Boolean Algebras in Analysis

613 Pages·2002·21.02 MB·English

by  D. A. Vladimirov (auth.)

Boolean Algebras in Analysis
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 540
Boolean Algebras
in Analysis
by
D.A. Vladimirov (t)
formerly of St. Petersburg State University,
St. Petersburg
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.l.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-5961-1 ISBN 978-94-017-0936-1 (eBook)
DOI 10.1007/978-94-017-0936-1
Printed on acid-free paper
Ali Rights Reserved
© 2002 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2002
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Contents
Foreword to the English 'Iranslation IX
Denis Artem'evich Vladimirov (1929-1994) Xl
Preface XVll
Introduction xix
Part I GENERAL THEORY OF BOOLEAN ALGEBRAS
o.
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 A utomorphisms 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
v
vi BOOLEAN ALGEBRAS IN ANALYSIS
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 (5!JC 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-Caratheodory 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
Contents vii
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
Appendices 562
Prereq uisites to Set Theory and General Topology 563
1 General remarks 563
2 Partially ordered sets 564
3 Topologies 565
Basics of Boolean Valued Analysis 569
1 General remarks 569
2 Boolean valued models 569
3 Principles of Boolean valued analysis 571
4 Ascending and descending 572
References
581
Index
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 represen
tative of the Russian mathematical school in functional analysis which
was founded by Leonid Vital' evich Kantorovich, a renowned mathemati
cian and a Nobel Prize winner in economics.
This school comprises two affiliations in St. Petersburg and Novosi
birsk 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 su
pervised; 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. Ta'lpale translated a few final sections and made the entire
book more readable. I. I. Bazhenov, I. I. Kozhanova, Yu. N. Lovyagin,
A. A. Samorodnitski'l, and Yu. V. Shergin helped me with the proofread
ing.
IX
x BOOLEAN ALGEBRAS IN ANALYSIS
The translation took much more time than planned: the reasons be
hind 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. There
fore, the battle against solecism and mistranslation was partly lost in
proofreading ...
Vladimirov was unhappy that he had no opportunity to include a chap
ter on Boolean valued analysis in this edition of his book. At the pub
lisher'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
Denis Artem'e vich Vladimirov
1929-1994
Vladimirov, the author of this book, died on August 2, 1994 after
a fatal illness.
Vladimirov was born on February 7, 1929 in St. Petersburg (the
former Leningrad). In the early 1940s he lived in the besieged city of
St. Petersburg and shared the heavy burdens and oppressions of war.
He left his reminiscences about the siege full of vivid observations and
personal recollections together with interesting facts about the routine
and realities of that time.
In 1950 he entered the Mechanics and Mathematics Department of
St. Petersburg State University and graduated from the university in
1955 to become an assistant of the chair of mathematical analysis in his
alma mater. His scientific preferences were formed under the influence
of L. V. Kantorovich, G. P. Akilov, and A. G. Pinsker. He was a dis
tinguished representative of the area of research in functional analysis
which was charted by 1. V. Kantorovich based on the concept of ordered
vector space.
The scientific interests of Vladimirov encircled not only the general
theory of ordered vector spaces but also measurable function spaces,
invariants of measurable functions under metric isomorphisms of their
domains of definition, the properties of integral operators, the theory
of Boolean algebras and measure theory as well as their applications to
general topology and probability.
The first publication of Vladimirov1 [1] is a small masterpiece. It
solves three difficult problems from the celebrated survey article of 1951
1 Within this introductory obituary, the references are cited in accordance with the list at the
end of the book.
Xl