There are two aspects to the theory of Boolean algebras; the algebraic and the set-theoretical. A Boolean algebra can be considered as a special kind of algebraic ring, or as a generalization of the set-theoretical notion of a field of sets. Fundamental theorems in both of these directions are due to M. H. STONE, whose papers have opened a new era in the develop ment of this theory. This work treats the set-theoretical aspect, with little mention being made of the algebraic one. The book is composed of two chapters and an appendix. Chapter I is devoted to the study of Boolean algebras from the point of view of finite Boolean operations only; a greater part of its contents can be found in the books of BIRKHOFF [2J and HERMES [IJ. Chapter II seems to be the first systematic study of Boolean algebras with infinite Boolean operations. To understand Chapters I and II it suffices only to know fundamental notions from general set theory and set-theoretical topology. No know ledge of latticetheory or of abstract algebra is presumed. Less familiar topological theorems are recalled, and only a few examples use more advanced topological means; but these may be omitted. All theorems in both chapters are given with full proofs.
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ISBN-13
978-3-662-01507-0 (9783662015070)
DOI
10.1007/978-3-662-01507-0
Schweitzer Klassifikation
Terminology and notation.- I. Finite joins and meets.- II. Infiinite joins and meets.- Append.- § 39. Relation to other algebras.- § 40. Applications to mathematical logic. Classical calculi.- § 41. Topology in Boolean algebras. Applications to non-classical logic.- § 42. Applications to measure theory.- § 43. Measurable functions and real homomorphisms.- § 44. Measurable functions. Reduction to continuous functions.- § 45. Applications to functional analysis.- § 46. Applications to foundations of the theory of probability.- § 47. Problems of effectivity.- List of symbols.- Author Index.
