Measure, Integration, and Functional Analysis

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. Cover......Page 1 Title Page......Page 2 Copyright......Page 3 Contents......Page 4 Preface......Page 6 1 Sets......Page 8 2 Real Numbers......Page 9 4 Topology......Page 10 5 Vector Spaces......Page 11 6 Zorn's Lemma......Page 12 1.1 INTRODUCTION......Page 14 1.2 FIELDS, igma-FIELDS, AND MEASURES......Page 16 1.3 EXTENSION OF MEASURES......Page 26 1.4 LEBESGUE-STIELTJES MEASURES AND DISTRIBUTION FUNCTIONS......Page 35 1.5 MEASURABLE FUNCTIONS AND INTEGRATION......Page 47 1.6 BASIC INTEGRATION THEOREMS......Page 56 1.7 COMPARISON OF LEBESGUE AND RIEMANN INTEGRALS......Page 66 2.1 INTRODUCTION......Page 71 2.2 RADON-NIKODYM THEOREM AND RELATED RESULTS......Page 76 2.3 APPLICATIONS TO REAL ANALYSIS......Page 83 2.4 L?SPACES......Page 93 2.5 CONVERGENCE OF SEQUENCES OF MEASURABLE FUNCTIONS......Page 105 2.6 PRODUCT MEASURES AND FUBINI'S THEOREM......Page 109 2.7 MEASURES ON INFINITE PRODUCT SPACES......Page 121 2.8 REFERENCES......Page 125 3.1 INTRODUCTION......Page 126 3.2 BASIC PROPERTIES OF HILBERT SPACES......Page 129 3.3 LINEAR OPERATORS ON NORMED LINEAR SPACES......Page 140 3.4 BASIC THEOREMS OF FUNCTIONAL ANALYSIS......Page 151 3.5 SOME PROPERTIES OF TOPOLOGICAL VECTOR SPACES......Page 163 3.6 REFERENCES......Page 180 4.1 INTRODUCTION......Page 181 4.2 THE DANIELL INTEGRAL......Page 183 4.3 MEASURES ON TOPOLOGICAL SPACES......Page 191 4.4 MEASURES ON UNCOUNTABLY INFINITE PRODUCT SPACES......Page 202 4.5 WEAK CONVERGENCE OF MEASURES......Page 209 4.6 REFERENCES......Page 213 A1 INTRODUCTION......Page 214 A2 CONVERGENCE......Page 215 A3 PRODUCT AND QUOTIENT TOPOLOGIES......Page 221 A4 SEPARATION PROPERTIES AND OTHER WAYS OF CLASSIFYING TOPOLOGICAL SPACES......Page 224 A5 COMPACTNESS......Page 226 A6 SEMICONTINUOUS FUNCTIONS......Page 233 A7 THE STONE-WEIERSTRASS THEOREM......Page 236 A8 TOPOLOGIES ON FUNCTION SPACES......Page 239 A9 COMPLETE METRIC SPACES AND CATEGORY THEOREMS......Page 243 A10 UNIFORM SPACES......Page 247 BIBLIOGRAPHY......Page 254 Solutions to Problems......Page 256 Subject Index......Page 292 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. 284 pp., Hardcover, some soiling to covers else text clean & binding tight (lacks dust jacket). *Buyer is responsible for any additional duties, taxes, or fees required by recipient's country* - If you are reading this, this item is actually (physically) in our stock and ready for shipment once ordered. We are not bookjackers.