Foreword to the English translation

Contents Foreword to the English Translation Denis Artem′ evich Vladimirov (1929–1994) Preface Introduction ix xi xvii xix Part I GENERAL THEORY OF BOOLEAN ALGEBRAS 0. PRELIMINARIES ON BOOLEAN ALGEBRAS 1 Lattices 2 Boolean algebras 3 Additive functions on Boolean algebras. Measures. Relation to probability theory 4 Automorphisms and invariant measures 3 3 10 1. THE BASIC APPARATUS 1 Subalgebras and generators 2 The concepts of ideal, filter, and band 3 Factorization, homomorphisms, independence, and free Boolean algebras 35 35 48 2. COMPLETE BOOLEAN ALGEBRAS 1 Complete Boolean algebras; their subalgebras and homomorphisms 2 The exhaustion principle and the theorem of solid cores 3 Construction of complete Boolean algebras 4 Important examples of complete Boolean algebras 5 The Boolean algebra of regular open sets v 23 29 55 83 83 89 95 102 103 vi BOOLEAN ALGEBRAS IN ANALYSIS 6 7 The type, weight, and cardinality of a complete Boolean algebra Structure of a complete Boolean algebra 105 111 3. REPRESENTATION OF BOOLEAN ALGEBRAS 1 The Stone Theorem 2 Interpretation of the basic notions of the theory of Boolean algebras in the language of Stone spaces 3 Stone functors 125 125 136 150 4. TOPOLOGIES ON BOOLEAN ALGEBRAS 1 Directed sets and generalized sequences 2 Various topologies on Boolean algebras 3 Regular Boolean algebras. Various forms of distributivity 181 181 184 223 5. HOMOMORPHISMS 1 Extension of homomorphisms 2 Lifting 3 Extension of continuous homomorphisms 4 Again on representation of a Boolean algebra 233 234 238 244 267 6. VECTOR LATTICES AND BOOLEAN ALGEBRAS 1 K -spaces and the related Boolean algebras 2 Spectral families and resolutions of the identity. Spectral measures 3 Separable Boolean algebras and σ -algebras of sets. Measurable functions 4 The integral with respect to a spectral measure and the Freudenthal Theorem. The space SX as the family of resolutions of the identity. Functions of elements 277 277 287 296 298 Part II METRIC THEORY OF BOOLEAN ALGEBRAS 7. NORMED BOOLEAN ALGEBRAS 1 Normed algebras 2 Extension of a countably additive function. The Lebesgue–Carathéodory Theorem 3 NBAs and the metric structures of measure spaces 4 Totally additive functions and resolutions of the identity of a normed algebra 5 Subalgebras of a normed Boolean algebra 6 Fundamental systems of partitions 7 Systems of measures and the Lyapunov Theorem 317 317 325 332 342 353 382 387 Contents 8. EXISTENCE OF A MEASURE 1 Conditions for existence of a measure 2 Existence of a measure invariant under the automorphism group 3 The Potepun Theorem 4 Automorphisms of normable algebras and invariant measures 5 Construction of a normed Boolean algebra given a transformation group vii 391 391 399 417 433 439 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 1 A system of two subalgebras 2 A test for metric independence 535 535 541 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 562 563 563 564 565 569 569 569 571 572 References Index 581 601 Foreword to the English Translation I am deeply honored to introduce this great book of a great author to the English language reading community. Denis Artem ′ evich Vladimirov (1929–1994) was a prominent representative of the Russian mathematical school in functional analysis which was founded by Leonid Vital′ evich Kantorovich, a renowned mathematician and a Nobel Prize winner in economics. This school comprises two affiliations in St. Petersburg and Novosibirsk which maintain intimate relations since the latter was set up by the former, so it is not astonishing that I enjoyed the wit and charm of Vladimirov for many years. Our contacts were usually established through the students we supervised; he, in St. Petersburg and I, in Novosibirsk. I always tried to arrange matters so that my students spent some time near Vladimirov to master Boolean algebras and ordered vector spaces. Probably one of the results of this cooperation is the fact that there is now an active group in Boolean valued analysis in Novosibirsk. Unfortunately, the only possibility of continuing this practice is offered by the present book... It was not long before Vladimirov’s death when he and his friends had asked me to help with the publishing and editing of the English translation of the book. I agreed readily and soon Kluwer Academic Publishers decided to print the book. The book was mostly translated by Professor A. E. Gutman and his students in Novosibirsk, all “descendants” of Vladimirov. E. G. Taı̆pale translated a few final sections and made the entire book more readable. I. I. Bazhenov, I. I. Kozhanova, Yu. N. Lovyagin, A. A. Samorodnitskiı̆, and Yu. V. Shergin helped me with the proofreading. ix x BOOLEAN ALGEBRAS IN ANALYSIS The translation took much more time than planned: the reasons behind this are understandable for anyone aware of the present standards of academic life in Russia. Unfortunately, capable mathematicians are not always experienced translators and knowledgeable grammarians. Therefore, the battle against solecism and mistranslation was partly lost in proofreading... Vladimirov was unhappy that he had no opportunity to include a chapter on Boolean valued analysis in this edition of his book. At the publisher’s request, I compiled a short appendix which is intended to serve as an introduction to this new and promising area for expansion and proliferation of Boolean algebras. Denis Artem ′ evich Vladimirov was one of the giants of the past who bequeathed us his insight into part of the future with this book. I hope the reader will enjoy it. S. S. Kutateladze August, 2001