Boolean algebras in analysis / by D.A. Vladimirov.

Includes bibliographical references (p. 581-599) and index.

Machine generated contents note: Part I GENERAL THEORY OF BOOLEAN ALGEBRAS -- 0. PRELIMINARIES ON BOOLEAN ALGEBRAS 3 -- 1 Lattices 3 -- 2 Boolean algebras 10 -- 3 Additive functions on Boolean algebras. Measures. -- Relation to probability theory 23 -- 4 Automorphisms and invariant measures 29 -- 1. THE BASIC APPARATUS 35 -- 1 Subalgebras and generators 35 -- 2 The concepts of ideal, filter, and band 48 -- 3 Factorization, homomorphisms, independence, and -- free Boolean algebras 55 -- 2. COMPLETE BOOLEAN ALGEBRAS 83 -- 1 Complete Boolean algebras; their subalgebras and -- homomorphisms 83 -- 2 The exhaustion principle and the theorem of solid cores 89 -- 3 Construction of complete Boolean algebras 95 -- 4 Important examples of complete Boolean -- algebras 102 -- 5 The Boolean algebra of regular open sets 103 -- 6 The type, weight, and cardinality of a complete Boolean -- algebra 105 -- 7 Structure of a complete Boolean algebra 111 -- 3. REPRESENTATION OF BOOLEAN ALGEBRAS 125 -- 1 The Stone Theorem 125 -- 2 Interpretation of the basic notions of the theory of -- Boolean algebras in the language of Stone spaces 136 -- 3 Stone functors 150 -- 4. TOPOLOGIES ON BOOLEAN ALGEBRAS 181 -- 1 Directed sets and generalized sequences 181 -- 2 Various topologies on Boolean algebras 184 -- 3 Regular Boolean algebras. Various forms of distributivity 223 -- 5. HOMOMORPHISMS 233 -- 1 Extension of homomorphisms 234 -- 2 Lifting 238 -- 3 Extension of continuous homomorphisms 244 -- 4 Again on representation of a Boolean algebra 267 -- 6. VECTOR LATTICES AND BOOLEAN ALGEBRAS 277 -- 1 K-spaces and the related Boolean algebras 277 -- 2 Spectral families and resolutions of the identity. -- Spectral measures 287 -- 3 Separable Boolean algebras and a-algebras of sets. -- Measurable functions 296 -- 4 The integral with respect to a spectral measure and -- the Freudenthal Theorem. The space 6y as the family -- of resolutions of the identity. Functions of elements 298 -- Part II METRIC THEORY OF BOOLEAN ALGEBRAS -- 7. NORMED BOOLEAN ALGEBRAS 317 -- 1 Normed algebras 317 -- 2 Extension of a countably additive function. -- The Lebesgue-Carath6odory Theorem 325 -- 3 NBAs and the metric structures of measure spaces 332 -- 4 Totally additive functions and resolutions of the identity -- of a normed algebra 342 -- 5 Subalgebras of a normed Boolean algebra 353 -- 6 Fundamental systems of partitions 382 -- 7 Systems of measures and the Lyapunov Theorem 387 -- 8. EXISTENCE OF A MEASURE 391 -- 1 Conditions for existence of a measure 391 -- 2 Existence of a measure invariant under -- the automorphism group 399 -- 3 The Potepun Theorem 417 -- 4 Automorphisms of normable algebras and -- invariant measures 433 -- 5 Construction of a normed Boolean algebra given -- a transformation group 439 -- 9. STRUCTURE OF A NORMED BOOLEAN ALGEBRA 445 -- 1 Structure of a normed algebra 445 -- 2 Classification for normed algebras 464 -- 3 Interlocation of subalgebras of a normed Boolean algebra 470 -- 4 Isomorphism between subalgebras 475 -- 5 Isomorphism of systems of subalgebras 531 -- 10. INDEPENDENCE 535 -- 1 A system of two subalgebras 535 -- 2 A test for metric independence 541.