Ash Measure, Integration, and Functional Analysis

1. Auflage 2014
ISBN: 978-1-4832-6510-0
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 300 Seiten, Web PDF

Measure, Integration, and Functional Analysis deals with the mathematical concepts of measure, integration, and functional analysis. The fundamentals of measure and integration theory are discussed, along with the interplay between measure theory and topology. Comprised of four chapters, this book begins with an overview of the basic concepts of the theory of measure and integration as a prelude to the study of probability, harmonic analysis, linear space theory, and other areas of mathematics. The reader is then introduced to a variety of applications of the basic integration theory developed in the previous chapter, with particular reference to the Radon-Nikodym theorem. The third chapter is devoted to functional analysis, with emphasis on various structures that can be defined on vector spaces. The final chapter considers the connection between measure theory and topology and looks at a result that is a companion to the monotone class theorem, together with the Daniell integral and measures on topological spaces. The book concludes with an assessment of measures on uncountably infinite product spaces and the weak convergence of measures. This book is intended for mathematics majors, most likely seniors or beginning graduate students, and students of engineering and physics who use measure theory or functional analysis in their work.

Autoren/Hrsg.

Ash, Robert B.

Inhaltsverzeichnis

1;Front Cover;1
2;Measure, Integration, and Functional Analysis;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;8
6;Summary of Notation;10
7;Chapter 1. Fundamentals of Measure and Integration Theory;18
7.1;1.1 Introduction;18
7.2;1.2 Fields, s-Fields, and Measures;20
7.3;1.3 Extension of Measures;30
7.4;1.4 Lebesgue–Stieltjes Measures and Distribution Functions;39
7.5;1.5 Measurable Functions and Integration;51
7.6;1.6 Basic Integration Theorems;60
7.7;1.7 Comparison of Lebesgue and Riemann Integrals;70
8;Chapter 2. Further Results in Measure and Integration Theory;75
8.1;2.1 Introduction;75
8.2;2.2 Radon–Nikodym Theorem and Related Results;80
8.3;2.3 Applications to Real Analysis;87
8.4;2.4 Lp Spaces;97
8.5;2.5 Convergence of Sequences of Measurable Functions;109
8.6;2.6 Product Measures and Fubini's Theorem;113
8.7;2.7 Measures on Infinite Product Spaces;125
8.8;2.8 References;129
9;Chapter 3. Introduction to Functional Analysis;130
9.1;3.1 Introduction;130
9.2;3.2 Basic Properties of Hilbert Spaces;133
9.3;3.3 Linear Operators on Normed Linear Spaces;144
9.4;3.4 Basic Theorems of Functional Analysis;155
9.5;3.5 Some Properties of Topological Vector Spaces;167
9.6;3.6 References;184
10;Chapter 4. The Interplay between Measure Theory and Topology;185
10.1;4.1 Introduction;185
10.2;4.2 The Daniell Integral;187
10.3;4.3 Measures on Topological Spaces;195
10.4;4.4 Measures on Uncountably Infinite Product Spaces;206
10.5;4.5 Weak Convergence of Measures;213
10.6;4.6 References;217
11;Appendix on General Topology;218
11.1;A1 Introduction;218
11.2;A2 Convergence;219
11.3;A3 Product and Quotient Topologies;225
11.4;A4 Separation Properties and Other Ways of Classifying Topological Spaces;228
11.5;A5 Compactness;230
11.6;A6 Semicontinuous Functions;237
11.7;A7 The Stone–Weierstrass Theorem;240
11.8;A8 Topologies on Function Spaces;243
11.9;A9 Complete Metric Spaces and Category Theorems;247
11.10;A10 Uniform Spaces;251
12;Bibliography;258
13;Solutions to Problems;260
14;Subject Index;296