Lebesgue Decomposition ......Page 152
3.14 The Vitali-Hahn-Sake Theorem ......Page 155
4.1 Differentiating Indefinite Integrals ......Page 158
4.2 Differentiation of Monotone Functions ......Page 163
4.3 Integrating Derivatives ......Page 169
4.4 Absolutely Continuous Functions ......Page 171
5.1 Normed Linear Spaces (NLS) ......Page 176
5.2 Linear Mappings between Normed Linear Spaces ......Page 185
5.3 The Uniform Boundedness Principle ......Page 189
5.4 Quotient Spaces ......Page 192
5.5 The Closed Graph/Open Mapping Theorems ......Page 194
5.6 The Hahn-Banach Theorem ......Page 197
5.6.1 Applications of the Hahn-Banach Theorem in NLS ......Page 200
5.6.2 Extension of Bounded, Finitely Additive Set Function s ......Page 204
5.6.3 A Translation Invariant, Finitely Additive Set Function ......Page 205
5.7 Ordered Linear Spaces ......Page 208
6.1 L^P-spaces, 1<p<\infty ......Page 218
6.2 The Space L^\infty ......Page 229
6.3 The Space of Finitely Additive Set Functions ......Page 234
6.4 The Space of Countably Additive Set Functions ......Page 237
6.5 The Space of Continuous Functions ......Page 239
6.6 Hilbert Space ......Page 246
6.6.1 Fourier Transform ......Page 260
Al. Functions of Bounded Variation ......Page 266
A2. The Baire Category Theorem ......Page 269
A3. The Arzela-Ascoli Theorem ......Page 271
A4. The Stone-Weierstrass Theorem ......Page 273
References ......Page 278
Index ......Page 283