There are two aspects in the theory of Boolean algebras: algebraic and set-theoretical. Boolean algebras can be considered as a special kind of algebraic rings, or as a generalization of the set-theoretical notion of field of sets. Fundamental theorems in the both directions are due to M. H. STONE whose papers have opened a new period in the development of the theory. This work treats of the set-theoretical aspect, the algebraic one being scarcely mentioned. 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 also in the books of BIRKHOFF [2J and HERMES [1 J. Chapter II seems to be the first systematic study of Boolean algebras with infinite Boolean operations. To understand Chapters land II it suffices to know only fundamental notions from General Set Theory and Set-theoretical Topology. No knowledge of Lattice Theory or Abstract Algebra is supposed. Less known topological theorems are recalled. Only a few examples use more advanced topological means but they can be omitted. All theorems in both Chapters are given with full proofs. On the contrary, no complete proofs are given in the Appendix which contains mainly a short exposition of applications of Boolean algebras to other parts of Mathematics with references to the literature. An elementary knowledge of discussed theories is supposed
CONTENT
Terminology and notation -- I. Finite joins and meets -- II. Infinite joins and meets -- ยง 39. Relation to other algebras -- ยง 40. Applications to Mathematical Logic. Classical calculi -- ยง 41. Topology in Boolean algebras. Applications to non-classical Logic -- ยง 42. Applications to Measure Theory -- ยง 43. Measurable functions and real homomorphisms -- ยง 44. Measurable functions. Reduction to continuous functions -- ยง 45. Applications to Functional Analysis -- ยง 46. Applications to foundations of the Theory of Probability -- ยง 47. Problems of effectivity -- List of symbols -- Author Index
