Theory of Charges: A Study of Finitely Additive Measures

321 Pages·1983·3.94 MB·English

by  K.P.S. Bhaskara Rao and M. Bhaskara Rao (Eds.)

This is a volume in
PURE AND APPLIED MATHEMATICS
A Series of Monographs and Textbooks
Editors: SAMUEELI LENBERAGND HYMANBA SS
A list of recent titles in this series appears at the end of this volume.
Theory of Charges
A Study of Finitely Additive
Measures
K. P. S. Bhaskara Rao
Indian Statistical Institute, Calcutta, India
M. Bhaskara Rao
University of Sheffield, UK
1983
Academic Press
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British Library Cataloguing in Publication Data
Bhaskara Rao, K.P.S.
Theory of Charges-(Pure and Applied Mathematics)
1. Algebraic Topology
I. Title 11. Bhaskara Rao, M. 111. Series
514' 2 QA612
ISBN 0-12-095780-9
Typeset and printed by J. W. Arrowsmith Ltd
Foreword
Many years ago, S. Bochner remarked to me that, contrary to popular
mathematical opinion, finitely additive measures were more interesting,
more difficult to handle, and perhaps more important than countably
additive ones. At that time, I held the popular point of view, but since
then I have come around to Bochner’s opinion. Apparently, many other
mathematicians have also done so, as is indicated by the large number of
papers listed in the bibliography of this book. I, for one, had not realized
how much research had been done on finitely additive measures, at least
partly because the material is scattered in isolated papers. The authors
have done the mathematical community a real service by providing easy
access to this research (to which they themselves have made significant
contributions). This service is all the greater in that not only is the material
that they cover interesting in itself, but the presentation is very clear and
is enlivened with many illustrative examples. Two especially valuable
features of their work are an annotated bibliography and a section of notes
and comments. But perhaps the most valuable feature of the work to the
working measure theorist or functional analyst lies in exhibiting clearly
where countable additivity of a measure is used, and what can and what
cannot be done without it. Roughly speaking, without countable additivity
most of the measure theoretic examples are “counter”, but a great deal of
functional analysis can be done-with more work!
No one book in an area as large as this can do justice to all the material
that deserves coverage, and I certainly do not blame the authors for omitting
or treating too briefly some topics which I think are important. I only regret
the necessity.
I understand that the authors expect to write a book on finitely additive
probability also, and perhaps they will include some of the topics so omitted.
In any case, I look forward to seeing a continuation of the excellent work
they have done in this book.
December 1982 Dorothy Maharam Stone
University of Rochester
Rochester, New York
According to S. Bochner, finitely additive measures are more interesting,
and perhaps more important, than countably additive ones (see Maharam
(1976)).F initely additive measures arise quite naturally in many areas of
analysis. Over the years, there has been a sustained growth of activity in
finitely additive measures propelled by mathematicians and statisticians.
The case for finitely additive probability is put forward strongly by Dubins
and Savage (1965) in their book “How to Gamble If You Must”. They
refer to de Finetti, who, in a large number of papers published as early as
1930, “has always insisted that countable additivity is not an integral part
of the probability concept but is rather in the nature of a regularity
hypothesis.” In fact, Dubins and Savage “view countably additive measures
much as one views analytic functions-as a particularly important special
case.” But not much attention is paid to finitely additive measures in
text-books on Measure Theory. (Books on Functional Analysis do a bit
better.) One reason could be that countably additive measures are more
tractable than finitely additive ones.
A need was felt to have a book on finitely additive measures which could
serve as a reference book as well as a text-book. Cultivation of our interest
in finitely additive measures started ten years ago. Our sustained interest
in this area over the years led us to write this book.
In this book we have made an attempt to present a systematic and
detailed study of finitely additive measures as we understand them, filling
in any gaps that we discerned. This study of finitely additive measures as
a mathematical object, in many of its manifestations, is like a study that a
botanist would carry on a particular plant, or that a zoologist would launch
on a particular species of mammals, or that a sociologist would initiate
about a certain tribe, delving deep into various facets of the subject of
interest. We look at the finitely additive measure (i) as a single entity
(extension, nonatomicity and purity); (ii) in relation to another of its own
kind (absolute continuity and singularity); (iii) in an introspective mood
(decomposition theorems); (iv) as a member of a community (Nikodym
theorem and Vitali-Hahn-Saks theorem); (v) as a member of a community
in motion (weak convergence); (vi) in interaction with objects of different
...
Vlll PREFACE
kind (integration); (vii) in association with related external communities
(Vp-s paces); (viii) and its behaviour in external environment (range); (ix)
in its internal environment (lifting).
Measure Theory (The Study of Countably Additive Measures) is an
integral part of this wider study and the contrast between finite additivity
and countable additivity is brought into sharp focus at various junctures
in this work.
This book contains a good number of examples illustrating various aspects
of finitely additive measures. A special feature of this book is the Selected
Annotated Bibliography provided at the end of the book listing research
papers we have come across in our pursuit of finitely additive measures.
We hope that this book serves practising analysts well and stimulates
further research.
K.P.S. Bhaskara Rao gave a series of lectures on some of the topics
covered in this book at the University of Lecce (Italy) in 1980 and at the
University of Naples in 1981. He acknowledges gratefully the help given
by these universities in making the visits possible. We also thank the Indian
Statistical Institute for rendering help in making reciprocal visits of the
authors possible in connection with this work.
Finally, a word of appreciation and gratitude to Surekha for her monu-
mental patience in putting up with one of the most taxing and demanding
spouses while this work was in progress. We also thank B. R. Marepalli
for typing the entire manuscript.
December 1982 K.P.S. Bhaskara Rao
Calcutta
M. Bhaskara Rao
Sheffield
CHAPTER 1
Preliminaries
The only prerequisite that is needed for understanding a substantial part
of this book is a knowledge of Real Analysis, Set Theory and General
Topology at a rudimentary level. The purpose of this chapter is to collect,
in succinct form, various basic notions and results that are needed in this
book. Section 1.1 presents various classes of sets and their properties.
Section 1.2 briefly touches on some notions from Set Theory. Section 1.3
makes a sojourn with General Topology. Section 1.4 briefly dwells on
Boolean Algebras. Finally, Section 1.5 presents vector lattices in some
detail adequate for our needs.
A word of advice; before entering the terrain of finitely additive measures,
the reader is urged to ensure a good degree of familiarity with the concepts
presented in this chapter.
1.1 CLASSES OF SETS
Various types of classes of sets are presented in this section. The most
important concept is the field of subsets of a set. This collection is, usually,
the domain of definition for finitely additive measures.
Throughout this book, R is always understood to be a non-empty set.
The set theoretic operations we use are standard. For the reader’s con-
venience, a list is appended at the end of this book.
1.1.1 Definitions. Let fl be a set and 8 a collection of subsets of R.
(1). 9 is said to be a lattice on R if the following conditions are satisfied.
(i). A, B E 9 3 A uB E g.
(ii). A , B E ~ ~ A ~ B E ~ .
(2). $is said to be a semi-ring on fl if the following conditions are satisfied.
(i). 0 E 9.
(ii). A,Bc93AnB€$.
(iii). If A, BE$ and AcB, then there exists a finite number
Ao, A1,. . . , A, of sets in $such that A=AocA1c Azc -* -cA , =B and
Ai -A~-I~ 8 f oi r= 1,2,. . . , n.
2 THEORY OF CHARGES
(3). 9 is said to be a semi-field on R if 9 is a semi-ring and R E 9.
(4). 9i s said to be a ring on R if the following conditions are satisfied.
(i). 0 EF.
(ii). A , B E ~ ~ A u B E ~ .
(iii). A, B E 9+ A- B E 9.
(5). 9i s said to be a field on R if 9 is a ring and RE9 .
(6). 9 is said to be an additive-class on R if the following conditions are
satisfied.
(i). 0 E 9.
(ii). A, B E 9 and A n B = 0+ Au B E 9.
(iii). A E 41I$A ' E 9.
(7). 9 is said to be a cr-ring on R if the following conditions are satisfied.
(i). 0 €9.
(ii). {A,,; n h 1)c 93UnzAl ,, E 9.
(iii). A, B E 9 3 A - B E 9.
(8). 9 is said to be a cr-field on R if 9 is a cr-ring on R and R E 9.
(9). 9 is said to be a a-class on R if the following conditions are
satisfied.
(i). 0 €9.
(ii). If A,, n 2 1 is a sequence of pairwise disjoint sets in 9,th en
UnzlA n E 9.
(iii). A E ~ + A ~ E ~ ,
One could form a comprehensive picture of the interrelations between
various types of classes introduced above. We shall not go into details. We
present some important ones in the following.
1.1.2 Remarks.
(1). A lattice of sets need not be a semi-ring.
(2). A semi-ring need not be a lattice.
(3). Every ring is a semi-ring.
(4). Every ring is a lattice.
(5). Every field is a ring.
(6). Every field is a semi-field.
(7). Every field is an additive-class.
(8). Every a-field is a field.
(9). Every cr-field is a cr-ring.
(10). Every a-field is a a-class.
The above statements can be easily verified. The converse of any of the
implications in (3) to (10)d oes not hold. Examples can easily be constructed.
A sample of examples to illustrate some of the definitions is presented
here. Many more are to come later.
1. PRELIMINARIES 3
1.1.3 Examples.
(1). Let R be any infinite set. A set A cR is said to be cofinite if A' is a
finite subset of R. Let 9 be the collection of all finite and cofinite subsets
of R. Then 9i s a field on R, but not a cr-field on a.
(2). Let R = [0, 1) and %' ={[a,6 );0 5 a Ib I1 ). Then %' is a semi-field
on R.
{uY=l
(3). Let R = [0,1) and 9= [ai,b i);[ ai,b i)n[ aj,b j)= 0 for all i # j,
0 5 ai Ib i I1 for all i, n 2 1). Then 9 is a field on R. 9 is precisely the
collection of all those subsets of R each of which is a finite disjoint union
of sets from %' of (2) above. It will follow from Theorem 1.1.9(2) that 9
is precisely the smallest field on R containing %'.
(4). Let P(n)d enote the class of all subsets of R. (P(R) is called the power
set of 0.)T hen S(R) is an example of each type of class presented in
Definition 1.1.1. P(R) is also called discrete field or discrete cr-field on a.
In the following, we give salient features of some important types of
classes presented in Definition 1.1.1. These are not hard to discern.
1.1.4 Properties.
u:='=,n y='=,
(1). If 9i s a ring on R, then Ai and Ai E 9for any finite number
AI,A Z, . . ,A , of sets in 9,i.e . 9 is closed under finite unions and finite
,
intersections.
(2). If 9 is a ring on R, then 9 is closed under symmetric differences, i.e.
A AB = (A - B) u (B -A) E 9w henever A, B E 9.
(3). If 9 is a semi-ring on R and is closed under finite disjoint unions,
then 9i s a ring on R.
(4). If 9i s an additive-class on R, then 9i s closed under proper differences,
i.e. A-BE 9w henever A, BE^ and BcA .
(5). If 9 is an additive-class on R and is closed under finite intersections
or differences, then 9i s a field on R.
(6). If 9 is a a-ring on R, then 9 is closed under countable intersections,
n,,,
i.e. A, E 9 whenever A,, n 2 1 is a sequence of sets in 9.
(7). If 9 is a cr-class on R and is closed under finite intersections or
differences, then 9 is a c-field on R.
A given collection %? of subsets of a set R may not be of a particular
type P listed in Definition 1.1.1. It is natural to enquire about the existence
of a smallest collection 9 of subsets of l2 of the type P containing %'T. he
following results are designed to answer this query.
1.1.5 Lemma. Let R be any set and P be any of the types listed in Definition
r
1.1.1 with the exception of P being a semi-ring or a semi-field. Let ga,a E
be a family of collections of subsets of R such that each ga is of type P.
naEr
Then gai s of type P.
4 THEORY OF CHARGES
Proof. The proof easily follows from the definitions of each type. 0
1.1.6 Remark. The following example explains why we have made excep-
tions of certain types of classes of sets in the above lemma. Let
Q = {1,2,3,41,
={0(1,1, (21, {I3219 (3,413 a1
9 1
and
9 2= 10,(1 1, {2,3,4), 0).
91an d g2a re semi-fields on R but
% = 91n 92 = {0,{ l),R } is not a semi-field on R.
1.1.7 Theorem. Let R be any set and P be any of the types listed in
Definition 1.1.1 with the exception of P being a semi-ring or a semi-field.
Let % be a collection of subsets of R. Then there exists a smallest collection
$of subsets of R of type P containing %.
r
Proof. Let 9=a, E be the family of all collections of subsets of R each
of which is of type P and contains %. rnfa 0r s.rin ce 9(R) is of type P and
contains %. Then, by Lemma 1.1.5,9 = Paris of type P and contains
%. It is not hard to see that 9 is the desired collection of sets. 0
In the above, % is called a generator of 9 with respect to the type P or,
simply a generator of 9 if P is understood.
1.1.8 Remark. Let R={1, 2,3,4) and % ={0{,l), R}. Then there is no
smallest semi-field 9on R containing %. See Remark 1.1.6.
In some special cases, we can explicitly construct the smallest collection
9 of subsets of R of a given type containing a given collection % of subsets
of R. The following theorem gives some examples of such cases.
1.1.9 Theorem. Let R be any set.
(1). Let % be a lattice on R and 0 E %.Let 9={F-E; E, FE% and EcF).
Then 9 is the smallest semi-ring on R containing %.
(2). Let % be a semi-ring on R and 9= {ul=Cli C;1 , CZ,. . . , C, are pair-
wise disjoint sets in % and n ? 1). Then 9'is the smallest ring on R contain-
ing %. {uy=l
(3). Let % be a semi-field on R and 9 = Ci; CI, C2, . . . , C, are
pairwise disjoint sets in % and n 2 1). Then 9 is the smallest field on SZ
containing %.
(4). Let % be a ring on R. Let %I ={A c a;A 'E %}. Then 9 = % v %I is
the smallest field on R containing %.
(5). Let % be a u-ring on R. Let %I ={A c R; A' E %}. Then 9= % v %I is
the smallest u-field on R containing %.