Boolean Algebras in Analysis

ISBN-13: 9781402004803 / Angielski / Twarda / 2002 / 604 str.

D. A. Vladimirov

Opis

Boolean algebras underlie many central constructions of analysis, logic, probability theory, and cybernetics. This book concentrates on the analytical aspects of their theory and application, which distinguishes it among other sources. Boolean Algebras in Analysis consists of two parts. The first concerns the general theory at the beginner's level. Presenting classical theorems, the book describes the topologies and uniform structures of Boolean algebras, the basics of complete Boolean algebras and their continuous homomorphisms, as well as lifting theory. The first part also includes an introductory chapter describing the elementary to the theory. The second part deals at a graduate level with the metric theory of Boolean algebras at a graduate level. The covered topics include measure algebras, their sub algebras, and groups of automorphisms. Ample room is allotted to the new classification theorems abstracting the celebrated counterparts by D.Maharam, A.H. Kolmogorov, and V.A.Rokhlin. Boolean Algebras in Analysis is an exceptional definitive source on Boolean algebra as applied to functional analysis and probability. It is intended for all who are interested in new and powerful tools for hard and soft mathematical analysis.

Recenzje

"This book consists of two parts. The first is devoted to the general theory of Boolean algebras. The main content of the chapters comprises those sections of the theory of Boolean algebras which relate to these applications. The author gives basic attention to complete Boolean algebras whose structure is described in detail. The first part of the book also contains the extension theorems for continuous homomorphisms. The author examines the topologies and uniformities related to the order and presents the theory of lifting, realizations of Boolean algebras, Stone functors between the categories of Boolean algebras and totally disconnected spaces. One of the chapters gives a sketch of the theory of vector spaces.

The second part of the book is devoted to the metric theory of Boolean algebras. Here measure algebras are studied, and traditional matters are described: the Lebesgue-Carathéodory theorem, Radon-Nikod\'ym theorem and Lyapunov theorem on vector measures, the algebraic and metric classifications of probability algebras and their subalgebras, theorems about automorphism groups and invariant measures. Much room is allotted to the problem of existence of essentially positive totally additive measure. The closing chapter is devoted to the problem of algebraic and metric independence of subalgebras..." 
(MATHEMATICAL REVIEWS)

Spis treści

Foreword to the English Translation. Denis Artem'evich Vladimirov (1929-1994). Preface. Introduction. Part I: General Theory of Boolean Algebras. 0. Preliminaries on Boolean Algebras. 1. The Basic Apparatus. 2. Complete Boolean Algebras. 3. Representation of Boolean Algebras. 4. Topologies on Boolean Algebras. 5. Homomorphisms. 6. Vector Lattices and Boolean Algebras. Part II: Metric Theory of Boolean Algebras. 7. Normed Boolean Algebras. 8. Existence of a Measure. 9. Structure of a Normed Boolean Algebra. 10. Independence. Appendices. Prerequisites to Set Theory and General Topology. 1. General remarks. 2. Partially ordered sets. 3. Topologies. Basics of Boolean Valued Analysis. 1. General remarks. 2. Boolean valued models. 3. Principles of Boolean valued analysis. 4. Ascending and descending. References. Index.