Ergebnisse cler Mathematik uncl ihrer Grenzgebiete

Transcription

1 Ergebnisse cler Mathematik uncl ihrer Grenzgebiete Band 25 Herausgegeben von P. R. Halmos. P. J. Hilton. R. Remmert. B. Szokefalvi-Nagy Unter Mitwirkung von L. V. Ahlfors. R. Baer. F. L. Bauer' R. Courant A. Dold J. L. Doob. S. Eilenberg. M. Kneser. G. H. Muller M. M. Postnikov. H. Rademacher' B. Segre. E. Sperner Geschaftsftihrender Herausgeber: P. J. Hilton

2 Roman Sikorski Boolean Algebras Third Edition Springer-Verlag Berlin Heidelberg. New York 1969

3 ISBN-13: DOl: / e-isbn-13: All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. by Springer-Verlag, Berlin Heidelberg 1960,1964,1969. Library of Congress Catalog Card Number Title No Softcover reprint of the hardcover 3rd edition 1969

4 To Professor Kazimierz Kuratowski

5 Preface 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 development 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 [1]. 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 knowledge of lattice theory 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. On the other hand, no complete proofs are given in the Appendix, which contains mainly a short exposition of some of the applications of Boolean algebras to other parts of mathematics with references to the literature. An elementary knowledge of the theories discussed is assumed. I am very much indebted to Professor PAUL R. HALMOS for suggesting that I write this book. I wish to express my thanks ~o H. BASS, A. BIALYNICKI-BIRULA and R. WHERRITT for the revision of the manuscript, and to J. BROWKIN, R. ENGELKING and T. TRACZYK for help in proofreading. Warsaw-New Orleans-Princeton ROMAN SIKORSKI

6 Preface to the second edition Chapter r and the Appendix are almost unchanged. On the contrary, many new results are included in Chapter II; some sections have been extended while others have been completely rewritten. However the general character of Chapter II has been preserved. r am very grateful to PH. DWINGER, H. GAIFMAN, A. W. HALES, J. D. HALPERN, C. R. KARP, K. MATTHES, R. S. PIERCE, Z. SEMADENI and F. M. Y AQUB for valuable information which helped greatly in bringing the material up to date. r am also obliged to A. E. FARLEY for the revision of the manuscript and to T. TRACZYK for help in proofreading. Aarhus, 1962 ROMAN SIKORSKI Preface to the third edition The third edition has been reprinted from the second by offset lithography. It is unchanged except for the correction of errors and the removal of misprints. Warsaw, 1968 ROMAN SIKORSKI

7 Contents 'ferminology and notation... Chaptet' I. Finite joins and meets 1. Definition of Boolean algebras 2. Some consequences of the axioms 3. Ideals and filters Subalgebras Homomorphisms, isomorphisms. 6. Maximal ideals and filters Reduced and perfect fields of sets 8. A fundamental representation theorem. 9. Atoms Quotient algebras Induced homomorphisms between fields of sets 12. Theorems on extending to homomorphisms. 13. Independent subalgebras. Products Free Boolean algebras Induced homomorphisms between quotient algebras. 16. Direct unions Connection with algebraic rings Chaptet' II. Infinite joins and meets 18. Definition Algebraic properties of infinite joins and meets. (m, n)-distributivity m-complete Boolean algebras m-ideals and m-filters. Quotient algebras m-homomorphisms. The interpretation in Stone spaces m-subalgebras Representations by m-fields of sets Complete Boolean algebras The field of all subsets of a set The field of all Borel subsets of a metric space Representation of quotient algebras as fields of sets A fundamental representation theorem for Boolean a-algebras. m-representability Weak m-distributivity Free Boolean m-algebras Homomorphisms induced by point mappings , Theorems on extension of homomorphisms Theorems on extending to homomorphisms Completions and m-completions Extensions of Boolean algebras m-independent subalgebras. The field m-product Boolean (m, n)-products

8 x Contents Appendix 39. Relation to other algebras Applications to mathematical logic. Classical calculi Topology in Boolean algebras. Applications to non-classical logic Applications to measure theory Measurable functions and real homomorphisms Measurable functions. Reduction to continuous functions Applications to functional analysis Applications to foundations of the theory of probability Problems of effectivity 210 Bibliography. List of symbols Author Index. Subject Index

9 Terminology and notation Capital latin letters are used to denote sets of points and their Boolean analogue, elements of Boolean algebras. Capital gothic letters denote classes of sets and their Boolean analogue, sets of elements of Boolean algebras (except for filters and ideals). In particular, Q( and Q3 (with indices, if necessary) always denote Boolean algebras or fields of sets. The letter 'J always denotes a field of sets. The symbol "v" is used both for the set-theoretical union and for the more general notion of Boolean join. In most cases, if both interpretations of "v" are possible, they coincide. In the opposite case, either it is explicitly stated, or it is evident from the text how the symbol "v" should actually be interpreted. The same remarks hold for the dual symbol "n" used both for the set-theoretical intersection and for the more general notion of Boolean meet. The same is true for the symbols " U " and" n " of the corresponding int.nite operations (see also notation on p for infinite Boolean joins and meets) and for the symbol "-" of complementation and the symbol" C " of inclusion. The empty set is denoted by /1, and so is its Boolean analogue, the zero element. The dual notion, the unit element in a Boolean algebra, is denoted by the dual symbol V. The letter LI denotes an ideal. The dual symbol 17 denotes a filter. Thus dual Boolean notions and operations are denoted by dual symbols. m always denotes an infinite cardinal. n denotes any (finite or infinite) non-zero cardinal (except when other hypotheses are explicitly stated). The cardinal of the set of all integers will be denoted both by Xo and a. The last notation will be used chiefly in expressions like "ameasure", "a-field", "a-algebra" etc. according to the generally adopted terminology. Sets of cardinality Xo are called enumerable or countable. Sets of greater power are called non-enumerable or uncountable. The cardinal of a set X is denoted by X. If we concentrate our investigation on subsets of a fixed set X, then X is often called a "space" (no additional structure of X is distinguished, unless it is explicitely stated). By a topological space we understand a set with a closure operation satisfying the well-known four axioms of KURATOWSKI (see p. 198). However in all cases (except, perhaps, 41) only Hausdorff spaces play an essential part. For any subset S of a topological space, CS and IS denote the closure and the interior of S, respectively. By an indexed set {AthET we shall understand a mapping which assigns, to every t E T, an element At. This notion should not be Ergebn. d. Mathern. N.F. Bd. 25, Sikorski, 2. Auf!.

10 2 Terminology and notation identified with the set of all At, t E T. This is essential, e.g., in 13, 16, 36 and 38 where indexed sets of Boolean algebras are examined. This is not essential in many other cases, e.g. when joins and meets of indexed sets of elements of a Boolean algebra are examined (Chapter II). The following abbreviation will be useful, especially in Chapter II: an indexed set {At}tET will be called an m-indexed set if T ~ m. The same terminology will be as~umed for d~ubly indexed sets; {A t,s}tet,8es is called an m-indexed set if f ~ m and 5 ~ m and it is called an (m, n)- = = indexed set if T ~ m and S ~ n. If Sand T are non-empty sets, then ST will denote the set of all mappings of T into S. If test and g E TU, then tg denotes the composite mapping given by tg(u) = t(g(u)) for u E U. If T' C T and test, then tit' is the mapping t restricted to T'. Formulas and examples are quoted by giving only their numbers if they are in the same section. Otherwise the number of the section is added.