Theory of Charges
This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: SAMUEL EILENBERG AND HYMAN BASS
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
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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
This Page Intentionally Left Blank
According to S. Bochner, finitely additive measures are more interesting, and perhaps more important, than countably additive ones (see Maharam (1976)).Finitely 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-spaces); (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 monumental 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
Contents
Foreword Preface
vii
CHAPTER 1 1.1 1.2 1.3 1.4 1.5
PRELIMINARIES Classes of sets Set theoretical concepts Topological concepts Boolean algebras Functional analytic concepts
1 1 13 15 18 23
CHAPTER 2 2.1 2.2 2.3 2.4 2.5 2.6
CHARGES Basic concepts The space of all bounded charges, b a ( Q 9 ) Measures The space of all bounded measures, ca(R, 9) Jordan Decomposition theorem Hahn Decomposition theorem
35 35 43 47 50 52 56
CHAPTER 3 3.1 3.2 3.3 3.4 3.5 3.6
EXTENSIONS OF CHARGES Real valued set functions and induced functionals Real partial charges and their extensions Extension procedure of Eos and Marczewski Extension of partial charges in the general case Miscellaneous extensions Common extensions
58 58 64 70 76 78 82
CHAPTER 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7
INTEGRATION Total variation and outer charges Null sets and null functions Hazy convergence D-integral S-integral L,- spaces b a ( O , 9 ) as a dual space
85 85 87 92 96 115 121 133
CHAPTER 5 5.1 5.2 5.3 5.4
NONATOMIC CHARGES Basic concepts Sobczyk-Hammer Decomposition theorem Existence of nonatomic charges Denseness
141 141 144 150 156
V
CONTENTS
X
CHAPTER 6 6.1 6.2 6.3
ABSOLUTE CONTINUITY Absolute continuity and singularity Lebesgue Decomposition theorem Radon-Nikodym theorem
159 159 166 169
CHAPTER 7 7.1 7.2 7.3 7.4 7.5
V,,-SPACES L,- spaces-An overview V,- spaces Duals of V,- spaces Strong Convergence Weak Convergence
178 178 185 193 197 200
CHAPTER 8 NIKODYM THEOREM, WEAK CONVERGENCE AND VITALI-HAHNSAKS THEOREM 8.1 Nikodym and Vitali-Hahn-Saks theorems in the classical case 8.2 Examples 8.3 Phillips' lemma 8.4 Nikodym theorem 8.5 Norm bounded sets in the presence of uniform absolute continuity 8.6 A decomposition theorem 8.7 Weak convergence 8.8 Vitali-Hahn-Saks theorem
203 204 205 206 209 213 216 218 226
CHAPTER 9 THE DUAL OF b a ( 4 9 ) AND THE REFINEMENT 23 1 INTEGRAL 23 1 9.1 Refinement integral 234 9.2 The dual of ba(R, 9) CHAPTER 10 10.1 10.2 10.3 10.4 10.5
PURE CHARGES Definitions and properties A decomposition theorem Pure charges on cr- fields Examples Pure charges on Boolean algebras
240 240 24 1 243 244 246
CHAPTER 11 11.1 11.2 11.3 11.4 11.5
RANGES OF CHARGES Ranges of bounded charges on fields Ranges of charges on cr-fields Cardinalities of ranges of charges Charges with closed range Charges whose ranges are neither Lebesgue measurable nor have the property of Bake
249 249 252 256 257 264
CHAPTER 12 ON LIFI'ING Appendix 1: Notes and Comments Appendix 2: Selected Annotated Bibliography Appendix 3: Some Set Theoretic Nomenclature Index of Symbols and Function Spaces Subject Index
268 272 282 305 306 309
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 o n 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 convenience, 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). 9is said to be a lattice on R if the following conditions are satisfied. (i). A, B E 9 3 A u B 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 , B c 9 3 A n B € $ . (iii). If A, B E $ and A c B , then there exists a finite number Ao, A1,. . . ,A, of sets in $such that A = A o c A 1 c A z c -* - cA, = B and Ai - A ~ - I~ 8 f o i r= 1 , 2 , . . . ,n.
2
THEORY OF CHARGES
(3). 9is said to be a semi-field on R if 9 is a semi-ring and R E 9. (4). 9is said to be a ring on R if the following conditions are satisfied. (i). 0 EF. (ii). A , B E ~ ~ A u B E ~ . (iii). A, B E 9+A- B E 9. (5). 9is said to be a field on R if 9is a ring and R E 9. ( 6 ) . 9 is said to be an additive-class on R if the following conditions are satisfied. (i). 0 E 9. (ii). A, B E 9and A nB = 0+ A u B E 9. (iii). A E 41I$A' E 9. (7). 9is said to be a cr-ring on R if the following conditions are satisfied. (i). 0 €9. (ii). {A,,; n h 1)c 9 3 U n zA,, l E 9. (iii). A, B E 9 3 A - B E 9. (8). 9 is said to be a cr-field on R if 9is 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, then
U n z l A 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)does 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 c R 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 9is 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. ( 3 ) . Let R = [0,1) and 9= [ai,bi);[ai,b i ) n [aj,b j )= 0 for all i # j , 0 5 ai Ibi 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)denote the class of all subsets of R. (P(R) is called the power set of 0.)Then 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.
{uY=l
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. (1). If 9is a ring on R, then Ai and Ai E 9 f o r any finite number AI,AZ,, . . ,A, of sets in 9, i.e. 9is closed under finite unions and finite intersections. (2). If 9is a ring on R, then 9is closed under symmetric differences, i.e. A AB = (A - B) u (B -A) E 9whenever A, B E 9. ( 3 ) . If 9 is a semi-ring on R and is closed under finite disjoint unions, then 9is a ring on R. (4).If 9is an additive-class on R, then 9is closed under proper differences, i.e. A - B E 9whenever A, BE^ and B c A. ( 5 ) . If 9 is an additive-class on R and is closed under finite intersections or differences, then 9is a field on R. (6). If 9 is a a-ring on R, then 9 is closed under countable intersections, A, E 9 whenever A,, n 2 1 is a sequence of sets in 9. i.e. (7). If 9 is a cr-class on R and is closed under finite intersections or differences, then 9is a c-field on R.
u:='=,ny='=,
n,,,
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 %'.The 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 1.1.1 with the exception of P being a semi-ring or a semi-field. Let ga, a Er be a family of collections of subsets of R such that each ga is of type P. Then gais of type P.
naEr
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 exceptions of certain types of classes of sets in the above lemma. Let Q = {1,2,3,41, 9 1
={0, (11, (21, {I3219 (3,413 a1
and 9 2=
10, (11, {2,3,4), 0).
91 and g2are semi-fields on R but % = 91 n92 = {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 %. Proof. Let 9=, a E r be the family of all collections of subsets of R each of which is of type P and contains %. r f 0 since 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 9is the desired collection of sets. 0
nar.r
In the above, % is called a generator of 9with 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 9 o n R containing %. See Remark 1.1.6. In some special cases, we can explicitly construct the smallest collection 9of 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, F E %and E c F ) . Then 9 is the smallest semi-ring on R containing %. (2). Let % be a semi-ring on R and 9= {ul=l C i C1, ; CZ,. . . , C, are pairwise disjoint sets in % and n ? 1). Then 9 ' i s the smallest ring on R containing %. (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 %.
{uy=l
1. PRELIMINARIES
5
Proof. (1). Let F1 -El and FZ-Ez E 9, where El, F1, E2,F2 E %, EI c F1 and El c F2. Then (F1-El) n(F2-Ez) = (FInFz)- (EluE2)n(F1nF2)E 9. So, 9 is closed under finite intersections. Let F1 -El, F2-Ez E 9 and F1-El c F2- E2, where El, F1, E2, F2E %, El c F1 and EZc Fz. Let C = F1 -El c C c Fz -E2, C- (F1-El) E 9 (F1nF2)- (FInE2). Then C E 9, and (FZ-E2)-C€ 9. Hence 9 is a semi-ring on R. Since 0 E %?, %? ~ 9 . It is not hard to see that 9is indeed the smallest semi-ring on Cl containing %. (2). From the definition of 9, it is clear that 9 is closed under finite disjoint unions. We show that 9is closed under finite intersections. let U:=I Ci E 9 and D j E 9, where C1, C2, . . . , Cm are pairwise disjoint sets in % and D1, D2, * ,D, are pairwise disjoint sets in %. Then
u,”=l
since Ci n D , i = 1 ,2 , . . . , m and j = 1 ,2 , . . . , n are pairwise disjoint sets in %’. Now, we show that F is closed under differences. Suppose E, F E % and E c F . Then there exist Eo,El,. .. , E n in % such that E = E ~ ~ E ~ ~ . . . ~ E n = F a n d E i - E for i - 1e v€e% r y i = l , 2 , . . . , n. So, FE= (Ei -EiPl)E 9, since Ei -Ei-l’s are pairwise disjoint. Let Ci Dj be any two sets in 9, where C1, Cz, . . . , C, are pairwise and disjoint sets from % and D1, Dz, . . . ,D, are pairwise disjoint sets from %. Note that
uYzl
ur=~ uy=l
n
=
m
u n (Dj- (Ci nDj)).
j=1 i=l
From what we have proved above, D j - (Ci n Dj) E 9for every i and j . Since 9is closed under finite intersections, (Dj - (CinDj))E 9 for every j . Since 9is closed under finite disjoint unions,
nk
u n(Dj-(CinDj))E9. n
m
j-1
i=]
If we show that 9 is closed under finite unions, it would imply that 9 is aringonn. LetA,BEF.ThenAuB=Au(B-A). SinceB-AE9and A and B -A are disjoint, it follows that A u B E 9. It is obvious that % c 9. It is not hard to see that 9is indeed the smallest ring on R containing %. (3). This is similar to (2). (4).We show that 9 is a field on R. It is obvious that 9 is closed under complementation. Let E, F E 9.Case (i). E, F E %. Then E u F E % c F.
6
THEORY OF CHARGES
Case (ii). E E % and F E % ~Then . E u F E % : , . For, (EuF)'=E'nF'= F - E E %. Case (iii). E E %I and F E%. This case is similar to Case (ii). Case (iv). E , F E % l . Then E u F E Y 1 . For, (EuF)"=E'nF'E%. In any Thus 9is a field on R containing %. It is obvious case, we have E u F E 9. that 9is the smallest field on R containing %. ( 5 ) . This can be proved as in (4). 0 Next, we describe a constructive procedure for obtaining the smallest field 9 on R containing a given class % of subsets of R in a finite number of steps. First, we introduce a special notation.
1.1.10 Notation. For any subset A of R, let A' = A and A'
= A'.
1.1.11 Theorem. Let % be any class of subsets of a set 0. Form the following classes of sets successively. q1= ( 0 ,R}u % u{Ac R; A'E %}.
W2= The collection of all subsets of R each of which is a finite intersection of sets from g1.
=(AAi;
I.
for i = 1 , 2 , . . . ,n and n 21
i=l
q3= The collection of all subsets of R each of which is a finite disjoint union of sets from %2. =
{ 6Bj; Bj€%2for all j , Bi n B j
= 0 for all
i Z j a n d m rl
j=1
I.
Then V3 is the smallest field on R containing %, Proof. We note the following obvious facts. (i). % C % ~ C % : Z C % ~ . (ii). 0 E Z3. (iii). V1 is closed under complementation, i.e. if A E q1,then A'E V1.(iv). "2 is closed under finite intersections. (v). %3 is closed under finite disjoint unions. (vi). If %3 is a field on R, then it is indeed the smallest field on 0 containing %. We show that g3is a field on R. This is carried out in the following steps. Step 1. Note that if a collection 9 of subsets of a set fl is closed under complementation and finite intersections, then 9 i s a field on R. (If A, B E 9, then A u B = (A"nB")' E 9.) Step 2. It is clear that Z3 is closed under finite intersections in view of the fact that Z2is closed under finite intersections. Step 3. Let A E Z2.We claim that A'E q3.We can write A = fy=l Ai for some A1,A2,. . . ,A, in '%I. Note that A' = A? nA > n* nA:n, where the union is taken over all S1,S2,. . . ,a, in (0,l) with the exception that
u
--
1, PRELIMINARIES
7
(S1,S2,..., & ) = ( l , l , ..., 1). Each A ? ' n A 9 n . - . n A > belongs to Wz and these sets are pairwise disjoint. So, A"EW3. Step 4. As a final step, let B E W3. Then B = u y = l Bi for some pairwise B:. Then each B:E q3, disjoint sets B1, B2, . . . ,B, in Wz.Then B" = by Step 3. By Step 2, B'E W3. This completes the proof. 0
ny=l
We obtain some important consequences of this result.
1.1.12 Corollary. Let W be any class of subsets of a set R and 9 the smallest field on R containing %. Then the following statements are true. (1). A E9 if and only if there exist sets A,, j = 1,2, . . . ,ni and i = 1, 2, . . . ,m such that each A, or AGE%? and
u n A,. m
A=
i=l j = l
(2). A E $if and only if there exist sets Bii,j such that each Bii or B;E %? and
. . . ,kiand i = 1 ,2 , . . . ,n,
k,
n u Bib n
A=
= 1 ,2 ,
j=l
j=1
Proof. (1). This follows from Theorem 1.1.11. (2). This follows from (1) and the distributive laws of the operations u and n.
1.1.13 Corollary. Let 9 be a field of subsets of a set R and A c a.Then the smallest field 9 1 on R containing 9and A is given by 9 1 = {(B
nA) u (Bz n A"); B 1, B2 E s}.
Proof. This follows from Corollary 1.1.12(1). (One can also show that is a field on R directly.)
9 1
1.1.14 Corollary. Let % be any countable collection of subsets of a set R. Then the smallest field 9on R containing %? is also countable. Proof. This follows from Theorem 1.1.11. (Note that each q iconstructed in the proof of Theorem 1.1.11 is countable.) 0 1.1.15 Remark. Given a class %? of subsets of a set R, there is no simple way of constructing the smallest cr-field on R containing Y.The collection of all countable unions of sets each of which either belongs to % or its complement belongs to % need not be a cr-field on 0. Now, we introduce the notion of an atom of a field.
1.1.16 Definitions. Let 9 be a field of subsets of a set R.
8
THEORY OF CHARGES
(1). A set A in 9 is said to be an atom of 9 if the following conditions are satisfied. (i). A f 0 . (ii). B E $ , B c A J B = 0 or B = A . (2). 9 is said to be atomic if every non-empty set in 9 contains an atom of 9. (3). 9 is said to be nonatomic if 9has no atoms. An atom A of 9 is, intuitively, a minimal non-empty element of 9. If A and B are atoms of 9, then either A = B or A n B = 0. The following remarks give some more information about these notions.
1.1.17 Remarks. (1). If R is the union of all atoms of 9, then 9 is atomic. But the converse is not true. As an example, let R = {1,2,3, . . . , CO} and 9 be the collection of all finite subsets of {1,2,3, . . .} and their complements. Then 9is atomic. But the union of all atoms of 9 = {1,2,3, . . .} # R. (2). If 9 is a finite field on R, then 9 is atomic. For each w in R, let A, =n,.,,,A. Then A, is an atom of 9 containing w . (3). For any set R, P(R) is atomic. (4). The field given in Example 1.1.3(3) is nonatomic. If %' is a finite collection of subsets of a set R, then the smallest field 9 on R containing W can be described in a simple way. The following proposition amplifies this point.
1.1.18 Proposition. (1). If % = {Al, Az, . . . , A,} is a finite partition of R, i.e. Ai n A j = 0 for Ai = R and A, # 0 for every i, then the smallest fieZd 9 on all i # j , R containing W is the collection of all possible unions of sets from W. The atoms of 9 are precisely A1, Az, . . . , A,,. (2). If W ={A1, AZ,. . . , A,} is any finite collection of subsets of a set R, then the smallest field 9 on R containing %' is the collection of all possible unions of sets from g1= {AY1n Ag'n * n A:-; S1,Sz, . . . ,S,, E (0, l}}.The atoms of 9are precisely the non-empty sets in %'I. (3). I f 9= {B1,Bz, . . . , B,,,} is a finite field on a set R, then the atoms of S u r e the non-empty sets from
uy=,
-
{BY'nBgZn...nB6,,;S1,S2 , . . . , S,E{O,l}}. (4). If 9 is a finite field on R, then the number of sets in 9 is 2k for some kzl.
The above statements are easy to check and the details are left to the reader.
1.
PRELIMINARIES
9
Now, we collate the two notions, an additive-class and a field, in relation to generators. The following results are in this direction.
1.1.19 Theorem. Let % be a class of subsets of a set 0, gothe smallest additive-class on R containing %' and 9 1 the smallest field on R containing %. Suppose V has one of the following properties. (i). A, B E % A nB E 9 0 . (ii). A, B E% + A - B E 90. Then 90= 91.
+
Proof. Since Sois the smallest additive-class containing % and S1is an additive-class containing %, it follows that g o c $1. If we show that 90 is a field, it would then follow that g1c goand hence go= gl. For this, it suffices to show that gois closed under finite intersections or differences. See Property 1.1.4(5). Suppose (i) holds. For each A in So,define . F A = {B E 9 0 ;A n B E go}. We show that is an additive-class. Clearly, @ € % A . If B E ~ A then , A n B c = A - B = A - ( A n B ) E s o as gois closed under proper differences. See Property 1.1.4(4).So, B'E $A. It is easy to check that g A is closed under finite disjoint unions. Thus F A is an additiveclass contained in 9-0. If A E %, then % c g A , by (if. Therefore, g A = g o . Now, if A E So,even then % c 9 A . For, if C E %, then .?Fc = 9 0 and SO, A nCE or C E F A . Consequently, FA = $0 for any A in F 0 . This implies that gois closed under finite intersections. One can show that the properties (i) and (ii) are equivalent using Property 1.1.4(4). This completes the proof. 0 An easy consequence of the above result is the following observation. If %' is a class of subsets of a set R closed under finite intersections or differences, then the smallest additive-class on R containing %? and the smallest field on R containing %? are identical. The following results are in the spirit of Theorem 1.1.19but in the setting of a-classes and a-fields.
1.1.20 Theorem. Let % be a class of subsets of a set R and gothe smallest a-class on R containing %. Suppose V has one of the following properties (which are equivalent anyway). (i). A, B E % + A n B E g o . (ii). A , B E V + A - B E ~ O . 0 Let glbe the smallest a-field on R containing V.Then 9 0 = F1. 1.1.21 Corollary. Let %' be a collection of subsets of a set R closed under finite intersections or differences. Let FO be the smallest a-class on R contain ing %' and gl the smallest a-field on R containing V.Then g o = $1. 0
10
THEORY OF CHARGES
A proof of Theorem 1.1.20 can be given by a slight modification of the proof of Theorem 1.1.19. Corollary 1.1.21 follows easily from Theorem 1.1.20. Next, we introduce two very useful notions, namely “Ideals and Filters” which are complementary to each other.
1.1.22 Definitions. Let 9be a field of subsets of a set a. (1). 4 c 9is said to be an ideal in 9 if the following conditions are satisfied. (i). R g 4 . (ii). A , B E ~ + A U B E $ . (iii). A € $ , B c 9 , B c A J B e 4 . An ideal 4 in 9 is said to be a maximal ideal in 9 if there is no ideal in 9properly containing 4. (2). 8;c 9is said to be afilter in 9if the following conditions are satisfied. (i). 0 g8;. (ii). A, Bc$+AnBEB;. (iii). A€$, BE^, A ~ B J B E ~ ; . A filter 8; in 9is said to be a maximal filter in 9 if there is no filter in 9 properly containing 8;.
1.1.23 Remarks. The following statements follow from the above definitions. (1). If 4 is an ideal in 9, then 8;={A E 9; A‘ E 4)is a filter in 9. If 4 is then the filter 8; defined above is a maximal filter a maximal ideal in 9, in 9. (2). If R = {1,2,3, . . .} and 4={A c R; A is finite}, then 4 is an ideal in 9= s ( R ) . 4 is not a maximal ideal in 9. (3). If %’ c 9 has the property that Ci# R for any finite number C1, Cz, . . . ,C, of sets in %’, then there exists a smallest ideal 4 in 9 containing %’.In fact, 4 is given by
uy=l
6 Ci for some C1, C2,. . . ,C, in %‘I. (4). If %’ c 9 has the property that n;=, C i# 0 for any finite number A E ~A ;c
i=l
C1, C2,. . . ,C, of sets in %’,then there exists a smallest filter 8; in 9 containing %’.In fact, 8; is given by
8;={A€$;
h Ci = A for some C1, C2,. . . ,C, in
i=l
%‘I.
(5). Let 9 be an ideal in 9 and A E 9. Suppose A u B # R for any B in 4. Then 9 1
={CE 9; C c A u B for some B in 4 )
1. PRELIMINARIES
11
is the smallest ideal in 9containing 4 and A. We call Y1the ideal generated by 4 and A. (6). Let 9 be a filter in 9 and D E 9. Suppose D nB # 0 for every B in 9.Then
91={CE9;DnBcCforsomeBin$} is the smallest filter containing 9 and D. We call the filter generated b y 2 and D. (7). An ideal 4 in 9is a maximal ideal in 9if and only if for every A in 9either A or A' E 4. (This follows from (5).) (8). A filter 9 in 9 is a maximal filter in 9 if and only if for every A in 9either A or A'€$. (This follows from (6).) (9). If 9l and $2 are two distinct maximal filters in 9, then there exists A in 9such that A ~9~ and A ' E ~ ~ . We define limit supremum and limit infimum of a sequence of sets and give some properties of these notions.
1.1.24 Definitions. Let A,, n 2 1 be a sequence of subsets of a set R. Define lim sup A,, =
n u Ak
nzl kzn
n+m
and lim inf A, n-m
=
u n Ak.
n z l kzn
1.1.25 Properties. (1). lirn sup,,oo A, = {w ; w E A, for infinitely many n's}. (2). lim inf,,, A, = {w ;w E A, for all but a finite number of n's}. (3). lim inf,,m A, c lim sup,,m A,. (4). (lim A,)" = lim inf,,m A:. (5). (lim inf,,m A,)" = lim A:. Now, we introduce some operations in the setting of real valued functions defined on a set. Let f and g be two real valued functions defined on a set R. f v g, f A g, f- and If1 are functions on CR defined by
a
(fv d ( w ) = max Mw),g(o)},
wE
( f A g>(w>= min {f(o>, g(w)},
w E Q,
f + = f vo,
f- = (-f)v 0
12
THEORY OF CHARGES
and
Iflb)= If(w>l,
CIJ
E a.
We give some identities involving these operations which actually stem from the corresponding properties of real numbers.
1.1.26 Identities. (1). f = f +-f-.
.
y-.
(2). If1 =f+ (3). f v g = d f + g + I f - d . (4). f g = 3 f + g - If-gl). ( 5 ) . f + g = (f v g>+ ( f g). ( 6 ) . If-g,' = (fv g ) - (f. g). (7). fg =4"f+g)2-(f-g)21. (8). I f l = f + v f - = f V (-f). (9). IIfl-lgll~If+gl~lJfl+lgl. (10). I(fl V gi) - (fz v g2)I 5 If1 -fzl+ lgl - gzI for any real valued functions f l , fz, g1, gz on a. (11). 1(f1 A 81) -(fz A gz)]5 If1 + Igl -gzI for any real valued functions f l , fz, g1, g2 on a. (12). - I f l - l ~ l ~ f ~ g ~ f v g ~ l f l + I g l .
-fzl
Finally, we end this section with some notes on measurable functions.
1.1.27 Definitions. (1). The BoreZ a-field on the real line R is defined to be the smallest a-field on R containing all intervals. (2). Let R be a set and % a c+-field on R. A real valued function f on R is said to be measurable with respect to % if f-'(B) = {w E R; f ( w) E B} E % for every Bore1 subset B of R. ( 3 ) . For any subset A of R, the indicator function I A of A is a map from to R given by I A ( w )= 1, if w E A, =0, if w €AC.
1.1.28 Properties, Let '% be a a-field of subsets of a set R. Then the following statements are true. (1). For A c R, la is measurable with respect to % if and only if A € % . (2). A real valued function f on R is measurable with respect to % if and only if f-'{[k, CO)} E % for every -a< k < a. ( 3 ) . If f , g are measurable functions with respect to 3 and c, d are real numbers, then cf + dg, I f l , fg, f2, f v g, f A g, f' and f- are all measurable functions with respect to %. (4). A real valued function f on R is a measurable function with respect to '% if and only if there exists a sequence fa, n I1 of functions on R such
1. PRELIMINARIES that f ( w ) = limn+mfn(oJ)for every w in
a, where each fn
13 is of the form
for some real numbers cnl, c n z , . . . ,C n k , and for some sets Anl,An2,.. . , A,k, from 3 .
1.2 SET THEORETICAL CONCEPTS
In this section, we present, in a concise form, some of the set theoretical notions we have used in this book. One of the basic concepts in set theory is the notion of a relation on a set X. A relation on X is any subset Z of X x X, where X x X is the Cartesian product of X with itself, i.e. X x X = {(x, y ) ; x, y E X}. A relation Z c X X X is said to be reflexive if (x,x) E Z for every x in X, symmetric if (x, y ) E Z whenever (y, x) E Z, antisymmetric if (x, y ) , ( y , x) E Z only when x = y, transitive if (x,z ) E Z whenever (x, y ) and (y, z ) E Z for some y in X. A partial order on a set X is any relation Z which is reflexive, antisymmetric and transitive. We usually denote a partial order Z c X X X by I, i.e. for x, y in X, x Iy if and only if (x,y ) E Z . In terms of the notation I, the axioms of a partial order can be restated as follows. (i). x Ix for every x in X. (ii). x Iy and y 5 x j x = y (antisymmetry). (iii). x 5 y and y Iz 3 x 12 (transitivity). The pair (X, I) is called a partially ordered set. A partial order Ion a set X is said to be a linear order on X if x I y or y 5 x for any x, y in X. In this case, we call (X, I) a linearly ordered set. Let (X, 5)be a partially ordered set. A subset C of X is called a chain in X if the partial order Irestricted to C is a linear order on C. A partial order Ion a set X is said to be a we@-ordering on X, if for every non-empty subset A of X, there exists an element a in A such that a IX for all x in A, i.e. a is the smallest element in A. In this case, the pair (X, I) is called a well-ordered set. A relation Z c X x X is said to direct X if the following are satisfied. (i). (x,x) E Z for every x in X (reflexivity). (ii). (x,z ) E Z whenever (x, y ) and (y, z ) E Z for some y in X (transitivity). (iii). For every x, y in X, there exists z in X such that ( z , x ) E Z and ( z , y ) Z.~ The relation Z c X x X satisfying (i), (ii) and (iii) above is usually denoted by L, i.e. x 2 y if and only if (x,y ) E Z. In terms of this notation I, the above three conditions can be restated as follows. The relation 2 on X is said to direct X if the following are satisfied. (i). x zx for every x in X. (ii). x L Z whenever x 2 y and y L Z for some y in X. (iii). For every x, y in X, there exists z in X such that z Ix and z L y. The pair (X, 2 ) is called a directed set.
14
THEORY OF CHARGES
A relation Z c X x X is said to be an equivalence relation if the following are satisfied. (i). ( x , x ) ~ Zfor every x in X (reflexivity). (ii). (x, Y ) E Z whenever ( y , x ) E Z (symmetry). (iii). (x, 2 ) E Z whenever (x, y ) and (y, z ) E Z for some y in X (transitivity). An equivalence relation on X is usually denoted by -. In terms of the notation -, the above conditions can be rephrased as follows. (i). x -x for every x in X. (ii). x - y if y -x. (iii). x - 2 whenever x - y and y -2. If is an equivalence relation on X, let [XI denote the equivalence class in X containing x €or any x in X, i.e. [ x ] = { y ~ X ; y - x } .For x , y in X, [ x ] = [ y ] or [ x ] n [ y ] = 0 .Under the on X, X thus can be written as the union of all its equivalence relation equivalence classes. On some occasions, we use cardinals and ordinals. The reader is advised to refer to Kamke (1950) or any other book on Set Theory. Transfinite induction is often used in inductive definitions, inductive constructions and in proofs. Let (X, I) be a well-ordered set. Let a. be the smallest element in X. For each x in X, let A(x) be a proposition.
-
-
1.2.1 Principle of Transfinite Induction. If ( i )A(ao)is valid, and (ii)for any a in X, A ( a ) is valid whenever A(x) is valid for every x < a (i.e. x Ia and x f a ) hold, then A(x)is valid for every x in X. The above principle is also adopted for inductive definitions and inductive constructions. We will be using Zorn's lemma in this book to exhibit objects with specified properties. This lemma which is equivalent to the Axiom of Choice is stated below.
1.2.2 Zorn's Lemma. Let (X, I) be a partially ordered set in which every chain C has an upper bound, i.e. there exists a in X such that x 5 a for every x in C. Then X has a maximal element ao, i.e. if a l E X and a o l a l , then al = ao. 1.2.3 A n Application of Zorn's Lemma. Let 9 be a field of subsets of a Then there exists a maximal ideal 4" in 9 set R and 9 an ideal in 9. containing 9. Existence of 9" can be established by Zorn's lemma as 92 follows. Let X be the collection of all ideals in 9containing 9.For 91, in X, say 9l ~9~if 9l~ 95 is~a partial . order on X. Let {,aa;CY E r}be E X and is an upper bound of the chain a chain in X. Then UaEr9a {9a; CY EF). Consequently, there exists a maximal element 9" in X. Similarly, if 3 is a filter in S, one can show that there exists a maximal filter 8;* in 9containing 3, using Zorn's lemma. Finally, we end this section with an important example of a directed set. Let 9be a field of subsets of a set a. A finite partition of 0 in 9is a finite
1. PRELIMINARIES
15
family P ={El, E2, . . . , Em}of pairwise disjoint sets in 9 whose union is a. Let B denote the collection of all finite partitions of R in 9. For P1, say PI L PZif every set in P1 is contained in some set of P2. Another PZin 9, expression that is commonly used in this case is that PI is a refinement of Pz. We claim that the relation 2 on B directs 8.It is obvious that P Z P for every P in 9. If P 1 2 P z and P22P3, then it is clear that P l z P 3 . If PI ={El, Ez, . . ,Em}E 9 and Pz = {F1,Fz, . . . ,F,} E 9, then P = {EinFj; 15 i Im, 15 j In }E 9 and P 2 P1, P L Pz. Hence the relation 2 directs 9. In other words, (9,2 ) is a directed set. If F E ~let, PFdenote the collection of all finite partitions of F in 9.Under the same relation 2 as above, (PF, 1)is also a directed set.
.
1.3 TOPOLOGICAL CONCEPTS In this section, we briefly review some of the topological concepts needed in the subsequent chapters. For any other unexplained terminology in the text which slipped inadvertently from our compilation here, the reader is advised to refer to Kelley (1955). A topological space is a pair (X, Y), where X is a set and Y is a collection of subsets of X closed under unions and finite intersections and contains 0 and X. Members of Y are called open sets and the complements of open sets are called closed sets. For any set A c X , the closure of A is defined to be the set {C; C closed, A c C } and is denoted by A. A is the smallest closed set containing A. Interior of A is defined to be the set {V; V open, V c A} and is denoted by A". A" is the largest open set contained in A. A subset A of X is said to be a G6-set if A is a countable intersection of open sets. A subset B of X is said to be an F,-set if B is a countable union of closed sets. A point x in X is an accumulation point of a subset A of X if (A-{x}) nU # 0 for every open set U containing x. A point x in A c X is an isolated point of A if there exists an open set U containing x such that A n U = {x}. is any function from a directed set A net in a topological space (X, 9) (D, L) into X. Nets are usually denoted by x,, a E D . A net x,, a E D is said to converge to an element x in X if given any open set V containing x, there exists a0 in D such that x, E V whenever a z a0.A subset A of X is closed if and only if x E A whenever there is a net in A converging to x. Let (D, L) be a directed set. Suppose for every a in D, there is a /3, in D such that Pa 2 a. If x,, a E D is a net in X, then xOa,a E D is called a subnet of x,, a E D. If a net x,, a E D is convergent in X, then every subnet of x,, a E D is also convergent and converges to the same limit.
n
u
16
THEORY OF CHARGES
A topological space (X, Y) is said to be a Hausdorf topological space if for every distinct x, y in X, there exist open sets V1 and V2 such that x E V1, y E V2 and V1 nV2 = 0. An open cover of X is any family of open is said to be compact if sets whose union is X. A topological space (X, 9) every open cover of X admits a finite subcover of X. A topological space (X, Y) is compact if and only if every net in X admits a convergent subnet in X. Equivalently, (X, Y) is compact if and only if for every collection of closed sets {A,; a! E r}with finite intersection property, i.e. A, # 0 for every finite subset rlof r, A, # 0. The notion of compactness of a subset A of X can also be introduced in the same way as above for X. If A is a closed subset of X and X is compact, then A is compact. If (X, 3)is any Hausdorff topological space, then any compact subset of X is closed. A subset A of X is said to be clopen if A is open as well as closed. A topological space'(X, Y) is said to be totally disconnected if the family of all clopen subsets of X forms a base for the topology of X, i.e. every open set is a union of clopen sets. Let X be any set. Let 8 be a family of subsets of X closed under finite intersections and containing X. Then there is a smallest topology on X for which 8 is a base. This smallest topology is precisely the collection of all unions of sets from 8. Let (X, 9) be a topological space. A real valued function f on X is said to be continuous if f-l(U) E T for every open subset U of the real line R, i.e. U is a union of open intervals. f is continuous if and only if f(x,), a! E D converges to f(x) whenever x, a! E D converges to x in X. Every real valued continuous function on a compact space X is bounded. If A is a compact subset of X and f is a continuous function on X, then the image of A under f,i.e. f(A), is a compact subset of the real line R. Let (X, F)be a topological space and A a subset of X. Then A n .Y ' = {A nV; V E Y} is called the relative topology on A. (A, A nY) is a topological space. A is a compact subset of X if and only if (A, A n Y) is a compact topological space. Let (X, Y) be a topological space. A subset A of X is said to be dense-in-itself if there are no isolated points in A. A subset A of X is said to be perfect if it is closed and there are no isolated points in A. X is said to be scattered if no non-empty closed subset of X is perfect, i.e. every non-empty closed subset A of X contains an isolated point of A. Let X be any set. A pseudo-metric on X is a map p from the Cartesian product space X x X to [0, co) satisfying the following. (i). p ( x , x) = 0 for every x in X. (ii). p ( x , y ) = p ( y , x ) for every x, y in X. (iii). p ( x , z ) i p(x, y ) + p ( y , z ) for all x, y , z in X. A pseudo-metric p on X is said to be a metric on X if x = y whenever p ( x , y ) = 0. A pseudo-metric space (X, p ) is said to be complete if every Cauchy sequence in X is convergent, i.e.
napr
n,,,,
1. PRELIMINARIES
17
whenever x,, n 2 1 is a sequence in X and limm.n+osp(xm, x,) = 0, there x ) = 0. If (X,p) is a pseudo-metric exists x in X such that limn+osp(xn, space, there exists a complete pseudo-metric space (X, p") with the following properties. (i). x c k . (ii). ~ ( xy, ) =p(x, y ) for all x, y in X. (iii). x is a dense subset of 2, i.e. for every x' in k,there exists a sequence x,, n 2 1 in X converging to x, i.e. limn-+os b(x,, 2) = 0. (2,p") is called a completion of the pseudo-metric space (X, p ) . Every pseudo-metric space (X, p ) induces a natural topology S on X. S is the smallest topology on X containing 8 = (B(x, r ) ; x E X, r > 0}, where B(x, r ) = { y E X; p ( x , y ) < r}. Let (X, S )be a topological space. A subset A of X is said to be nowhere dense if (A)" = 0. A subset B of X is said to be of first category if B is a countable union of nowhere dense subsets of X. Baire Category Theorem : If (X, p ) is a complete metric space, then X is not of first category. A subset A of X is said to have the property of Baire if there exists an open set V suyh that AAV is of first category. The collection of all subsets of X each of which has the property of Baire is a a-field on X containing all open subsets of X. The product space C = (0, l}Ho = {(XI, x2, . . .); x i = 0 or 1 for every i 2 1) is called Cantor Set. There is a natural metric p on C defined by p ( ( x 1 , x2, . . J, ( Y l , Y 2 , *
*
.I) =
c
1
F I X "
-Ynl
n2l
for ( x I , x ~ , .. .) and ( y l , y 2 , . . .) in C. (C,p) is a compact Hausdorff totally disconnected perfect metric space. Every clopen subset of C is a finite union of sets of the form {xl} X (x2} x . x (x,} x (0,1} x ( 0 , l ) x * *, for xl, x2,. . . ,x, in (0, 1) and n 2 1. Let (X, T)be a compact Hausdorff space. The Baire a-field 90on X is the smallest a-field on X with respect to which every real valued continuous function on X is measurable. The Borel a-fieEd 93 on X is the smallest cr-field on X containing all open subsets of X. Obviously, Boc B. We now show that 3 0 is the smallest a-field on X containing all compact Gs subsets of X. Let B o o be the smallest a-field on X containing all compact Gs subsets of X. Let A be a compact Gs subset of X. By Urysohn's lemma, there exists a real valued continuous function f on X such that A = f-'({O}). Therefore, A E 90. Consequently, B o o c Wo. Iff is a real valued continuous function on X, then f-'{[k,oo)} is a compact Gs subset of X and so f-'([k, a)} E Woo. Therefore, f is measurable with respect to Boo. Hence Boc Boo. Thus we have shown that Bo= ao0. Let (X, F) be a compact Hausdorff totally disconnected space, % the field of all clopen subsets of X and Bothe Baire a-field on X. We show that the smallest a-field 9;on X containing % is Bo.Since every C in % is a compact Gs subset of X, it follows that $236 cBo. We now show that
-
18
THEORY OF CHARGES
every real valued continuous function f on X is measurable with respect to 936.For n 2 1 and x in X, let U, = {y E X; I f ( x ) -f(y)I < l/n}. U, contains a clopen set V, containing x. {V,; x EX} is an open cover for X. Let {V,,, V,,, . . . , VXk}be a sub-cover of X. Let D I = V,,, DZ= V,, -V,,, D3 = V,, - (Vxlu V,,), . . . ,Dk = V,, - (V,, U V,, U * * * U VXk-,). D1, Dz, . . . , Dk are pairwise disjoint clopen subsets of X whose union is X. Choose and fix y i in Di for each i = 1 , 2 , . . . ,k. Let k
ffl= i c f(Yi)lD*. =l Then I f n ( x ) - f ( x ) I 5 l / n for everyx in X. Consequently, limfl+mf,,(x)= f ( x ) for every x in X. Each f,,is measurable with respect to 3;.Therefore, f is Hence 9 3 6 =ao. measurable with respect to 96.
1.4 BOOLEAN ALGEBRAS The aim of this section is to present some basic ideas on Boolean algebras and prove Stone Representation Theorem for Boolean algebras. 1.4.1 Definition. A Boolean algebra is a non-empty set 5 in which two binary operations v, A (join and meet) and one unary operation " (complementation) are defined satisfying the following identities. (i). a v b = b v a , a A b = b A U for all a, b in 5. (ii). a v ( b v c ) = ( a v b ) vc, a A (b A C ) = ( a A b ) A C for all a, 6, c in 5. (iii). (a A 6 ) v b = b, ( a v b ) A b = b for all a, b in 5. (iv). a A ( b v c ) = (a A b ) v ( a AC), a v (b A C ) = (a v 6 ) A ( a v c ) for all a, 6 , c in 5. (v). ( a A a") v b = 6, ( a v a") A b = b for all a, b in B. One can show that a A a' does not depend on a in 5 and b v 6' does not depend on b in 5. Denote a A a" by 0 and a v a" by 1. If 9 is a field of subsets of a set R, then 9 is a Boolean algebra in the above sense if we identify the operation of join by union of sets, meet by intersection of sets and complementation by complementation of sets with respect to R. 0 identifies with the empty set 0 and 1 identifies with the whole space R. We introduce a partial order 5 on any given Boolean algebra 5 as follows. For a, b in B, say a 5 b if a A b = a, or equivalently, a v b = 6. The relation 5 is a partial order on B, i.e. it is reflexive (a 5 a for every a in B), antisymmetric (if a 5 b and b Ia, then a = b ) , and transitive (if a 5 b and b SC, then a IC). With respect to this partial order, a v b is the smallest
1. PRELIMINARIES
19
element in B r a and 6, a A b is the largest element in B - ( y A z ) ~ = ~ x - y ~ . (28). F o r x y , x l , y 1 in L , I(x v y ) - ( x l v y l ) l ~ ~ x - x l ) + l y - y ~ l . (29). F o r x , y , x l , y l in L, I ( x ~ y ) - ( x l A y l ) l s I x - x l l + l y - Y 1 I . Proof. (l),(2),(3), (4)and ( 5 ) follow directly from the definitions involved. (by (3)) (6). x + y -(x A y ) = x + y +[(-x) v ( - y ) ] = [(x
+ Y ) -x 1 v "x + Y 1- Y 1
=yvx=xvy. (7). x + - x - = ( x v 0 ) - " ( - x ) v 0 ]
(by (1))
26
THEORY OF CHARGES
(8). (x+ v X- ) + (x+ A X- ) = X + + XIt suffices to show that x + v x - = x + + x - . Observe that x++x- = x +x- +x-
+ 2[(-x) v 01 = x + [(-2x) v 01 =x
= (x -2x)
vx
= x v (-x)
= x v (-x) v o = (x v 0) v [(-x) v 01 + =x v x .
(Note that x v (-x)zO. For, x s x v (-x) and -x
XI
v (-x),
and so 0 4
2[x v (-x>l.)
(9). The proof of this is included in the proof of (8). (10). x = y - z implies that y -x = z 2 0 . So, y r x . Since y 2 0 , y z-x v O = x + . Similarly, one can show that z r x - . (11). Y A Z = O implies that y r O and 2 2 0 . By (lo), O ~ y - x + s y and OZSZ-X-CZ. Since y ~ z = O , ( Y - x + ) A ( z - x - ) = O . Since x = y - z = x + - x - , y -x+= z -x-. Hence y - x + = 0 = z -x-. (12). This follows from what we have proved above. (13). This follows from (7) and the definition of 1x1. (14). It is enough to observe that x v y I Ix I + Iy 1 which is obvious. (15). BY (I), (2) and (31, x +(y - x ) + = x +[(y - x ) v O ] = [x
+ (y -x)]
v (x
+ 0)
=xvy. X
- (X - )J)'=X
- [(X - y) V 01 = X
[(y -X)
=[X +(y -X)]A (X+o)
= X AY.
Consequently, (X V y ) - (X
A
y ) = (y -X)'+
(X
- y )'
=(x-y)++(x-y)= Ix -yI.
(16). This is a consequence of (6) and (15). (17). This is a consequence of (6) and (15).
A
01
(20). x 5 y implies that (x v 0) 5 (y v 0). This means that x + Iy+. Similarly, one can show that y - s x - . The converse is trivial. (21). This follows from (19). by (61 122). (1x1 v lY I>+(lxl A IY I ) = I X l + l Y I, From this identity, (22) follows obviously. (23). If x I y , then (by (6)) 0 = 1x1A IY I = 1x1+ lY I - (1x1 v lY I)
I + IY I -%lx + Y I + Ix - Y I) = 1x1+ IY I - Ix + Y I Hence, we have the equality Ix I + Iy 1 = lx + y I. Further, (x + y)' = (x + y) v 0 = $(x + y + Ix + y I) = Ix
=ax +Y
(by (18)) (by (21))
(by 16))
+IXI+lYl)
(by what we have proved above) =x++y+. (24). Let y =.VaErx,. Then x A X , 5 x A y for every a in r. We show that Let w E L and x AX, 5 w for every a in r. By (6), x + x , - ( x v x , ) ~ w for every a in r. By (l), w + ( x v y ) r x + y . So,
VaEr(x AX,)= x A y.
28
THEORY OF CHARGES
w z x + y - ( x vy)=x AY. Hence V a , r ( X AX,)=X A Y = X ~ ( V ~ ~ r x ~ ) indeed. The other law can be established analogously. (25). This follows from (18). . z r O and z s (26). Case ( i ) . I A l r l . Let z = I A x l ~ l y l = l A l I x I ~ l y JThen IAllx] and s l y l . From this, it follows that (l/lAI)z 11x1 and (l/IA])z s l y ] . Since 1x1A IyI = 0, z I0. Hence z = 0. Case (ii). l A l < 1. If A = 0, there is nothing to prove. Let \A 1 > 0. Find c > 1 such that c IA I = 1. Then x I c y , by case (i). This means that Ix 1 A c Jy1 = 0 = c JA 11x1A cly I = c (Ih1 Ix 1 A ly 1) which implies that [A 1 (xI A ly I = 0. (27). Using (15) and the distributive laws, we obtain I(x VZ)--(Y v z ) l + l ( x Az)-(Y A Z ) I
=(x v z ) v ( y V Z ) - ( X V Z ) A ( Yv z ) + ( x A Z ) V ( YA Z ) - ( X A Z ) A ( Y A Z ) = (X v y v 2)-(X
A
y ) v z +(x v y ) A z -(x
=[(X Vy)VZ+(X VY)AZ]-[(X
"(x
vy>+zI-[(x
=(X Vy)-(X
A
y
A2)
AY)VZ+(X A y ) A Z ]
AY)+ZI
AY)=IX-Y(
(28). I(x v y ) - ( x i v y i ) I = I ( x
(by (15 ) ) .
vy)-(xivy)+(xivy)-(xivyi)I
s I ( x VY)-(X1 VY)l+l(Xl VY)-(X1 V Y d l SIX
(by (6))
-x1l+Iy -Y1l
(29). This is similar to the above. This completes the proof of all the identities.
(by 12)) (by (27)).
E l
An important example of a vector lattice is the space of all bounded finitely additive measures to be presented in the next chapter. The above theorem brings into focus many facets of the main object of study in this book. Now, we introduce some important concepts such as normal sublattices and boundedly complete lattices.
1.5.5 Definition. Let (L, 5)be a vector lattice and W c L. W is called a sublattice of L if W is a vector subspace of L and x v y E W, x A y E W for every x, y in W. 1.5.6 Definition. A vector sublattice W of a vector lattice (L, 5)is said to be normal if the following conditions are satisfied. (i). x EW, O s l y ) ~ I x I + yEW. (ii). If V a E r x , exists for a given non-empty (xp; (Y E r}cW, then Vaar xa EW.
1.
29
PRELIMINARIES
1.5.7 Definition. Let (L, I) be a vector lattice and S c L. The orthogonal complement of S is denoted by S’ and is defined by
sL= { y EL; y IX
for every x in s}.
It is easy to check that for any S c L, S nSL= 0 or = (0).If SI c SZ c L, then S: c S:. The following theorem shows that normal sublattices arise in a natural way. 1.5.8 Theorem. Let (L, 5) be a vector lattice and S c L. Then SL is a normal sublattice of L.
Proof. First, we show that S’ is a vector subspace of L. Let y , z c, d real numbers. Then for any x in S,
E S’
and
0 s KCY +dz)l A 1x15 (2lCllY I v 2ldl lzl) A 1x1
I I A Ix I) v (14Jz I A Ix 111.
= 2HlC IY
The second inequality above follows from Theorem 1.5.4(25) and the equality from Theorem 1.5.4(24). Since y I x and z Ix,cy Ix and d z Ix. See Theorem 1.5.4(26). Hence (cy + d z ) I x . So, c y + d z ES’. Next, we show that for y , z ES’, y v z ES’ and y A Z ESI. But these follow from Distributive laws. Thus we have proved that S’ is a vector sublattice of L. The normality of S’ again follows from distributive laws.
1.5.9 Definition. A vector lattice (L, 5)is said to be boundedly complete if for every non-empty subset {xa ;a E r}of L bounded above, i.e. there is an element x in L such that x, IX for every a in r, Vol.rX, exists. If (L, 5)is a boundedly complete vector lattice and if {x, ; a E r} is a non-empty subset of L bounded below, then A a E r x aexists. This follows from Theorem 1.5.4(3). The following is the main theorem of this section.
1.5.10 Riesz Decomposition Theorem. If S is a normal vector sublatfice of a boundedly complete vector lattice (L, a),then any element x in L can be written as a sum x f + x ” for some X I in S and XI’ in S’, and this decomposition is unique. Further, if x 2 0 , then x f = VYEs(x A Iyl). For x in L, in general, x’= (x+)’- (x-)’. Proof. First, assume that x 2 0. Then the set {x A J y1; y E S} is bounded (x A ly 1) exists. Define x f = above, and as L is boundedly complete, VYEs V,,S (x A Iyl). As S is normal, x A I y l S~ for every y in S and consequently, x f E S. Now, we show that x - X ’ E S’. Let y E S and u = (X - x f ) A Iy(. The problem reduces to showing that u = 0. Since 0 5 u 5 Iy 1 and y E S, u E S. Since S is a vector space, u + x ’ E S . Note that u + x l ~ ( x - x f ) + x l = x .
30
THEORY OF CHARGES
Consequently, u + x ~ = ( u + x ' ) A = x I U + X ' ~ A= X x A I u + x ' I ~ x 'from the definition of x'. This implies that u 5 0 . Hence u = 0. Thus x - X ' E S'. Denote x - x r by xN. Thus, we have achieved in writing x =xr+xr' with X ' ES and xrrE S'. Now, we obtain a decomposition for any x in L. Write
x = x + -x
-=
(X+)'+ (X+)r'- ( f ) ' - (x-)"
= [(x +)' - (x-)'I
+ [(x
+)'r
- (x-)"] = x r + X I ' ,
say.
It is obvious that X ' E S and X " E SI. Thus x = x ' + x B is a decomposition of x with the desired properties. To prove uniqueness, let x = X I + x2 = y1+ y 2 with x l , y l in S and x2, y 2 in SL.Then x l - y l € S and x 2 - y 2 ~ S ' . Since xl-yl=-(x2-y2), (x1-yl)ES' as well. This implies that Ixl-yll=O or x l = y l . See Theorem 1.5.4(12). We also obtain x 2 = y 2 . Hence the decomposition obtained above is unique with the stated properties. 0
1.5.11 Corollary. If S is a normal vector sublattice of a boundedly complete vector lattice (L, s),then (S')' = S . Proof. The result follows from the following relations. (i). L= SOS'. (ii). L = s'o(s')'. (iii). s c (S')'. (Here, 0 denotes the direct sum of the sets involved.)
0
Now, let y E L be fixed. We characterize the smallest normal vector sublattice of (L, 5)containing y .
1.5.12 Theorem. Let (L, 5)be a boundedly complete vector lattice and y E L . Then the smallest normal vector sublattice of L containing y is (S')', where S = { y } . Further, x E (S')' if and only if x=
V
n>l
(x+AnlyI)-
V (x-Anlyl). fl2l
Proof. Since S c (S')', (S')' is a normal vector sublattice of L containing y . If S1 is a normal vector sublattice of L containing y , then S c S 1 . This implies that S: c S'. From this, it follows that (S')' c (S:)'= S1. Hence (S')' is the smallest normal vector sublattice of L containing y . Thus the first part of the theorem is proved. Since (S')' is a normal vector sublattice of L, it follows that x=
V fl2l
(x+AnlyI)-
V
(x-Anlyl)E(S')'.
flZl
Conversely, let x E (S')' and assume that x 2 0 . Let u = Vfl21(x A nlyl). We show that u = x. If we show that ( x - u ) A Iy I = 0 , it would then follow that x --u E S' from the definition of S'. Since (S')' is normal, u E (S')'.
1. PRELIMINARIES
31
Since x E (S')I, x - u E (S')'. Thus we find that x - u is available in both S' and (S')I. This implies that x - u = 0 or x = u. In view of this argument, let us show that (x - u ) A 1y(= 0. To this end, we note (X - U ) A
IY 15((X
- U ) Ax) A
IY I = (X - U )
A (X A
IY 1)
5 (X - U ) A U.
By Theorem 1.5.4(2), we also note that [(x - u ) A u ] + u = x (X - U ) A U = -24 +(X A
A
2u. So,
2U) by Theorem 1.5.4(4)
=-U+(
v (XA2XnIyl))
nz1
=-u+u=o.
Hence 0 5 (x - u ) A IyI 5 (x - u ) A u = 0. Thus (x - u ) A IyI = 0. So, if x E (S')I and x 2 0 , the above representation is valid. If x E (S')' is any general element, write x = x + - x - . Note that x+, X - E (S')I. By what wehaveprovedabove,x+=V,,I ( x + ~ n l y l ) a n d x - = V , , ~( x - ~ n ( y l ) . T h i s 0 completes the proof of the theorem. Now, we provide a brief review of other relevant functional analytic concepts used in the book. First, we introduce vector spaces over the field of rational numbers which is specially required in Chapter 3.
1.5.13 Definition. Let Q be the field of all rational numbers. A vector space or a linear space over Q is an additive group L (with the additive operation denoted by +) together with a map m from the product space Q x L into L written as rn (r, x) = rx for r in Q and x in L, which satisfies the following four conditions. (i). r ( x + y ) = rx + r y , r E Q , x, y EL. (ii). (r +s)x = rx +sx, r, s E Q and x E L. (iii). r ( s x )= (rs)x, r, s E Q and x EL. (iv). l ( x) = x, x E L. The usual concept of a vector space over the real line R has the same description as the above with Q replaced by R. If L is a vector space over Q, a linear functional on L is any map T from L to R satisfying T(rx + sy) = r T ( x )+ s T ( y ) for all r, s in Q and x, y in L.
32
THEORY OF CHARGES
Let L be a vector space over Q. A susbset A of L is said to be a Hamel Basis for L if every non-zero x in L admits a unique representation x =rlxl+r2x2+* *+r,x,
for some non-zero r l , r2, . . . , r, in Q , xl, x2, . . . ,x, in L and n 5 1.A subset B of L is said to be linearly independent if rl = r2 = * * * = r, = 0 whenever rlxl+ r2x2+ * +r,x, = 0 for some rl, r2, . . . ,r, in Q, xl, x2, . . . ,x, in B and n L 1. Given any linearly independent subset B of L, one can find a maximal linearly independent set A containing B. A subset A of L is a maximal linearly independent set in L if and only if A is a Hamel Basis for L. Consequently, given any linearly independent subset B of L, one can find a Hamel Basis for L containing B.
--
The following is a version of Hahn-Banach Theorem.
1.5.14 Theorem. Let L be a vector space over Q. Let L1 be a subspace of L, i.e. rx +sy E L1 whenever x, y E L1 and r, s E Q. Let x be an element of L which is not a member of L1. Let c be any real number. Let T1be any linear functional on L1. Then there exists a linear functional T on L with the following properties. (i). T ( y )= T l ( y )for all y in L1. (ii). T ( x )= c. Proof. Let B1 be a Hamel Basis for L1 and B = B1 u {x}. Note that B is a linearly independent set. Let A be a Hamel Basis for L containing B. Define T on A as follows. T ( y )= T,(y), if y E B I , = c,
if y =x,
= 0,
otherwise.
T can be extended to L in the obvious fashion and this extension, denoted 0 again by T, is the desired linear functional. The following is generally known as Hahn-Banach Theorem.
1.5.15 Theorem. Let L be a vector space over the field of all real numbers R. L e t p be a real valued function on L satisfying p ( x + y ) l p ( x ) + p ( y )for all x , y in L and ~ ( c x=) c p ( x )for all c 2 0 and x in L. Let L1 be a subspace of L and T1a linear functional on L1 satisfying T l ( x )l p ( x ) for every x in L1. Then there exists a linear functional T o n L having the following properties. (i). T ( x )s p ( x ) for every x in L. (ii). T ( x )= T l ( x )for every x in L1.
1. PRELIMINARIES
33
Now, we introduce normed linear spaces. Let L be a vector space over R. A pseudo-norm on L is a map p from L to [O,m) with the following properties. (i). p ( x ) = 0 if x = 0. (ii). p ( c x )= I c l p ( x ) for every c in R and x in L. (iii). p ( x + y ) ~ p ( x ) + p ( yfor ) all x , y in L. A pseudo-norm p on L induces a pseudo-metric p on L as follows. For x , y in L, let p ( x , y ) = p ( x - y ) . A vector space L equipped with a pseudo-norm p is said to be complete if the pseudo-metric space (L, p ) is complete, i.e. every Cauchy sequence in L is convergent under the pseudo-metric p. In what follows, we assume that all vector spaces are over R. A pseudonorm p on a vector space L is said to be a norm if x = 0 whenever p ( x )= 0. If p is a norm on L, it is customary to denote p ( x ) by llxll for x in L. The is called a normed linear space. If L is complete under the pair (L, ~~~~~) norm 11 * 1 , then (L, 11 * 11) is called a Banach space. Let (L, 11 * 11) be a Banach space. Then the topology induced by the norm II.II on L is called strong topology on L. If x , x,, n 2 1 is a sequence in L such that x,, n z 1 converges to x in the norm II-II, i.e. lim,+m IIx, - X I ] = 0, we say that x,, n 2 1 converges to x strongly or x,, n 2 1 converges to x in L. If (L, II-II) is a Banach space, then the space of all continuous linear functionals on L is called the dual space of L and is denoted by L*. For any linear functional T on L, one defines IITII = Sup { I T ( x ) l ;x EL, IIxII I1). T E L* if and only if llTll< 00. In fact, T is continuous if and only if there exists a positive constant k such that l T ( x ) l s kllxII for all x in L. If T E L*, then IT(x)l ~IITllllxllfor all x in L. The function II.II on L* is indeed a norm on L". (We use the same symbol 11 11 for both L and L* and it should be clear from the context which norm we are dealing with.) Moreover, (L*, 11 * 11) is a Banach space. If (L, II-II) is a Banach space, the weak topohgy on L is the smallest topology on L with respect t o which every T in L* is continuous. A net x,, a E D in L is said to be weakly convergent to a x in L if T(x,) = T ( x ) for every T in L". A sequence x,, n 2 1 in L is a weak Cauchy sequence if lim,,,,m T ( x , - x m ) = 0 for every T in L*. The following is an important result which we have an occasion to use.
1.5.16 Theorem. Let (L, 11 * 11) be a Banach space. Then every weak Cauchy sequence x,, n 2 1 in L is norm bounded, i.e. Sup nzl IIx,]]< 00. Let (L, II.II) be a Banach space. A subset A of L is said to be weakly closed if A is a closed set in the weak topology on L. Equivalently, if x,, a E D is a net in A converging weakly to x in L, then x E A. In this context, we quote a result.
-
1.5.17 Theorem. Let (L,11 11) be a Banach space and LI a closed (in the strong topology on L) subspace of L. Then L1 is weakly closed.
34
THEORY OF CHARGES
A Banach space (L, II-II) is said to be weakly complete if every weak Cauchy sequence in L is weakly convergent. Finally, we close this section with Banach lattices.
1.5.18 Definitions. A vector lattice (L, 5)is said to be a normed vector lattice if there is a norm 11 )I defined on L such that IIx 11 IIIy 11 whenever x , y E L and 1x1Ily 1. A Banach lattice is a normed vector lattice (L, I) with a norm 11 * 11 such that (L, 11 11) is a Banach space.
-
-
1.5.19 Theorem. In any normed vector lattice (L, 5,(1 II), the maps (x, y ) + A y from LX L to L are uniformly continuous. Hence for any subset S of L, SL is a closed subspace of L. In particular, any normal vector sublattice of L is closed. x v y and (x, y ) + x
Proof. For any x in L, observe that IIx)]= )IIx 11). Now, the uniform continuity of the maps (x, y ) + x v y and ( x , y ) + x A y from L x L to L follows from Theorem 1.5.4 (28) and (29). The rest of the assertions also follow from the.same observations. 0
CHAPTER 2
Charges
In this chapter, we introduce the main concern of this book, namely charges. They are usually known as finitely additive measures in the literature. We chronicle most of the rudimentary facts about charges in this chapter. In Section 2.1, basic concepts about charges are presented. The space of all bounded charges on a field of sets is shown to be a boundedly complete vector lattice in Section 2.2. Sections 2.3 and 2.4 deal with measures. Jordan Decomposition theorem and Hahn Decomposition theorem for charges not necessarily bounded are covered in Sections 2.5 and 2.6, respectively.
2.1 BASIC CONCEPTS This section is mainly devoted to the study of various properties of charges.
2.1.1 Definitions. Let 9be a field of subsets of a set 0. (1). A map p : 9+[-m, m] is said to be a charge on 9 if the following conditions are satisfied. (i). p ( 0 )= 0. (ii). If A, B ~ . F a n d A n B = 0 ,t h e n p ( A u B ) = p ( A ) + p ( B ) . (2). A charge p on 9 is said to be a real charge if -m*) is partial order L* as follows. For C, D ~ dsay, C z*D if C 2 D . (d, a directed set. For each C in d, let pc = VaeCpa. pc, obviously, exists in ba(R, 9)as C is only a finite set. Then the net { p c ;C E d }is an increasing net, i.e. if C, D E & and C ?* D, then p c l p D in ba(R,9). Further, p c s A
2.
45
CHARGES
for every C in d. Let T be the pointwise limit of pc, C E d, i.e. T(F)= limc.dpc(F), F E ~This . limit obviously exists for every F in 9. T is also Since p, 5 T 5 A for any fixed a in r, T is also bounded. It a charge on 9. s)is a boundedly complete is obvious that T = VaErpa. Hence (ba(n, 9), vector lattice. (10). If p = 0, then lp I = 0. So, llpII = 0. Conversely, if IlpII = 0, Ip I(n)= 0. Ipl = 0. Hence p = 0. See Theorem Since 1 ~ is1 a positive charge on 9, 1.5.4(12). If a is any real number and p E ba(R, then llap)I= IapI(C2)= IaIIpI(s2)= IaIIIpII. See Theorem 1.5.4(12). Finally, let pl, p 2 c b a ( a , F ) . Then IIcLI+cL~~I= 1~~1+1l.21(SZ)51p~~1I(R)+\1*.2)(n> =Ilpc~Il+lp211.See Theorem is a norm on ba(R, 9). 1.5.4(12). Hence ((.(( (11). If p, vEba(n, 9)and I p l 5 I v I , then ~ p ~ ( l l ) 5 ~ v Consequently, ~(fl). \\p\Isl\v\\.Hence ba(O,9) is a normed vector lattice. Now, we show that ba(n, 9)is complete under the norm 11 11. Let p n , n I1 be a Cauchy sequence in ba(O,9), i.e. limm,,,+mIIpm-pnll = 0. Since pm- p n s Ipm-pnl for all m,n 1 1 , we have for any F in 9,
m,
-~nl(n)
I ~ m ( F ) - ~ n ( F ) I ~ I ~ m - ~ n I ( F ) ( I ~ m
= llkm - ~ n l l .
This shows that p m(F), m 2 1is a uniform Cauchy sequence of real numbers Let p (F) = limm+mp m(F), F E 9. Thus p m , m 2 1 converges to p over 9. uniformly over 9. p is obviously a charge on 9. Since the convergence is uniform, p is bounded. Since II*(I is a norm, lllp,, -pII-IIpm -pIII 5 II(p,,-p)-(pm-p))II=llpn -pmll,iffollowSthatlimm,, IIpm-pll=O.Hence ( b a ( n , 9 ) , 5,11 * 11) is a Banach lattice. 0 The above theorem in conjunction with Theorem 1.5.4 brings into focus various aspects of bounded charges. Of particular interest, are the charges p c , p - , and lp I associated with p E ba(n, 9). First, we isolate these entities and write formally their computational equivalents. Let p ~ b a ( f l , 9 ) .Then p f = p v O and p - = ( - p ) v O . By Theorem 2.2.1(5),
'(F) = SUP{@(B);B C F ,B E 8, FE and
p-(F) = -1nf
{ p (B); B c F, B E 9}, F EF.
and p - ba(n, ~ and they are called the positive and negative variations of ICL respectively. The charge Ipl = p L c + p -is called the total variation of p. Combining Theorem 1.5.4 and Theorem 2.2.1, we chronicle some salient features of charges in the following theorem for future reference. pc
46
THEORY OF CHARGES
2.2.2 Theorem. Let p E ba(R, 9).Then the following are true. (1). p = p + - p - a n d p + A p - = O . (2). l p l = p + + p - = p + v p - = p v ( - p ) . A n important consequence of this is )(F)= SUP { p (B) - p (F- B); B c F, B E g}, F E 9. (3). I f p = A - v with A, vEba(R,5"), A r O and v 2 0 , then A z p + and
IF
v r p
-
.
Y E ba(R, 9)andA A v = 0 , then A = p + and v = p - . ( 5 ) . p++=$(Ip I + p 1 and p - = $(IP I - p j . + (6). If Y E ba(R, 9)and p Iv, then p 5 v and p - 2 v-. (7). I f v E b a ( R , m and ~ A V = Othen , ( p + v ) + = p + + v + and I p + v l =
(4).I f p = A - v with A,
IPI
+I4
2.2.3 Remark. Theorem 2.2.2( 1) is precisely Jordan Decomposition theorem for bounded charges. Any bounded charge p can be expressed as the difference of two positive charges with a certain minimality property in the following sense. If p = A - v also, where A, v 2 0, then A L p + and v 2 p -. We will take up the issue of writing a given charge as a difference of two positive charges in Section 2.5. In the following theorem, we give an alternative description of lpl, in addition to the one described in Theorem 2.2.2(2).
2.2.4 Theorem. Let p E ba(R, 9).Then for any F in 9, i=l
where the supremum is taken ouer all finite partitions F1, Fz, . . . ,F, of F in 9.Further, IpI(R)52 S u p { l p ( F ) I ; F ~ 9 } . Proof. Let F1,F2,. . . ,F, be any partition of F in 9. Let I = (1 5 i 5 n ; p(Fi)L 0) and J = (1Ii ~ np(Fi) ; d , then (pa v pa,)(fl)> d for some n 2 1. This contradicts the definition of d . So, (pa v r * ) ( f l )5 d = r * ( f l )5 (pa v ~*)(fl).So, (pa v ~ * ) ( f = l )r*(fl).Since r * s p a v r * and ~ * ( f l=) (pa v r*)(fl),it follows that T* = p, v r * ! (Check this.) This implies that pa ST*. Since this is true for every a in r, V a E pol r = r 5 r * . On the other hand, since pan:s r for pan= T* 5 T. Hence T = r * . This shows that r is ameasure. every n 2 1, Vn2? In the above, we have shown that r is a measure under the assumption that each p, is a positive measure. Now, we treat the general case. Choose and fix a0 in r. Let v, = polv p,, + Ipa0l,a E r. Each Y, is a positive measure since I/, 2 pa0+ Ip,,l = 2pL0 2 0. Observe also that
v
I/,
=
a E r
v
[(Pa v
+l ~ ~ , I l
11.010)
,Er
by Theorem 1.5.4(1)
By what we have proved above, r+lp,,l is a measure. Since (paOlis a is a normal sublattice measure, it follows that 7 is a measure. Hence ca(R, 9) of ba(fl, It follows from Theorem 1.5.19 that c a ( f l , 9 ) is a closed sublattice of bdfl, 0
a.
w.
52
THEORY OF CHARGES
2.4.3 Corollary. ca(R, 9 )is a boundedly complete vector lattice and also a Banach lattice in the usual norm. 2.4.4 Remark. Given any field 9 of subsets of a set R, one can find a set X and a c+-field d on X such that ba(Q, 9) and ca(X, d)are isometrically isomorphic as Banach lattices, i.e. there is a linear map T from ba(R, 9 ) onto ca(X, d)such that IIT(p)ll=llpll, p E ba(R, 9) and T ( p )1 0 whenever p 2 0 . This can be proved using the Stone Representation theorem for Boolean algebras given in Section 1.4.
2.5 JORDAN DECOMPOSITION THEOREM Jordan Decomposition theorem for bounded charges has been discussed in Remark 2.2.3. It essentially says that every bounded charge can be written as a difference of two positive charges in a minimal way. The main theorem of this section gives a simple necessary and sufficient condition for such a decomposition to prevail for charges not necessarily bounded. First, we introduce the lattice operations v and A for general charges.
2.5.1 Definitions. Let p and v be two charges on a field 9 of subsets of a set R such that either both p and v avoid the value -a or both avoid the value +a.Define the set functions A and T on 9 by
+
A (F)= SUP{ p (E) v(F-E); E c F, E E
a,
FE9,
and T(F)= Inf { p (E)+ v(F- E); E c F, E E 9}, F E9. The above set functions were defined for bounded charges in Theorem 2.2.1(4) and (6).We used the same formula for general charges in the above definition. We denote A by p v v and T by p A v. This is consistent with the notation used for bounded charges.
2.5.2 Proposition. Let p and v be two charges on a field 9of subsets of a set Q such that either both p and v avoid the value -a or both avoid the value +a.Then the set functions A ( = p v v ) and T ( = p A v ) defined above are charges on.9. Proof. The proof given for Theorem 2.2.1(4) and (6) carries through essen0 tially here. The following theorem is the main result of this section.
2. 2.5.3
of a set
53
CHARGES
General Jordan Decomposition Theorem. Let 9 b e a field of subsets n. Let p be a charge on 9.Define p + and p - by p+(F)= Sup { p (E);E c F, E E9}, F E g,
and y-(F) = -1nf {p(E);E c F, E E
w,
F E 9.
Then the following statements are true. (1). p + and p - are positive charges on $. (2). I f p does not take the value +a,then p + - p = p - . (3). If p does not take the value -00, then p + p - = p c . (4). I f p does not take the value +OO and ~ 1 p- = p 2 for some positive c h a r g e s p l , p 2 0 n 9 ,t h e n p ~ ~ p + a n d p ~ ? p - - . ( 5 ) . I f p does not take the value --CO and p + hl = A 2 for some positive charges h l , h2 on 9,then h 1 z p - and h 2 ? p C . (6). p = p + - p - if and only if p is either bounded below or bounded above. More generally, we can write p = p l - p 2 for some positive charges p l , p2 on 9 if and only i f p is either bounded below or bounded above. (7). p + A p - = 0 if and only if p is either bounded below or bounded above. ( 8 ) . I f pl and p 2 are positive charges on 9 satisfying p = p l - p 2 and pI ~ p 2 = 0 then , p~ = p+ and p2 = p - . (9). If p is a real charge, then p = p + - p - holds if and only if p is bounded. In such a case, both p + and p - are bounded. More generally, i f p is a real charge, then we can write p = p1 - p 2 for some positive charges p l and p2 on 9 if and only if p is bounded. Proof. (1)follows from Proposition 2.5.2 if we observe that p + = p v 0 and p - = ( - p ) v 0 as per Definition 2.5.1. (2). Let F E $. Suppose p(F) = -a.Then, from the definition of p - , p-(F) = co.So, p+(F)- p (F)= a = p-(F). Suppose p (F)> -a.By the given hypothesis, -a< p (F)< a.Consequently, --CO < p (E) < 00 for any E in 9 such that E c F . See Proposition 2.1.2(vii). So, CL +(F)- W
(F)= SUP{ P
(a;E c F, E
= SUP{ p (E)- p
E
SI(F)~
(F);E c F, E E $}
=SU~{-~(F-E);ECF,EEF} = -1nf {p(C);C c F, C E = p-(F).
This completes the proof. (3). This can be proved as above.
54
THEORY OF CHARGES
(4).Since p l - p = p 2 and p 1 and p2 are positive, it follows that p l z p . So, for any F in 9, p+(F)=Sup{p(E);E c F , E ~ 9 } s p ~ ( FHence ). p+5 p1. For the second part, observe that p + - p s p l - p = p 2 .By (2), p + - p = p . This completes the proof. ( 5 ) . The proof is analogous to that of (4). (6). Suppose p is bounded above. By (2), p + - p = p - . Note that pLfis a bounded charge. Consequently, - p = p - - p + or p = p + - p - . A similar argument works when p is bounded below. Conversely, if p = p + - p - , then either pLcis bounded or p - is bounded. In the former case, p is bounded above and in the later case, p is bounded below. The more general version can be established using (4). (7). Suppose p is bounded below. We show that p + A p - = 0. Since p is bounded below, p - is a positive bounded charge. So, for any F in 9, +I(
A
p-)(F) = Inf {p+(E)+p-(F-E); E c F, E E fl
+
= Inf { p (E) p-(E) +p-(F-E); E c F, E E 9},
(by (3)) =Inf{p(E)+p-(F); E c F , E € . F } = Inf { p (E); E c F, E E -+p-(F),
(since p - is bounded) = -p-(F)+p-(F)=O.
If
is bounded above, a similar argument shows that p + A p - = 0. Conversely, suppose p is neither bounded below nor bounded above. We show that p+r\ p - # O . Since p cannot take both the values +a and -a,assume, without loss of generality, that p does not take the value -a.Note that pL-(n) = a.By (3), p + p - = p + . So, p
( F +A
p - ) m = u p + p - > A p-l(n2) = Inf { ( p +p-)(F)+p-(F");
FE9}
= Inf {p(F)+p-(R); F E 5F)= a.
This completes the proof. (8). If p = p1 - p ~ where , p l and p2 are positive charges, then p is either bounded below or bounded above. See ( 6 ) .Assume p is bounded above. This implies that p l is a bounded charge. Since p = p l - p 2 = p + - p -, p1-p:-p2=-p-50. Therefore, p 1 - p + 5 p 2 . By (4),p l ? p + . So, 0 5 p1-p +5p2. Since p l ~ p ~ =and O O s p l - p c s p l , it follows that 0 s (PI - p ) 5 p1 A p2 = 0. Hence p l= p + . The other equality follows easily now.
2.
55
CHARGES
(9). If p is a real charge and p = p1- p 2 , where p 1 and p2 are positive charges, then p 1 and p2 are bounded. So, p is bounded. If p is bounded, of course, we can write p as a difference of two positive charges. This completes the proof of the theorem. 0 Some comments are in order on the above theorem.
2.5.4 Remarks. (i). The charge p described in Example 2.1.3(2) is neither bounded above nor bounded below. There is no way we can write p as a difference of two positive charges. For this charge, p + - p = p - and p f p - = p + . p + and p - work out as follows. p +(A)= n, = 03,
p-(A) = 0, = 03,
if A is finite and has n elements, if A is cofinite. if A is finite, if A is cofinite.
= 03. Observe also that (p’ A &*.-)(a) (ii). Theorem 2.5.3 goes through in toto if we replace the word “charge” by “measure”.
An interesting result emerges for measures on a-fields in contrast to the Remark 2.5.4(i), i.e. every measure on a a-field can be written as a difference of two positive measures. First, we need a lemma.
2.5.5 Lemma. Let p be a measure on a u-field % of subsets of a set R. Then p is either bounded below or bounded above. Proof. First, assume that p is real valued. In this case, we show that p is bounded. Suppose p is unbounded. By Theorem 2.1.6, there exists a sequence B,, n 2 1 of pairwise disjoint sets in % such that Ip (Bi)l r 1 for every i 2 1. Then, either there exists a sequence n1 < n2 < * * such that p (B,,) 2 1 for every i L 1 or there exists a sequence k l< k2 < such that p ( B k l ) % - l for every i r l . Assume that the former holds. Then p(UiZ1 B,,) = 03 contradicting the fact that p is real valued. If the latter holds, we do still get a contradiction. Hence, if p is a real valued measure, then it is bounded. 031 or in In the general case, observe that p takes values either in (-a, [-a, 03). Assume that p takes values in (-O3,03]. Then we show that p is bounded below. Suppose p is unbounded below. We find A1 in % such that p (A1) 5 -1. Since p is real valued on A1 n%, by what we have proved above, p is bounded on Al. Consequently, p is unbounded below on A?. So, we can find A2 in 2l such that A2 c A; and p (A2)I-1. By the same reasoning given above, we can show that p is unbounded below on Af - A2.
--
56
THEORY OF CHARGES \
Continuing this way, we obtain a sequence A,,, n 2 1 of pairwise disjoint A,,) = -a.This sets in % such that p (A,) 5 -1 for every n 2 1. So, p (UnZl is acontradiction. Hence p is bounded below. This completes the proof. 0
2.5.6 Jordan Decomposition Theorem for Measures on a-fields. Let be a measure on a u-field % of subsets of a set R. Then we can write f
F=P
p
-
-cL
with the property that p + A p - = 0. Further, the above decomposition has the following optima& property: if p = p l- p 2 , where p I and p2 are positive measures on %, then p 1 2 p + and p2 L p -. Proof. By Lemma 2.5.5, p is either bounded below or bounded above. Remark 2.5.4(ii) completes the proof. 0 2.5.7 Remark. The above theorem is not valid for charges on a-fields.
2.6 HAHN DECOMPOSITION THEOREM In this section, we prove Hahn Decomposition theorem for charges. Hahn Decomposition theorem for measures on a-fields will be proved in Section 6.1.
2.6.1 Definition. Let 9 be a field of subsets of a set R and p a charge on 5 Let E > 0. A partition {D, D? of s1 with D in 9is said to be a E-Hahn decomposition of p if the following are satisfied.
+p (B) C c D" +p (C)
B E 9, B c D C E 9,
5 E. 2 -E.
2.6.2 Hahn Decomposition Theorem for Charges. Let p be a charge on a field 9of subsets of a set R which is either bounded below or bounded above. Then for any E >0, there exists a E-Hahn decomposition for p. I f p is neither bounded below nor bounded above, there exists E > O for which there is no E-Hahn decomposition of p . Proof. We give a direct proof of this result. This can also be proved using Jordan Decomposition theorem. Assume that p is bounded below. Let d = Inf (r-L (A); A E 9). Then d is a finite number. Let E > 0. We can find D in 9 such that d 5 p (D) Id + E . This implies that -CO < p (D) < co and for any B in 9,B c D , we have -oo
i=l
C p1(Di).
i=l
Hence
f
i=l
fi(Ci)+
fi(Di)+
fi(Ei)z
i=l
i=l
f. fi(Fi).
i=l
From this, it follows that (using Proposition 3.1.5), m+n
P+cq
C fi(Ai)? C fi(Bi)*
i=l
i=l
84
THEORY OF CHARGES
Hence fi is a positive real partial charge on % u9.By Theorem 3.2.10, we can find a positive bounded charge p on 9 which is an extension of 6 from % u $3to 9. This completes the proof. Now, we take up the case of real charges.
3.6.2 Theorem. Let % and 9be two fields on a set R and p l and p2 two real charges on % and 9respectively. Let 9 be a field on R containing both V and 9. A necessary and sufficient condition for the existence of a real charge p on 9 which is a common extension of both p 1 and p z is that p1(A) = pz(A)for every A in % n9. Proof. The proof of this theorem is similar to the one given for the previous theorem, It may be remarked that these two theorems remain valid if p l and take infinite values.
pz
The two theorems proved above are not extendable to the case when Here are the relevant examples. there are more than two subfields of 9.
3.6.3 Example. Let R={l,2,3,4}; %1=
9=P(R);
(0, {1,21,{3,41, R1; V2 = (0, K 3 1 , (2341, n1; %3 = (0, (1,419 (2,313 n1;
pi({1,2))=& p2({2,41) =
t;
pi({3,4})=% p3({1,41) =
a,
@2({1,3))=$, p3({2,31) = .:
Each pair of p1 and p 2 , p l and p3, kz and p 3 satisfies the condition of Theorem 3.6.1. But there is n o positive charge on 9 which is a common extension of all the three charges p l , p2 and p 3 .
3.6.4 Example. Let
R = {1,2,3}; %1=
9=P(R);
(0, (11, (2,319 R1; %2 = i0, (21, {1,31, R1; V3 = (0, (31, (1,219 a);
@11({1I)=i=E.L1({2,31); p2({21)=&p2({1, 31)=f; 113({31)=
a,
p3({1,21) =
t.
Note that p l= p 2 on V l n V 2 , p l= p 3 on V 1 n V 3 and p 2 = p 3 on V2nV3. But there is no real charge p on 9 which is a common extension of pl, PZ and P3.
CHAPTER 4
Integration
In this chapter, we develop the theory of integration for real valued functions with respect to charges. Integration with respect to charges requires a good deal of tact, patience and circumspection to get around measurability problems. The treatment of this topic given here is fairly comprehensive. After presenting the preliminaries in the first three sections, we develop D-integral as presented by Dunford and Schwartz in Section 4.4. In Section 4.5, we introduce S-integrals which are of Stieltjes type and make comparisons with D-integrals. L,-spaces are introduced and studied in Section 4.6. Finally, in Section 4.7, ba(n, 9)is realized as a dual space.
4.1 TOTAL VARIATION AND OUTER CHARGES Let p be a charge defined on a field 9 of subsets of a set n. Recall the definitions of positive and negative variations, p + and p - , of p as expostulated in Section 2.5. p+(A)=Sup{p(B);BcA,B~fl,
A E ~ ,
and
@-(A)= -1nf {p(B);B c A, B ~ f l ,A E P .
As has been noted in Theorem 2.5.3, p + and p - are positive charges on g.The total variation lpl of p has been defined for bounded charges p on $. This notion can be introduced for any charge p. 4.1.1 Definition. For any charge p on a field 9 of subsets of a set 12, the total variation lp I of p is defined by
IF [(A)= p + ( ~+ )p - ( ~ ) , A E 9. Clearly, lp I is a positive charge on 2E lp I can also be described following way as in Theorem 2.2.4.
in the
86
T H E O R Y OF CHARGES
4.1.2 Theorem. For any charge A in 9,
p
on a field 9of subsets of a set R and
I~I(A)=sUP
f: Ip(Bi)l
i=l
holds true, where the supremum is taken over all partitions {Bt, B2, . . . , B,} of A in 9. Proof, If ~ ~ / ( A ) < cthen o , p is a bounded charge on the field A n 9 = { A n B; B E fl on A and the above equality follows from Theorem 2.2.4. If Ipl(A) = m, then either p+(A)= co or p-(A) = 00. From the definitions of p + and p - , it follows that SupCy=, Jp(Bi)l=a,where the supremum is taken over all partitions {B1,B2,. . . , B,} of A in 9. This proves the theorem. 0 Now, we introduce the concept of an outer charge.
4.1.3 Definition. Let 9be a field of subsets of a set SZ and p a positive charge on 9. The set function p * :P ( R ) [0, co] defined by --f
p*(A) = Inf {p(B);A c B, B E F}, is called the outer charge induced by
A cR
p.
The following proposition chronicles some of the properties of outer charges.
4.1.4 Proposition. Let F b e a field of subsets of a set SZ and p a positive charge on 9. Then the following are true. (i). p *(D)= 0. (ii). p *(A)s p*(B) if A c B c 0. (iii). p *(A) = p (A) if A E 9. (iv). p * ( A u B ) s p * ( A ) + p * ( B )if A, B c R . (v). I f 9 is a c+-field and p is a measure on 9, p*(Unat A,)S CnZlp*(A,) for any sequence A,, n 2 1 of subsets of R. Proof. Properties (i), (ii) and (iii) are obvious. To prove (iv), we proceed as follows. If either p*(A) = 00 or p*(B) = 00, the inequality obviously follows. Suppose p *(A)< co and p*(B) 0. There exist At, B1 in 9 such that A c A l , B c B t , p ( A l ) s p * ( A ) + & / 2 and p ( B t ) s p * ( B ) + ~ / 2 . Consequently, ~ * * ( A W B ) S ~ * * (=Ap ~ ( A~ lBu B ~ t) ) s p (AI)+ p (B,) s p *(A)+ p *(B) + E. Since E > 0 is arbitrary, the desired inequality follows. The proof of (v) is analogous to that of (iv). 0 4.1.5 Remark. Proposition 4.1.4(v) is not valid if 9 is only a field. Let R={1,2,3,. .. ,a}and 9 = { A ; either A or A" is a finite subset of
4.
87
INTEGRATION
{1,2,3, . . .}). Define p on 9 by p (A) = 0, = 1,
if A is a finite subset of {1,2,3, . . .}.
,
otherwise.
is a measure on the field 9. Note that p*({l,2 , 3 , . . .)) = 1. If we let A, = { n } ,n 2 1, then ps(Un2l A,,)= 1 and p*(A,,)= 0.
p
Finally, we end this section with a result on the outer charge induced by the sum of two charges. 4.1.6 Proposition. Let subsets of a set 0. Then
p1
and p2 be two positive charges on a field 9 of
(p1+pz)*=pT+CLK
Proof. It is obvious that ( p l + pz)*(A)2 p (A)+ p ; (A) for every A c a. If eitherp?(A)=ooorp:(A)=oo, then ( p 1 + p Z ) * ( A ) s p T ( ( A ) + p $ ( A is ) true. Assume that p (A)< oo and p: (A) < CO. Let E > 0. There exist B1, BZin 9 such that A c B1, A c BZ,
T
T
and ~ nB2)5 p ? (A)+ p: (A) + E . Since E > 0 Hence ( p I + pz)*(A)5 ( p +p2)(B1 is arbitrary, the result follows. 0
4.2
NULL SETS AND NULL FUNCTIONS
In this section, we formalize the notions of a null set and a null function. We first introduce the notion of a charge space. 4.2.1 Definition. A charge space is a triple (0,9 p ), , where R is a set, 9 is a field on R and p a charge on 9. 4.2.2 Definition. Let (R, 9,p ) be a charge space. A subset A of R is said to be a p-null set, or simply, a null set if p is understood, if lp ["(A)= 0.
The following properties of null sets follow from Proposition 4.1.4. 4.2.3 Proposition. Let (R, 9, p ) be a charge space. Then the following statements are true. (i). 0 is a null set. (ii). B is a null set if B c A and A is a null set.
88
THEORY OF CHARGES
(iii). u7-1A iis a null set if Al,A2,. . . , A, are null sets, where n is any positive integer. (iv). UnzlA, is a null set if A,, n 2 1 is a sequence of null sets, p is a measure and 9 is a a-field. The concept of a null set leads to the concept of a null function. Definitions. Let (R, 9, p ) be a charge space. (i). A real valued function f on R is said to be a p-null function, or simply a null function if p is understood, if
4.2.4
IP I*GJE a; If(w>l>E N = 0 for every E >O. (ii). Two real valued functions f and g defined on R are said to be equal almost everywhere iff - g is a null function. In this case, we use the notation f = g a.e.[~]. (iii). A real valued function f on R is said to be dominated almost everywhere by a real valued function g on R, if there exists a null function h on R such that f 5 g + h. In such a case, we use the notation f 5 g a.e.[p 3.
A sufficient condition for f 5 g a.e. [ p ] to hold is that lp I*({w E R; f ( w ) > g(w)})= 0 , though not necessary. The following proposition follows easily from Proposition 4.1.4.
Proposition. Let (R, 9, p ) be a charge space. The following statements are true. (i). cf +dg, IflP(p>0) and f g are null functions whenever f and g are null functions on R and c and d are real numbers. (ii). g is a null function if l g l s If I a.e. [ p ] and f is a null function. (5).For real fimctions f , g, h on R, f = h a.e. [ p ] whenever f = g a.e. [ p ] and g = h a.e. [ p ] . (iv). For real functions f l , f 2 , g l , g2 on R and c, d real numbers with f 1 = g l a.e. [ p ] and f 2 = g2 a.e. [ p ] ] cfi+df2=cgl-t-dg2 , a.e. [ p ] and I f l [ = lgll a.e. 4.2.5
[PI.
(v). On the space C(R, 9,p ) of all real functions on R, the binary relation defined by f g i f f = g a.e. [ p ] is an equivalence relation. (vi). For real functions f , g, h on R, f 5 h a.e. [ p ] whenever f 5 g a.e. [ p ] and g Ih a.e. [ p ] . (vii). A subset A of R is a null set if and only if I, is a null function. (viii). For real functions f , g on R, f = g a.e. [ p ] i f f 5 g a.e. [ p J and g f a.e. [ p ] .
-
-
4.2.6 Remark. If f l = g l a.e. [ p ] and f 2 = g2 a.e. [ p ] , it is not true that f i f 2 = g l g 2 a.e. [ p 1. It is not even true that f 2 = g 2 a.e. [ p ] iff = g a.e. [ p ] . The following example explains this.
4. INTEGRATION
89
Example. Let R = { l , 2 , 3 , . . .), 9 the finite-cofinite field on R and p the charge on 9 defined by p (A) = 0, if A is finite, = 1, if A is cofinite.
Letf a n d g o n R b e d e f i n e d b y f ( n ) = n + ( l / n ) , n 21 a n d g ( n ) = n - ( l / n ) , n 2 1. Then f = g a.e. [ p ] . But f z = g z a.e. [ p ] does not hold.
A sufficient condition for a real function f on R to be a null function is that lp I*({w E R; f ( w ) # 0 ) )= 0, though not necessary. The following proposition amplifies this point. 4.2.1 Proposition. Let (R, 9, p ) be a charge space. Let f be a real valued function defined on R. (i). I f IpI*({o E R; f ( w )# 0))= 0, then f is a null function. (ii). The converse of (i) is not true. (iii). If 9 is a u-field and p is a measure on 9, then f is a null function if and only if lp I*({w E R; f ( w )# 0 ) )= 0. Proof. (i). Note that for any E > 0, {oE
a;1f ( w ) l >
E}
c {w E R; f ( w ) # 0).
The monotonicity of Ip I* completes the proof. CL)be as in Remark 4.2.6. Let the function f on (ii). Example. Let (R, 9, R be defined by f ( n ) = l / n , n 21. Then f is a null function. For, {w E R ; If(w)I > E } is a finite set for any E >O. On the other hand, IpI*({w € 0; f (w 1 # 0))= 1. (iii). Observe that
u
E
>O
{WEn;If(o>l>E)=
u {OEn;If(w>l>l/n>
nzl
= { w E R; f ( w ) # 0).
By Proposition 4.2.3, the result follows.
0
Now, we come to the concept of essential boundedness.
4.2.8 Definition. Let (R, 9, p ) be a charge space and f a real valued function on R. f is said to be essentially bounded if there exists a null set A contained in R such that f is bounded on A'. Iff is essentially bounded, the essential supremum of f is denoted by (1film and is defined by
llfllm
= Inf SUP{If(w)l; w E A"),
where the infimum is taken over all null sets A contained in R. The following proposition gives equivalent conditions for essential boundedness of a function.
90
THEORY OF CHARGES
4.2.9 Proposition. Let (R,9, p ) be a charge space and f a real valued function on R. Then the following statements are equivalent. (i). f is essentially bounded. (ii). There exists k > 0 such that lp I*({w E R; If(w)I> k } ) = 0. (iii). There exists k > O such that If l Ik a.e. [ p ] . (iv). There exists a bounded function g on R such that f = g a.e. [ p ] . The following proposition gives the properties of essentially bounded functions which are easily established.
4.2.10 Proposition. Let (R,9,p) be a charge space. (All functions considered below are real valued functions on R.) (i). I f f and g are essentially bounded and c and d are real numbers, then cf + dg is essentially bounded. Further, (ii). 11f Ilm = 0 if and only i f f is a null function. (iii). I f f is essentially bounded and f = g a.e. [ k ] ,then g is essentially bounded and 11film = llgIlm. (iv). If B(R, 9,p ) is the collection of all essentially bounded functions on R, then the map 11 * llm :B(R, .fF, p ) + [0,00) is apseudo-norm on B(R, 9’’ p).
-
4.2.11 Remark. Introduce an equivalence relation on B(R, g, p ) by f g for f , g in B(R, 9, p ) iff = g a.e. [ p ] .Let Cm(R, 9,p ) be the collection of all equivalence classes of B(R, 9, p ) under -. For f in B(R, 9, p ) , let [f ] denote the equivalence class containing f . The map 11 Ilm :Cm(R, 9, p )+ [O, a) defined by Il[f]ilm = l\fllm for [ f ] in Cm(R, 9, p ) is a norm on cm(ag,p ).
-
The following concept of a simple function is important for the development of D-integral.
4.2.12 Definition. Let 9be a field of subsets of a set R. A real valued function f on R is said to be a simple function if it can be written in the form
f
=
c CiIFt
i=l
for some real numbers c l , c2, . . . ,c, ; F1, F z , . . . ,F , in 9with Fi nF j = 0 for every i # j and I Fi = R.
uY=
The following properties of simple functions are clear.
4.2.13 Proposition. Let 3 be a field of subsets of a set R. I f f and g are simple functions on R and c and d are real numbers, then cf +dg, f g and If Ip(p > 0 ) are all simple functions.
4.
INTEGRATION
91
Now, we introduce smooth functions.
4.2.14 Definition. Let (R, 9, p ) be a charge space. A real valued function f on R is said to be smooth if for every E > 0 there exists k > 0 such that IPI*hJE n ; If(u)l>kl) 0 ) are all smooth functions. (ii). I f f is essentially bounded, then f is smooth.
92
THEORY OF CHARGES
4.3 HAZY CONVERGENCE In this section, we introduce the notion of hazy convergence in the space of all real valued functions on R of a charge space ($ p )I . This ,$ notion , is commonly known as “convergence in measure” which is a misnomer in the present context of charges.
4.3.1 Definition. Let (a,9, p ) be a charge space. A sequence f n , n 2 1 of real valued functions on R is said to converge to a real valued function f on R hazily if lim l p l * ( { ~ ~ IR f n ;( W ) - f ( u ) I > E I ) = O
n-tw
for every
E
> 0.
The following proposition establishes that the limit function in hazy convergence is essentially unique.
Proposition. Let (R, 9,p ) be a charge space. If a sequence f,, n 2 1 of real valued functions on R converges to a real valued function f on R 4.3.2
hazily and f = g a.e. [ p ] , then f n , n L 1 converges to g hazily. Conversely, if f n , n 2 1 converges to both f and g hazily, then f = g a.e. [ p ] . Proof. For the first part, observe that for any E > 0, (W
lfn(W)-g(w)l>E)C{W
lfn(~)-f(~)I>~/2)
U { W En;
If(w)-g(w)l>E/21.
For the second part, observe that for any E > 0, {W
If(w)-g(W)I>E)C{W U{W
€0;I f n ( u ) - f ( a ) l > E / 2 ) Ifn(W)-g(W)I>E/2).
The monotonicity and the sub-additivity properties of lp I* complete the proof. 0 The following theorem gives algebraic properties of hazy convergence.
4.3.3 Theorem. Let (R, 9, w ) be a charge space. Let fn, n 2 1 and g,, n L 1 be two sequences of real valued functions on R converging hazily to real valued functions f and g on R respectively. Then the following statements are true. (i). cf, +dg,, n 2 1 converges to cf +dg hazily for any two real numbers c and d. (ii). If , I, n 2 1 converges to If I hazily.
4.
INTEGRATION
93
(iii). f:, n L 1converges to f’ hazily, i f f is a null function. (iv). fnh, n 2 1 converges to f h hazily, if h is a smooth function on R. (v). I f f is smooth and $ is a real valued continuous function defined on the real line, then $ ( f n ) , n 2 1 converges to $ ( f ) hazily. In particular, If n I p , n 2 1 converges to If Ip hazily for any p > 0. (vi). f,g,,, n 2 1 converges to f g hazily i f f and g are smooth functions. (vii). f:, n 2 1 converges to f’ and f i , n 2 1 converges to f - hazily. (viii). f n v g,, n 2 1 converges to f v g and f n A g,, n 2 1 converges to f A g hazily.
Proof. (i), (ii) and (iii) are easy to prove. (iv). Let E >O. Since h is a smooth function, there exists a real number k >0 such that Ip I*({w E R; Ih ( w ) ]> k } ) < ~ 1 2Since . f,, n L 1 converges to f hazily, there exists an integer m 2 1 such that IF I*({o E R; If n ( w ) - f ( w ) l > S / k } )< ~ / whenever 2 n 2 m, where S is a given positive number. So, if nzm,
IP I*({w
If n (0) h ( w ) - f ( w ) h (w )I > 6 ) ) 5 IP I*(b E 0;Iffl ( w ) - f (w)l Ih (w)l> 8,Ih (@)I 5 k } ) + IP I * ( b E a;I f n (0)- f (w)I Ih (0)I > 8,Ih (w )I > k } ) < IP I*({w E a;If n ( w ) - f ( w )I > ~ / k )+)~ / 2 E
< & / 2 + & / 2= E . This proves (iv). (v). Let E~ > 0 and E:! > 0. Since f is smooth, there exists a real number k > 0 such that Ip I*({w E R; If ( w ) l > k } )< ~ ~ / Since 2 . $ is uniformly continuous on [-2k, 2k], there exists S > 0 such that I $ ( x ) - $(y)l< ~2 whenever 1x1, IyI 5 2 k and Ix -yI < S . Without loss of generality, assume that S < k . Since f,, n 2 1 converges to f hazily, there exists m 2 1 such that lp I*({w E Q; If, ( w )-f ( w ) l > 8)) < e1/2 whenever n 2 m. Now, if n 2 m ,
IP I*({w E I $ ( f n (w 1) - (I,( f (w ))I > ~ 2 1 ) 5 IP I*({w E a;I $ ( f n ( w ) ) - $ ( f (w))I > E Z ~ If(w)l>kI) +I~l*({w +IPI*({W
I1Cl(fn(w)>-$(f(w))I>&*, E n ; I$(fn(w))-$(f(O))I>E2,
I f ( w ) l 5 k ,I f n ( w ) - f ( w ) I > S ) ) I f ( w ) l s k ,I f n b ) - f ( W ) I ~ S H
< E1/2+&1/2+0 = E l . Thus $ ( f , ) , n 2 1 converges to $(f) hazily. (vi). Observe that for any n 2 1,
94
THEORY OF CHARGES
Since f and g are smooth, f + g and f - g are smooth. See Proposition 2 4.2.18(i). By (i) and (v), ( f n +gn)2,n 2 1and ( f n -g,) ,n 2 1converge hazily to ( f + g ) 2 and ( f - g ) 2 respectively, Again, by (i), fngn, n 2 1 converges to fg hazily. and f- = $(f The assertion now fol(vii). Observe that 'f = i(f+ lows from (i) and (ii). f, v g, = f ( f , + g, + Ifn - g,l) and f, A g n = (viii). Observe that i(fn+gn-Ifn-gnI)for a l l n r l . 0
If[).
Ifl)
4.3.4 Remark. The assertion of Theorem 4 . 3 . 3 6 ~ is ) not valid unconditionally. We need to impose some conditions on h to ensure the hazy convergence of fnh, n 2 1 to fh. The following is a relevant example. Let (0,9, p ) be as in Remark 4.2.6. For n k 1, let fn
( k )= k, = 1-l/n,
if l s k s n , if k > n ;
f ( k )= 1
for all k in R;
h ( k )= k
for all k in R.
It can be checked easily that f n , n 2 1 converges to f hazily. But fnh, n 2 1 fails to converge to fh = h hazily. Let ( R , 9 , p ) be a charge space and C(n, 9, p ) the collection of all real valued functions on R. One can introduce a pseudo-metric p on C(R, .F,p ) such that convergence in the pseudo-metric space C(R, 9, p ) coincides with hazy convergence. p ) , Let For f in C(R, 9,
$(f, c 1 = c + Ip I * ( b E 0; If(w )I > cl),
c
> 0.
$(f, c ) is a nonnegative number and could be equal to 00. If p is bounded, then $(f,c) is a real number for all c > 0. Now, define
If $(f,c ) = 00 for every c > 0, we define llfll= 1. Now, we give the properties of the function 11 * 1 .
4.3.5 Proposition. The function I\.\\ defined on C(R, g,p ) above has the fdlo wing properties. = 0 if an d only i f f is a null function. (i). (ii). IF+gII 5 llfIl+llgll. (iii). The function p defined by p(f, g) = Ilf-gll for f, g in C(n, 9, p ) is a pseudo-metric on C(R, g,p ) .
]If]
4. INTEGRATION
95
(iv). fn, n L 1 converges to f hazily if and only if p ( f n ,f),n 2 1 converges to zero.
Proof. (i). If f is a null function, then +(f, c ) = c for every c > 0. Consequently ,
MI=
fril+c C
- 0.
Conversely, let llfll= 0. Let k be any positive number. In order to show that Ip I*({w E SZ; If(w)I > k } ) = 0, it suffices to show that lgl*({w E SZ; If(w)I > k } )< E for any 0 < E < k . Let 0 < E < k . Since llfll= 0, there exists c > O such that +(f, c ) / ( l ++(f, c ) ) < E / ( ~ + E ) . This implies that +(f, c ) < E . So, c < E and lp I*({w E SZ; I f ( @ ) [ > c } ) < E . Consequently, Ip I*({w E SZ; If(w)I >k } )I:IP I*(bE a;If(w)I > E ) ) 5 lp I*({w E a; I f ( w)I > c } ) < E . This shows that f is a null function. (ii). Observe that
= Inf r>O.s>O
Inf
I :
r>o,s>o
I Inf XI
+(f+g,r+s) l++(f+g, r+s)
+(f,r) $(g,s) l++(f,r) l++(g,s) +
+(f, r ) +Inf %4 $1 1 + +(f,r ) s>o 1 + +(g, s )
Ilfll+llgll.
I
(The first of the above inequalities can be proved as follows. The function y ( x ) = x / ( l +x), 0 I:X I:co is an increasing function having the additional property y ( x + y ) Iy ( x )+ y ( y ) for all 0 S X , y ICO. We use the convention that y(00) = 1. If c 1 = ICL I * ( b E ; If (w ) + g (w )I > r + sl), c2
=
I@I * ( b E a;If(w)I > r } )
and c3 =
IcLI*(b E a;Ig(0)l >sH,
then c1 I:c2+c3. Further,
96
THEORY OF CHARGES
From this, the first of the above inequalities follows. The rest of the equalities and inequalities are obvious.) (iii). This follows from (i) and (ii). (iv). Suppose fn, n 2 1 converges to f hazily. Then for any c > 0 and E > 0, there exists m L 1 such that IPI*({W
E n ;Ifn(W)-f(W)I>CH 0 is arbitrary, it follows that f n , we have lim n L 1 converges to f in the pseudo-metric space (C(R, $, p ) , p ) . l+c
Conversely, let p ( f n , f ) , n 2 1 converge to zero. Let k be any positive number. Let 0 < E < k be arbitrary. There exists m L 1 such that p(fn, f)< ~ / ( 1E+) whenever n L m.Now, let n L m be given. Since
This shows that fn, n L 1 converges to f hazily.
0
4.4 D-INTEGRAL In this section, we develop the basic ideas concerning D-integrals. We start with simple functions.
4.4.1 Definition. Let (R, 9, p ) be a charge space and f a simple function on R with a representationf = I:=,cJFifor some real numbers c1, CZ, . . . ,c, and partition {F,, F 2 .. . ,F,} of R in $, f is said to be D-integrable if Ip I(Fi)< CX, whenever ci # 0, and the D-integral of f, denoted by D 5 f dp, is defined to be the real number I;=, cip(Fi). (We adopt the convention = 0.) that 0 (*a)
4.
97
INTEGRATION
We settle the question of unambiguity of the above definition in the following proposition. Proposition. Let (a,.IF, p ) be a charge space. (i). Let f be a simple function with a representation f =ELl C ~ Ifor E ~some real numbers cl, c 2 , . . . ,cm and partition {El, Ez, . . . ,Em}of R in $such that lpl(Ei) 0 be any number such that E < Icil. Then {w E R: lh(w)l>E } = Uj., Fj, where J = { j ; 1 5j 5 m and
98
THEORY OF CHARGES
Icjl>~}. Obviously, i E J. Since h is a null function, IpI*(uj.J F j ) = l p l ( u j p FJ j ) = 0. Consequently, IpI(Fi)= 0. This proves (iii). (iv). The condition of Definition 4.4.1 for I$ is easily verified from the 0 hypothesis that f is D-integrable and simple. Part (iv) of the above proposition enables us to define D D-integrable simple function f and E in 9.
sEf d p for any
4.4.3 Definition. If f is a D-integrable simple function and E E 9, then D jEf d p stands for D 5 I$ dp. The following theorem gives the properties of integrals of simple functions.
-
4.4.4 Theorem. Let (R, 9, p ) be a charge space.
(i). I f f is a simple function on R and D-integrable with respect to p , then f is D-integrable with respect to p + and p - also. Further, for any E in 9, fdp+-D
J
fdp-. E
(ii). I f f and g are D-integrable simple functions on R, c and d are real numbers and E E 9, then cf + dg is a D-integrable simple function and
D
JE
(cf+dg)dp=c
(iii). I f f is a simple function on il and D-integrable with respect to p, then If l is a simple function on R which is D-integrable with respect to Ip I and for any E in 9
(iv). I f f is a D-integrable simple function on 9, i.e. I$? 0 a.e. [ p ] , then
and f 2 0 a.e.
(v). I f f and g are D-integrable simple functions on R and f E in 9, i.e. I $ s I E g a.e. [ p ] , then
(vi). I f f and g are D-integrable simple functions on R, then
[ p ] on
5g
If l ,
E in
a.e. [ p ] on
Igl,
If
+gl,
99
4. INTEGRATION
11f I - lgll are all D-integrable simple functions and for any E in 9,
(vii). Iffis a D-integrable simple function on R and c 5 f I d a.e. [ p ] on E in 9, i.e. c I ~ Id5 5 dIE a.e. [ p ] for some real numbers c and d, then c IP 103 5 D
I
E
f dlk I5dill. KE).
(viii). Iffis a D-integrable simple function on R, then the set function A on 9defined by c
A(F)=DJfdp,
F
E
~
F
is a bounded charge on 3? Also, IA [(F)= D 5, If l dlp 1, F E97 Further, A is absolutely continuous with respect to p in the following sense. Given E > 0, there exists S >0 such that \A (E)I < E whenever E E 9and Ip l(E) g(w)}) = 0. (The set {w E 0;f ( w ) >g(w)} does belong to 9.With ) this observation, we proceed as follows. (i). Let f = ciIEi be a representation of f. Obviously, p+(Ei)< co and p-(Ei) < 03 whenever ci # 0. Hence f is D-integrable with respect to p + as well as with respect to p - . Further, if we let J = (1s i Im ;ci# 0}, then r
D
J
m
cip(Ei) = C cip(Ei)
f dp = i=l
i=l
xi"=,
ieJ
i=l
(ii). Letf = c i E andg , = dJF, be representations off andg respectively such that lp I(Ei)< co whenever ci # 0 and f pI(Fj) < co whenever dj # 0.
100
THEORY OF CHARGES
Then
Suppose cci + ddj # 0. Then, either cci # 0 or ddj # 0. This implies that either ci# 0 or d j # 0. This means that either IpI(Ei) 0, there exists a simple function g on such that J pI*({w E R; I f ( @ ) - g ( w ) l > ~ / 2 } ~ / 2 } . Since Ip (*(G)= Inf {lp ((F);G c F, F E9 1, there exists Fo in 9 such that G c F o a n d I p I ( F o ) < e . L e t F i = E i n F & i = 1 , 2 , ..., m. Now,if l s i l m and o,U ’ E Fi, then If ( w )- g ( w ) l s ~ / 2If,( w ’ )- g ( w ’ ) l s ~ / and 2 g ( o ) = ci= g(w‘). Therefore,
zEl
If(w ) -f(w
’11 IIf(w ) - ci I+ lci -f(o ’)I 5
I f ( w >- g(w)l+ I f (a’)- g(w ’)I 5 E .
This shows that f is Tp-measurable. Conversely, let f be T2-measurable. For each n 2 1, let {Fno, Fnlr Fnz,. . . ,Fnk,}be a partition of n in 9such that Ip I(Fno) < l / n and If ( w ) f(w’)I < l / n for every w , w ’ in Fni and for every i = 1,2, . . . ,k,. For each n 2 1 and 1 Ii 5 k,, choose and fix wni in F,i. For each n 2 1, let k, fn
=
cf
i=l
+o
(wni)lFm,
*
IFn~*
Each f n is a simple function and f n , n 2 1 converges to f hazily. For, let E > O and m 2 1be such that l / m < E . If n L m , {w €0;I f , , ( o ) - f ( w ) I > ~ ) c F,o. Consequently, Ipl*({w en; I f n ( w ) - f ( w ) I > e } ) I I p I * ( F n o ) < l / n . This 0 establishes the result. 4.4.8
Corollary. Every TI-measurablefunction is smooth.
102
THEORY OF CHARGES
Proof. From the definition of T2-measurability, observe that every T2measurable function is smooth. The result now follows from Theorem 4.4.7. 0 Relations between TI-measurable functions, Smooth functions and bounded functions can be described as follows, T1-measurable function
%\Smooth function 4.4.9 Corollary. Let (a,9,p ) be a charge space. (i). I f f and g are TI-measurable functions on SZ and c and d are real numbers, then cf + dg and f g are all TI-measurable. (ii). If I,4 is a real valued continuous function defined on the real line and f is TI-measurable,then I++( f ) is TI-measurable. In particular, I f 1’ ( p > 0 ) is TI-measurable i f f is T1-measurable. (iii). If f n , n 2 1 is a sequence of TI-measurable functions converging to f hazily, then f is T1-measurable and fcl(fn), n 2 1 converges to $ ( f ) hazily, where I,4 is as in (ii).
Proof. (i) and (ii) follow from Corollary 4.4.8 and Theorem 4.3.3(i) and (vi). (iii) follows from the Remark following Definition 4.4.5, Theorem 4.3.3(v) and Corollary 4.4.8. The following proposition is instrumental in establishing the unambiguity in the general definition of D-integral. 4.4.10 Proposition. Let (a,9,p ) be a charge space. Let f n l , n 2 1 and f n 2 , n 2 1 be two sequences of D-integrable simple functions on SZ converging to a real valued function f on R hazily. Suppose
for i = 1,2. Then D JE f n i d p converges uniformly over E in 9 for each i = 1, 2, and the limits coincide. Proof. By Proposition 4.4.4(iii), for any m , n 2 1 and i = 1, 2
for every E in 9. Consequently, D jEfni dp, n 2 1converges uniformly over E in 9for each i = 1, 2. It remains to be shown that for each E in 9, fnl
d p = lim D n-tm
f n z dp.
4.
103
INTEGRATION
This is carried out in the following steps. 1". Let g , = l f n ~ - f n 2 1 , n 2 1, and p , ( F ) = D jFgndlpl for F in 9and y1 2 1. Each pn is a positive bounded charge on 9. 2". We claim that p,, n 2 1 converges uniformly over 9. For any F in 9 and n, rn 2 1, by Theorem 4.4.4(vi),
5~
J
Ifnl-frnlI
dIFI+D
I
Ifnz-frnzl
dbI-
Thus the From this, it follows that p,, n 2 1 is uniformly Cauchy over 9. claim is established. 3". Let A(F)=lim,,+mpn(F), F E ~It .suffices to show that A = O . For, for any E in 9,
If
p,(E) = 0, then
4". Now, we claim that A is absolutely continuous with respect to p. (See Theorem 4.4.4(viii).) Let E >O. There exists N 2 1 such that ]k,(E)-A (E)(< ~ / for 2 every E in 9 and n 2N.Since p N is absolutely continuous with respect to p, there exists S > O such that ~ N ( E O , there exists 6 > O such that IA(E)I < E whenever E E and~ IF I(E)EI2). ;
From these set inclusions, it follows that IpI*({w ER; I h , ( w ) - f ( w ) l > ~ } ) < : ~ / 2 + ~ / 2 = ~
whenever n LN. Next, observe that
+D
J Ifm-hmldl/l.l
for all m, n z 1. From this, it follows that limn.m+a,D 5 )h, -hml dip( = 0.
4.
INTEGRATION
115
Hence h,, n 2 1 is a determining sequence for f . So, f is D-integrable. Further, for n 2 1,
The second term on the right above converges to zero as n + CO, by Lemma 4.4.12. Hence lim D n+m
I Ifn
-f
I dlp I = 0.
This completes the proof.
4.5
S-INTEGRAL
In this section, we introduce S-integrals which are of Stieltjes type in the framework of charge spaces. We also show that D-integrals and Sintegrals coincide in the case of positive bounded charges and bounded functions. I n what follows, we assume that all the charges are positive bounded unless otherwise specified. For a given field 9 on a set R, let 8 denote the collection of all finite partitions of R in 9. On 8, we define a partial order by PI 2 P2 for PI, PZ in 8 if PI is a refinement of P2, i.e. every set in P2 is a union of sets in PI. Indeed, ( 8 , ~ is a)directed set. 4.5.1 Definition. Let ( R , 9 , p ) be a charge space. Let f be a bounded real valued function on R. For P = {Fl, F2, . . ,F,} in 8, let
.
and
L(P) is called the lower sum associated with P and U(P) is called the upper sum associated with P. (Since f and p are bounded, L(P) and U(P) are real numbers.) The following proposition gives some inequalities between these sums.
Proposition. Let (R, 9,p ) be a charge space and f a bounded real valued function on R. Then for any PI L P in ~ 8) 4.5.2
116
THEORY OF CHARGES
Proof. Since P 1 z P Z , every set in Pz is a union of sets in PI. Hence the above inequalities easily follow. Of course, we use the fact that p is positive in proving the above inequalities. Thus, we observe that the net {U(P);P E P} defined on the directed set (9,L) is a decreasing net of real numbers bounded below and hence has a limit. The net {L(P); P E 9)defined over the directed set ( 9 ,is~an) increasing net of real numbers bounded above and therefore, has a limit.
4.5.3 Definitions. Let (fl, 9, p ) be a charge space and f a bounded real valued function on R. Let
J
f d p = Inf U(P) = lim U(P) PEP
PEP
and
-
f d p = Sup L(P) = lim L(P). PE B
-
j f d p is called the upper integral of f with respect to
p
and
5f
d p the
lower integral off with respect to p. The following proposition is obvious in view of Proposition 4.5.2.
4.5.4 Proposition. Let (fl, 9, p ) be a charge space and f a bounded real valued function on fl. Then
Now, we define the S-integral.
4.5.5 Definition. Let (O,?F, p ) be a charge space and f a bounded real valued function on fl. f is said to be S-integrable if
If f is S-integrable, the S-integral of f is denoted by S I f d p and is
-
defined to be the common number 5 f d p = f d p . -
4.5.6 Remark. I f f is a simple function, then S j f d p = D f d p .
117
4. INTEGRATION
We link S-integrability and D-integrability of a function in the following result. 4.5.7 Theorem. Let (a, 9, t ~ be ) a charge space and f a bounded real valued function on Q. Then the following statements are equivalent. (i). f is TI-measurable. (ii). f is T2-measurable. (iii). f is S-integrable. (iv). There exists a real number a with the following property. For every E >0, there exists a partition Po in B such that for every partition P in B with P = {Fl, F2, . . . , F,} 2 POand for every wi in Fi, i = 1 , 2 , . . . , n,
holds. (v). For every E > 0, there exists a partition Po in B such that for every partition P in 9 with P ={F1, F 2 , . . . , F,}?Po and for every w i l , wi2 in Fi, i = l , 2, .. . , n ,
IC
I
I n
i = l ( f ( w i i ) - f ( W i Z ) ) ~ ( F i ) l< E
holds. (vi). For every E >0 , there exists a partition Po in B such that for every partition P in 9 with P = { F I , F2, . . . ,F,) 2 PO,
i[
i=l
SUP
If(wi1)-f(Wi2)11tL
(Fi) < E
w,l,w,zeFi
holds. (vii). For every E > 0 , there exists a partition PO= { E l , E2. . . . ,E m } in $9’ such that for any partition { E l l , E12,. . . , Elkl, E21, E22,.. ., E2k2,. .., Eml, Em2,. . ., Emk,} in 9 with E i = U f ~ , E i i , i = 1, 2, . . . ,m and for every choice Aii, j = 1 , 2 , . . . , ki, i = 1 , 2 , . . . ,m, of real numbers satisfying
holds. (viii). f is D-integrable.
118
THEORY OF CHARGES
Proof. (i)+(ii). This follows from Theorem 4.4.7. (ii) (iii). Let E > 0. We show that
+
This then would prove that f is S-integrable. Let M = SupwEnlf(w)l. Since f is Tz-measurable, there exists a partition P = {Fo,F1, Fz, . . . ,F,} in 9 such that p(Fo) 0. There exists a
partition POin P such that for every P z Po, U(P)- a < E and a - L(P) < 8. Let P = {FI,Fz, . . . ,F,} be any partition in 9 such that P z PO.Let wi in Fi, i = 1, 2, . . . ,YE be arbitrary. Then n
n
C f(Wi)P(Fi)-a i=l
5
C (Supf(w))P(Fi)-a
i=l
w€Fi
=U(P)-a
0 ; Ip I*({@E R; If(w)I > k}) = 0).
(We use the convention that the infimum over an empty set is co.) If f is a null function, obviously, l f 1 1, = 0 for any 1~p I03. If f and g are such that f = g a.e. [ p ] , f c L P ( f l , 9 , p ) for some 1 s p S c 0 , then g E L,(R, 9, p ) and Il f l , = llgllp We want to show that the nonnegative function II.IIp defined on L,(fl, 9 , p ) for 11p1co is a pseudo-norm on L,(fl, 9,p ) . We need the following inequalities for this purpose. The first of these is Holder’s inequality. 4.6.2 Theorem. Let (R, 9,p ) be a charge space and p and q be two positive numbers satisfying l / p + l / q = l . 1 f f € L p ( f l , 9 , p )and g E L q ( R , 9 , p ) , then fg E Ll(fl, 9, p ) and
llfgll1 5 llfllPllg11q. Proof. Assume, first, that p > 1 and q > 1. The function $(t) =
t P t-+-,
P
9
t >o
has a global minimum at t = 1. Therefore, for every t > 0, $ ( t ) 2 $(1)= l / p + l / q = 1. Let a and b be any two positive numbers and t = (al’q)/(bl’p). Then
This implies that ab s a P / pfbq/q. This inequality is valid even if a = 0 or b = 0. Now, we turn to the proof of the theorem. Iff or g is a null function, then fg is a null function. This can be proved as follows. Suppose f is a null function. Since any T1-measurable function is smooth, g is smooth. So, for a given E > O , there exists k > O such that Ip I*({@ E R; Ig(w)l> k}) I E . Consequently, for any s > 0 , IPI*({@Efln; If(@)g(w)l>S})~ICLI*({@ E n ;I f ( ~ ) l > S l W
+ IPl*({@E 0; Ig(w)l > kl) SO+&
=&.
This shows that fg is a null function. In this case, the theorem is evidently true.
4.
123
INTEGRATION
Since f and g are T1-measurable, fg is TI-measurable. See Corollary 4.4.9(i). By Theorem 4.4.18, fg is D-integrable. It is now obvious that llfgll1 5 ( l / P + ~/q~llfllPll~ll~ = IlfllPllgllq. If p = 1, then q =a.Therefore, lfgl s k l f l a.e. [ p ] for any number k > llgllm. Since fg is TI-measurable, by Theorem 4.4.18, it follows that fg is D-integrable and lFgll1s kllflll for any k >llgllm. Consequently, llfgllls Ilflllllgllm. The case p = 00 and q = 1 can be disposed of in a similar vein. 0
A more general version of the above theorem is the following result.
4.6.3 Corollary. Let (a,9, p ) be a charge space and p , q, r be numbers satisfying 1s p , q, r 5 00 and l / r = l / p + l/q. If f E L p ( R , 9 ,p ) and g E Lq (a,9, P ), then f g E Lr (a,9, CL 1 and llfgllr 5 Ilfllpllgllq* Proof. There are only three possibilities involving 00. Case (i). p = 1,q = 00, r = 1. Case (ii). p = 00, q = 1, r = 1. Case (iii). p = 00, q = 00, r = 00. In Cases (i) and (ii), the result follows from Theorem 4.6.2. For the case (iii), we proceed as follows. For any k > 0 and t > 0 satisfying k > llfl1m and t > llgllm, we have Ir.Ll"({w E n ; I f ( w ) g ( w ) l > k t } ) ~ I l l l * ( { w E n ;
If(w)l>kH
+lPl*(b ER; Ig(4l>tN = 0.
This shows that fg is essentially bounded. Further, llfgllmskt for any k > IIfIIm and t > IIgIIm. Hence IIfgIIm 5 IIfIImIkII.o. Let us look into the case 1< p (lf+gop~q.
so, I f + g ) p - ( p / q ) = If+g1521’q(lf(p+ ( g l p ) l / pThus, . we obtain the inequality If+gl” 52p/q(1flp+ Igl”). By Corollary 4.4.9 and Theorem 4.4.18,
4.
125
INTEGRATION
This follows from Holder's inequality. Therefore,
From the above inequality, it follows that (D
[ If+gl" dlul)
'-(l/q)
=Ilf+gllp ~llfllP+llgllP.
0
This completes the proof.
Theorem. Let (fl, 9, p ) be a charge space. Then,for each 15 p 5 00, (Lp(fl,9, p ) , 1) [Ip) is a linear space with a pseudo-norm 11 IIp.
4.6.7
-
Proof. It is now obvious that each Lp(fl,9, p ) is a linear space. Further, if f = 0, then Ilfl , = 0. For any real number c and f in L,(fl, 5 .F, p ) , it is ~ 9,p ) and that Ilcfll, = Ic I l fl1,. The inequality obvious that c f L,(fl, Ilf+gllp 5 l f11, +llgllp for f,g in L,(R, 9, E L ) follows from Theorem 4.6.6.0
Remark. If p is a 0-1 valued charge on a field 9 of subsets of a p ) / - is isometrically isomorphic to the real line R set fl, then Lp(fl,9, for any 15 p 5 CO, where Lp(fl,9,p ) / - is the collection of all equivalence induced by the classes of L,(fl, 9,p ) under the equivalence relation notion of a null function. (See Example 4.4.14.) Consequently, L,(fl, 9,p ) is complete. In general, (Lp(fl,9,p ) , 11 )1, need not be complete. Let fl = {1,2, . .}, 9the finite-cofinite field on fl and p the charge on 9defined by 4.6.8
-
.
p(A) =
1
12"'
if A is finite,
noA
= 2-
1 ".AC
1 2"'
-
if A is cofinite.
126
THEORY OF CHARGES
.
Let A,, = { 1 , 2 , 3 , . . ,n } , n 2 1. We claim that lim D m,n+m
J lIA,,-
This claim is established if we observe that D J IIA, -IA,/ dp = p(A,AA,) which converges to zero as m, n + co.Suppose la,, n L 1 converges to some function f on R hazily. We show that f- 1. For every k 2 1, there exists nk 1 1 such that p*({w E a; IIA,,(W)-~(WI > l/2k}) < 1/2k. Let Bk in 9be ~ ER;l I ~ , ~ ( ~ ) - f ( w )1l/>2 k } ~ B kfor any set such that ~ ( B k ) < l / 2and{w k 2 1. Assume, without loss of generality, that nl < 112 < n3 < * . We now give the properties of the sets Bk, k L 1. (i). Each Bk is a finite set. For, for any infinite set A, p*(A) 2 1. (ii). B k ~ { k + l , k + 2 , .. .}, k 2 l . (iii). I I ~ , , ( w ) - f ( w ) l 5 1/2&,if w&Bk,for k 2 1. (iv). k &Bk,for each k 2 1. Now, let ko in R be fixed. Let E > 0. We show that lf(ko)- 11 < E . This then would imply that f ( k o )= 1. Let N 2 1 be such that 1/2N < E . Let p 2 max{N, ko}. Since B, c { p + 1, p +2, . . .} and p 2 ko, ko & B,. Further, ko E A h c Anko c Anp. Therefore, If(ko)- I A , ~ ( ~=oI f)( ~ k 0 )- 11 5 1/2' 5 1/2N< B . This shows t h a t f s 1. Next, we show that IAn,n 2 1 does not converge to the constant function identically equal to 1 hazily. Let E = i. Then the set {w E 0;[IA,(W) - 11 >b} is a cofinite set and consequently, p ({w E R; IIA,(w)- 11 >b}) 2 1. So, IA., n 2 1 fails to converge to hazily. Thus, we have a Cauchy sequence in L1(R, 9 , p ) not convergent in L1(R, 9, p ) . Hence L l ( R , 9 , p ) is not complete.
4.6.9 Remark. In L,(R, $, p ) , if we introduce the equivalence relation by f - g for f, g in L,(R, 9, p ) if f = g a.e. [ p ] ] ,then the collection of p ) / - of L,(R, 9, p ) equipped with the all equivalence classes L,(R, 9, norm
-
Il[fIIIP = IlfIIP for f in L,(R, 9, p ) is a normed linear space, where [f]is the equivalence p ) containing f. class in L,(R, 9, Next, we aim at proving Lebesgue dominated convergence theorem. For this, we need the following theorem on convergence in L,-spaces.
4.6.10 Theorem. Let (a,9, p ) be a charge space and 1 S p 0, there exists S > 0 such that A, (E)< E for every n 2 1 whenever E E 9 and IP I(E)< 6.
(iii). For each E > 0, there exists a set E, A,(Ez) < E for every n z 1.
E 9 such
that
I(E,) < 00 and
Proof. The proof is carried out in the following steps. 1”. “Only if” part. If h is a nonnegative simple D-integrable function on 0, the following inequality known as Chebychev’s inequality is easy to establish. D IPl({wEa;h(W)>r))s
J h dlPl r
for any r > 0. 2”. In order to show that f,, II 2 1 converges to f hazily, it suffices to show that for any given E > 0 and s 2 > 0, there exist N z 1 and sets A, in 9 for n 2 N such that \pI(A,)< E~ and If , ( w ) -f(w)I < E~ for w in A: whenever n 2N. 3”.Let r and E be two positive numbers satisfying (2r)”’ < s2 and 3 ~ / I2 TI)< &IT.
See Lemma 4.4.12. Let B , in 9 be any set such that Ipl(B,)r}, n z 1. Then, by lo,lpl(C,) 5 (D 5 h , d(FI)/r for every n z 1. Let A, = B , u C,, n z 1. Then
J
I P I ( A . ) ~ l P I ( B f l ) + I P l ~ c n ~ ~ ~ /hr,+dlPl)/r (D
0, there exists 61 > O such that d l p l < ~ / 2wheneverFE9andIpI(F)N. Since A l , AZ, . . . ,AN are all absolutely continuous with respect to p , by Theorem 2 F E 9 and 4.4.13(xi), there exists SZ> 0 such that A, (F) < ~ / whenever Ipl(F) < & for n = 1 , 2 , . . . ,N . Let S =min (61,Sz). If F E and~ IpI(F)N .
This proves (ii). 7”. We now prove (iii).Let E > 0. There exists Eoin 9such that lp ~(EO) < 00 and D jE; dlpl< ~/2’. See Lemma 4.4.15. Further, there exists N 2 1 such that D d l p l < ~ / 2for ~ every n > N . Also, for each n = 1,2, . . . ,N, there exists E, in 9 such that ( pI(E,) < 00 and A,(E‘,) < E . See Lemma 4.4.15. Let E, = EOu El u uEN. Obviously, lp I(EE) < 00. If 15 n s N , then A,(ES)IA,(E;) N , then also,
Ifl”
If, -fl”
-
< [(&/2”)’/”+ (&/2”)’/”1”= E . This completes the proof of “only if” part. 8”. Now, we prove “if” part. First, we show that g = is D-integrable. Let g , = Ifnlp, n L 1. In view of Theorem 4.4.20, it suffices to show that g,, n 1 1 converges to g hazily and
Ifl”
lim D
m,n-co
J
lg, -gml dlpl= 0.
Hazy convergence can be proved as follows, Since each f , is TI-measurable and f,, n L 1 converges to f hazily (by (i)),f is T1-measurable and so, n 2 1 converges to hazily. See Corollary 4.4.9(iii).
Ifl”
lf,lp,
4.
129
INTEGRATION
9". It remains to be shown that the above limit is indeed equal to zero. For this, first, we show that if E E 9 and I(E)< 00, then
Let E > 0. By (ii), there exists 6 > 0 such that A,(F) < ~ / ( + 2 ( p((E))for every n z 1 whenever F E 9 and lp [(F)< 6. Since g,, n 2 1 converges to g hazily, we have m,n+m lim l~l*(b l ~ n ( ~ ) - g , ( ~ ) J ~ ~ / ( ~ + l ~ l ( E ) ) ~ ) = ~ . Consequently, there exist N L 1 and sets En, in 9for n, m r N such that lp I(E,,) < 6 and lg, (0)- gm(w)l< ~ / ( + 2 1~ I(E)) for w in Ei,,, whenever n, m 2 N. Now, if n, m 2 N,
D / lgn-gm/dlPI=D/ E
Ign-grnIdlPI+D E,,nE
J
Ez,nE
Ign-grnIdlPI
This establishes the desired assertion. 10". Next, we show that Iimm,n+w D lg,, -g,l d(pI= 0. Let E >O. By (iii), there exists a set E in 9 such that l(E)< co and A,(E') < E for all n 2 1. Consequently,
sD/EIgn-gmIdIPI+E+E l . < 00, for all n, m 2 1. Taking limits as n, m -f CO, we obtain, by 9" as ( ~ I(E) that limm,,+wD (g,, - g, 1 d ( pI 5 2.5. Since E > 0 is arbitrary, the above limit is indeed equal to zero. Hence g is D-integrable. = 0. Let A. on 9be defined by 11". Finally, we show that limn-tml f, - f [ l , Ao(F)= D jFIflp dip[, F in 9.By Theorem 4.4.13(xi) and Lemma 4.4.15, (ii) and (iii) hold for the sequence Ao, AI, AZ, . . . also.
130
THEORY OF CHARGES
Let E >O. By (iii), there exists E, in such that Ipl(E,) 0 such that (An(F))””< E for every n 2 0 whenever F E 9 and lp I(F)< 6. Since f n , n 2 1 converges to f hazily, there exist sets A, in 9such that limx+m[yl(A,) = 0 and \ f , ( w ) - f ( ~ ) I< r for every w in A‘, and n 2 1. So, there exists N 2 1 such that IpI(An)O. By Theorem 4.4.13(xi), there exists 8 > O such that D jF lglp dlp I < E whenever FE9 and Ip [(F)< 8. Since Ifnl c Igl a.e. [ p ] for every n 1 1,it follows that D jFIf n 1’ dlp I < E whenever F E 9and lp I(F) < 8. Thus (ii) holds. Next, we show that (iii) of Theorem 4.6.10 holds. By Lemma
132
THEORY OF CHARGES
4.4.15,thereexistsasetFEi n 9 s u c h t h a t IpI(F,) a for every (Y in D such that I l f p , -f[Ip > E . Then fpQ, a E D , being a subnet of fa, a E D , also converges to f hazily. So, again, there exists a sequence & < pm2< * * such that fpQ , n 2 1 converges to f hazily. But, by what we have proved for sequencGs, limn+mI l f p Qn -flip = 0. This is a contradiction. Hence [If, -fll, = 0. Thus for nets, (a)J(b) holds. Similarly, ( b ) j(a) can be established for nets too. We come to the second part. The equivalence of (a’) and (d’) follows from the first part. The implications (d’)3 (c’) (b’) are clear. We prove a E D. Then the uniform (b’)+ (d’). Let A,(F) = D 5, ( f , - f ) dip I, F E 9, convergence of A,, a E D to zero implies that IA, l(R), (Y E D converges to 0 zero, because ]A, l(R>= SUPFESIA, (F)-A, (Fc)l.
+
We end this section with a result on the denseness of D-integrable simple p ) for 1 5 p < a. functions in L, (R, 9, 4.6.15 Theorem. Let ( R , 9 , p ) be a charge space. Let Sim(R,9, p ) be the space of all D-integrable simple functions on R. Then Sim(R, 9, p ) is dense in L, (R, 9,p ) for every 1 5 p < CO.
Proof. Let 1 s p < a be fixed and f~ L,(R, C ,F , p ) . For a given E >O, by Lemma 4.4.15, there exists a set A in 9 such that IpI(A)
I 5 If(4 -f”
+lfN(W)
-fdw’)I
+IfN(W’)
-f(w’>I
< &/3+ &/3+&I3= E. This shows that f is 9-continuous. It is obvious that limn+mllfn -f l l = 0. II.II) is a Banach space. Hence (%‘(R,
134
THEORY OF CHARGES
(ii). Let f be $-continuous. For n rl, there is a partition {F,I, Fn2, . . . , Fnk,} of R in 9 such that for every i E {1,2, . . . , k,}, I f ( w ) - f ( w ' ) l < l / n for all w , w' in Fni.For each n 2 1 and i = 1,2, . . . , k,, choose and fix wni in Fni.Let
If(@)
-f,(w)I = We claim that f,, n 2 1 converges to f uniformly. If w E F,i, If(w)-f(wni)< l l / n for all n 2 1 and i = 1,2, . . . ,k,. Consequently, Ilf-f,ll< l / n for all n 2 1. Hence Ilf-f,ll = 0. This shows that the class of all simple functions on R is a dense subset of %(a,8. (iii). This is now obvious. 0
We now give a characterization of $-continuous functions based on D-integrability.
Theorem. Let 9be a field of subsets of a set R. Then f E %'(a,fl if and only i f f is D-integrable with respect to every bounded charge A on $.
4.7.3
Proof. Let f~ %(R, 9). Note that if A is a bounded charge on 9, then f is Tz-measurable with respect to A and hence is D-integrable. See Theorem 4.5.7. Conversely, suppose f is D-integrable with respect to every bounded Suppose f is not $-continuous. There exists E > O such that charge on 9. there exists 1Ii In satisfying given any partition {F1,F2, . . . ,F,} of R in 9, O(f,Fi) = Sup{lf(w)-f(w')l; w , O'E Fi}> E . For each partition P = {F1,F2, . . . ,F,} of R in 9, let A(P) = U{Fi; 1 si I n , O(f, Fi)> E } . Let B denote the collection of all finite partitions of R in 9. Then {A(P); P E 9 ) is a filter base in 9. To see this, let P1, Pz E 8 and P any partition in B finer than both P1 and Pz. Suppose for some F in P, O ( f ,F) > E . Then F is contained in some G1in PI and in some GZin P2. Consequently, O ( f ,GI)> E and O ( f ,Gz)> E . Thus F c G1nG2 and from this, it follows that A(P) c A(P1)nA(P2). Further, note that A(P) # 0 for every P in 8.Thus we have proved that {A(P); P E P } is a filter base in $. Hence there is a maximal Define A on 9by filter 8 in 9containing {A(P); P E 8}. A(E)=l, ifEE'iY, =0, ifEg'iYandEE9. A is a 0-1 valued charge on 9.
We claim that f is not D-integrable with respect to A. Suppose f is D-integrable with respect to A. Then f = c a.e. [A] for some constant c. See Example 4.4.14. Then A*({w E R; if(w)-cl > s / 2 } ) = 0. Consequently, there exists B in 9 such that {w E R; I f ( w ) - c I > ~ / 2 c} B and A(B)=O. Now, we look at the partition P={B,B"} in B. Then
4.
135
INTEGRATION
B ' c ( w ~ 0 ;I f ( w ) - c l 5 & / 2 } . For any w , w' in B', I f ( w ) - f ( w ' ) l I If(w)-cl+)f(w')-cI S E / ~ + E / ~ = E . Therefore, O ( f ,B')IE. Hence A(P) = B E 8. By the definition of A, A (B) = 1.This contradiction proves the 0 desired assertion. Now, we characterize the continuous linear functionals on (%(a, 9), II-II). 4.7.4 Theorem. Let 9 b e a field of subsets of a set 0.Let T b e a continuous linear functional on %(a, Then there exists a unique bounded charge p on 9 s u c h that
e.
for every f in %(a, 9).Further, llTll= Sup {IT(f)l; llflls1}=Ipl(a). Conversely, for any given bounded charge A on 9, the functional T' on %'(a, 9) defined by T ' ( f )= D J f dA, f in %(a, 9) is a continuous linear functional on %'(a, 9-with ) IlT'll= IA /(a). If T is a nonnegative linear functional on %?(a, 9), i.e. T (f ) 2 0 i f f 2 0 , then p is a positive charge. I AE %?(a, 9)and S O , define p (A)= T(IA).It is obvious Proof. For A in 9, that p is a charge on 9.Also, IpLA)I= IT(IA)I 511TllllrAl1511Tll
for every A in 9. This shows that p is a bounded charge on 9.Now, we 9), T ( f )= D J f d p . Iff ciIAi is a simple claim that for any f in %?(a, function in %(a, then
=xi"=,
a,
T ( f )=
i=l
ciT(IAi)=
f
i=l
c i p ( A i )= D
J f dp.
For any f in %(a, 9), by Theorem 4.7.2, there exists a sequence f n , n 5 1 of simple functions in %(a, 9) converging to f uniformly. It is obvious that f,,, n 2 1 converges to f hazily. Further, since p is bounded,
Thus, f n , n 2 1 is a determining sequence for f with respect to p. Consequently,
D
J'
f dp
= lim n+m
D
as T is continuous on %(a, 9).The claim is thus established.
136
THEORY OF CHARGES
Now, if fE%‘(R,.F) and llfilsl,then ~ T ( f ) ~ = ~ D ~ f d p ~ ~ D ~ Ipl(R). See Theorem 4.4.13 (iii). Hence ~ ~ T \ ~ I ~ Since ~ ~ Ipl(R) ( R ) =. Sup Ip (Ai)/,where the supremum is taken over all finite partitions {Al, A2, . . . ,A,} of R in 9, for any given E >0, there exists a partition (FI,F2, . . . ,F,,,} of R in 9such that m
Since E > 0 is arbitrary, we have Ip I(R) IIlTll. This shows that llTll= Ip [(a). The rest of the theorem is obvious. 0
4.7.5 Corollary. The dual of %‘(a,9)= %‘*(a, 9) = ba(R, F). 4.7.6 Corollary. Let X be a compact Hausdorff totally disconnected space, 9 the field of all clopen subsets of X, 93 the Bore1 u-field on X, %(X)the space of all real continuous functions on X and ca(X, 93) the space of all bounded regular measures on 93. (%(X) is a Banach space under supremum norm and ca(X, 93) is a Banach space under total variation norm.) Then the following statements are true. (9. %(X, 9) = The space of all .%continuous real functions on X = %(X).
(ii). (Riest Representation Theorem.) If T is a continuous linear functional on %(X), there exists a unique bounded regular measure p on 93 such that
T(f)=D
If
dp
for f in %(X) having the property that llTll= lpl(X). (iii). The dual of %(X)= %*(X)= ca(X, 93).
Proof. (i). Suppose f is a continuous real valued function on X. For E > O and for each x in X, there exists a clopen set C, containing x such that
4.
137
INTEGRATION
I f ( y ) - f ( x ) l < ~ / for 2 all y in C,. {C,; x EX} is an open cover for X. Since X is compact, there exists a finite subcover {C,,, C,,, . . . ,CXn}of X. For each 11i%n and for every x , y in CXi, we have I f ( x ) - f ( y ) l I I f ( ~ ) - f ( ~ i ) l + I f ( ~ i ) - f ( y ) ( < ~ / 2 + ~ / 2 = ~ . Let Di=Cxl, D2= C,, - C,,, . . . ,D, = CXn-(GIu C,, u . * u CXn-,). Then {Dl,D*,. . . ,D,} is a partition of X in 9and for this partition, we still have I f ( x ) -f(y)I < E for all x , y in Di and for all i = 1,2, . . . ,n. Hence f is $-continuous. . Conversely, if f is $-continuous, it follows easily that ~ E % ( X )This proves (i). (ii). Let T be a continuous linear functional on %(X).By Theorem 4.7.5, there exists a bounded charge C; on 9 such that T ( f )= D J f dC; for every f in %(X). C; can be extended as a measure on the Baire a-field Boof X. can be extended as a regular measure p on B. See Theorem 3.5.5. This completes the proof of (ii). (iii). This is obvious now. 0
-
4.7.7 Remark If 9 is a field of subsets of a set R, then %(a, 9) can be realized as %(X)for some compact Hausdorff totally disconnected space X.
Now, we obtain some natural subspaces of ba(Cl, 9) as dual spaces when a-field.
$ is a
We consider the following set-up. Let R be a set, 8 a a-field of subsets (ii) I. C, E $ if C c A, C E %?I of R and 9 a proper cr-ideal in 8, i.e. (i) 4 c ? and A E 4 , (iii) R @ 4 and (iv) Unal A,, E 9 if A,, n 2 1 is a sequence in 9. A real valued function f on R is said to be measurable if f-'(B) E 8 for every Bore1 set B c R. A measurable function f on R is said to be essentially bounded if there exists k > O such that {w E R; \ f ( w ) \> k } E 9 . If f is an essentially bounded measurable function on R, define
Ilfllm
= Inf
{ k ;k > 0 and
{w E R; If(w)I > k} E 9).
Obviously, 0 Illfllm < a. Let Lm(RZ, a, 9)denote the collection of all essentially bounded measurable functions on R. If f E Lm(Q%, 9), then A={w =
u
nzl
If(w)l>IIfllmI @En;
If(w>l>llfllm-tl/n}
€9.
Further, we show that A has the following properties. (i). SUP{lf(w)t; 0 E A'} = IFll==. ; E A'- B} = llfllm for any B in 4. (ii). Sup { l f ( w ) / w Obviously, Sup { I f ( w ) l ;w E A'} 5 llfllm. If Sup {If(o)(; w E A'} < llfllm, then {w'ER; If(w')l>Sup{If(w)l; w E A ' } } ~ A and hence {w'ER;If(w')I> Sup {If(w)l; w E A'}} E 9.But this contradicts the definition of llflloo. (ii) can be proved analogously.
138
THEORY OF CHARGES
-
4.7.8 Proposition. The function 11 Jlmdefined on L,(R, M, 9)above has the following properties. (i). llcfllm = IcI llfllm for any real number c and f in L,(R, a, 9 ) . 6).Ilf + gllm 5 llfllco + llgllm for allf, g in Lm(R, M, 9). Proof. (i). This is obvious. (ii). Note that {w €0;l f < ~ ~ + ~ ( ~ ) l > I l f l l m + + l ~ I I ~ ~ C{W
€0;If(~)l>llfil~I
u l w En; Id~I>llSll~}E.a.
Consequently, Ilf + gllm 5 llfllm + llgllm-
0
4.7.9 Proposition. (L,(R, M, 9 ) ,1)- llm) is a h e a r space and pseudo-norm under which L,(R, M,9) is complete.
11.1)m
is a
Proof. It is clear the L m ( R , M,9) is a linear space and that Il*llm is a pseudo-norm on Lm(R, a,9).We show that Lm(R, %,9)is complete. Let fn, n 2 1 be a Cauchy sequence in Lm(R,Vl, 9). For every n, m 2 1, let Anrn={w Ea;I f n ( ~ ) - f r n ( ~ ) l > l l f n - - f r n I l m } . Then for any B in 9, SUP{Ifn ( w )-frn(w>I; w E A;rn--B}
=
llfn -frnll,*
LetA=Un,,Urn,,Anrn. T h e n A E 9 and lim Sup(~fn(w)-frn(w)~;o €A'}=
rn,n-m
lim
llfn-frnllm=O.
rn,n-m
Let f ( w ) = limn,m f n ( w ) for w in A'. Then f n , n L 1converges to f uniformly on A". For w in A, define f ( w ) = 0. We claim that l f n -film, n L 1converges to zero.
llfn -fIL
= IKfn - f ) l A c + Il(fn -f)lAcllm
(fn
+ Il(fn
-f)lAllm
0 This completes the proof.
5 SUP {Ifn(w)-f(w)I;
which converges to zero as n + 00.
-fV*lIm w E A?+
In the present context, we define a null function as follows. A real valued measurable function f on R is said to be a null function if {w E R; If(o)I> k } E 9 for every k > 0. This is equivalent to the condition that )lfllrn = 0. The space of all null functions is a linear subspace of Lm(R, M, 9).We introduce an equivalence relation on Lm(R, M, 9)by f-g if f - g is a denote the collection of all equivalence null function. Let 2?m(R,8,9) classes of Lm(R, M, 9). The pseudo-norm 11. llm defines unambiguously a norm on Z,,(R, a, 9) which is again denoted by 11 llm. Now, it is apparent that Ym(R,a,9)is a Banach space. We work out its dual.
-
6
4.
139
INTEGRATION
4.7.10 Theorem. Let T be a continuous linear functional on (Lm(R,%, 9), 11 llm). Then there exists a unique bounded charge p on % with the following properties. (i) T ( f ) = D J f d p f o r e u e r y f i n Lm(R,%,9). (ii). llTll= IpI(W. (iii). p (A) = 0 i f A E 4;.
Proof. For A in 8,let @(A)= T(IA). p is obviously a charge on a. Note that T(f) = 0 whenever f is a null function. If A E 9,then IAis a null function and so, p (A)= 0. This proves (iii). The boundedness of p follows from the inequalities IpL(A)I= IT(IA)l
5
~
~
~
~
~
~
~
~
A
~
for any A in a. If f is a simple function, it is obvious that T(f)= D jf dp. Let f be any function in Lm(R, %, 9) with llfllm>O. We show that T(f)= D Jfdp. Let B = {w E R; If(w)I 5 Ilfllm}. Note that B“E 4. Let d = 211fllm. For each n z 1, define (similar to the construction given in the proof of Corollary 4.5.9)fn as follows. For w in R,
for i = 1 , 2 , . . . , 2 ” -1,
It is easy to check that f,,,n z 1 converges to f uniformly on B. We claim that f,,,n z 1 is a determining sequence for f with respect to p. It is obvioils that each fn is a simple function. For any E > 0, Ip
I({@
E
a;Ifn
(@)-f(@)l>
Elf =
l({w
E
B; Ifn (@> -ff@)l>
+ IPKb E B‘;
Ifn(4
I))
-f(d > &I)
= 0,
if n is sufficiently large. This assertion follows from lp [(B‘) = 0 and fn, n 2 1 converges to f uniformly on B. This shows that fn, n 2 1 converges to f
~
m
5
~
\
140
THEORY OF CHARGES
hazily. Also,
= 0.
Hence f n , n L 1 is a determining sequence for f. Since IIfn - fllm IIl(fn - f)I~ll~ + Il(fn - f ) I ~ c l l ~ , it follows that limn+m\Ifn - film = 0. Since T is a continuous linear functional on Lm(fl, %, $1,
T ( f )= lim T ( f n = ) lim D n+m
n-m
I
fn
d p =D
I
f dp.
Next, we show that ~ ~ T ~ ~ = ~For p ~any ( f lf )in. Lm(R,%,9),ITl= ID d p I I D 1 dlp I s Ip I(fl)llfllm. Hence, it follows that llTll5 Ip I(fl). Just as in the case of Theorem 4.7.4, we can show that Ipl(fl)crllTll. Thus 0 llTll= )pl(fl)holds true. This completes the proof.
If
If1
For the following corollary, let ba(R, %, 9) stand for the space of all bounded charges on % vanishing on 9. We equip ba(fl, %, 9)with the total variation norm.
4.7.11 Corollary. The dual of Ym(fl,%, 9 ) is isometrically isomorphic to ba(fl, %, 9).
CHAPTER 5
Nonatomic Charges
Classification of charges is an obvious pursuit one embarks on for a good understanding of charges. We have already come across 0-1 valued charges in Chapter 2. In this chapter, we examine what could be an antithesis of the notion of a 0-1 valued charge. Section 5.1 develops the relevant classification of charges. The Sobczyk-Hammer decomposition theorem is presented in Section 5.2. We prove some existence theorems for nonatomic charges in Section 5.3. Finally, in Section 5.4, we consider the plentitude of nonatomic charges.
5.1 BASIC CONCEPTS The notion corresponding to nonatomicity of measures on cr-fields can be introduced for charges in three different ways. We show that these three ways are actually distinct for charges.
5.1.1 Definition. Let 9 be a field of subsets of a set R. A set F in 9 is said to be a p-atom if the following conditions are satisfied. (i). p( F ) # 0. (ii). If E E 9 and E c F, then either p (E)= 0 or p (F- E) = 0.
If F is a p-atom, then the restriction of p to F n9 is a 0-p (F) valued charge on the field F n9of F. However, if p restricted to the field F n9 for an F in 9 is two valued, F need not be a p-atom. Also, it is not difficult to check that an F in 9 is a p-atom if and only if F is a Ipl-atom. For, if F is a p-atom, E E 9, E c F and p (E)= 0, then Ip I(E) = 0. Note that if F E 9, F is an atom of 9 and p (F)# 0, then F is a p-atom. However, a p-atom need not be an atom of 9. Non-existence of p-atoms is one way of defining a class of charges. More formally, we give the following definition. 5.1.2 Definition. Let 9 be a field of subsets of a set R and p a charge on 9. p is said to be nonatomic on 9 if there are no p-atoms in 9. Equivalently, if F E 9 and p (F) # 0, then there exists E in 9 such that EcF,p(E)#Oandp(F-E)#0.
142
THEORY OF CHARGES
In view of the remarks made after Definition 5.1.1, it follows that p is nonatomic if and only if Ip I is nonatomic. If p is a positive bounded charge on 9, then p is nonatomic on 9, if for every F in 9 with p (F)> 0, there exists E in 9such that E c F and 0 < p (E)< p (F).Now, we prove a simple property of nonatomic charges.
5.1.3 Proposition. Let 9 be a field of subsets of a set R and p a positive bounded nonatomic charge on 9. Then given F in 9 with p(F)>O and E > 0 , there exists E in 9 such that E c F and 0 < p (E)< E .
Proof. Since p is nonatomic, there exists El in 9 such that El c F and 0 ~ / 2 for every i 1 1 or there exists a sequence kl< k z < * such that .-(Aki)> ~ / for 2 every i 2 1. Assume that the former holds. Since v+(A,,) > ~ / 2 , there exists Bi in 9 such that Bi CAniand v(Bi)>&/2for every i L 1. Bi, i 2 1 is a sequence of pairwise disjoint sets in 9and limi+a v(Bi)f 0. This contradiction proves that Iv 1 is s-bounded. The converse is trivial. (ii). If v < < p , then v < < 11.1.1. By (i), lpl is s-bounded. Therefore, v is sbounded. (iii). Every s-bounded real charge is bounded. See Corollary 2.1.7. (iv). This follows from (iii) and the fact that every bounded charge is s-bounded. 0
-
Now, we study the countable additivity property of charges in the presence of absolute continuity.
6.1.11 Theorem. Let p and v be two charges defined on a field 9 of subsets of a set 0. If v 0, there exists a set B(T,S) in 9such that T(E)2 -6
whenever E E 9 and E c B(7, S ) ,
7(E)I6
whenever E E 9 and E c (B(T,8))'.
7
and Let m be any positive integer
For k
= 1 , 2 , . . . , m, define
Let A = B,, A, = B,-l -Bm, A,-* = Bm-2-(Bmp1u B,), . . . , A 2= B1 -(B2uB3 u - - uB,) and Al = R-(Bl u B z u . - * uB,). Note that A, AI, A2, .. . ,A, are pairwise disjoint sets in 9with union = R. Let
(i-1)e
f = i=12()p.l(n)+&) c We show that A and f are the desired ones. Let E E 9and E c A = B,. Then
6.
ABSOLUTE CONTINUITY
171
Also, by the given hypothesis, p (E)2 - ~ / 2> -e. Consequently, Ip (E)]< e. Thus (i) is proved. Now, let E E g a n d E c A ' = A l u A 2 u . * . u A m .So, E=
m
m
i=l
i=l
u ( E n A i ) =u Ei,
whereEi=EnAifori=1,2,. . .,m.Notethatforeach2sism,EicAic Bi-l and so
On the other hand, for each 15 i 5 m, Ei c B; and so
Consequently, from the above two inequalities, we obtain
and
m
"
5 &/2+&/2 =E.
Consequently, Iu(E)-D j, f d p l s e . This proves (ii). The following theorem is a generalization of the above theorem.
0
172
THEORY OF CHARGES
6.3.2 Theorem. Let 9be a field of subsets of a set R. Let p and v be two bounded charges on 9 and E >O. Then there exists a set A in 9 and a simple function f on R such that (i). Ip (E)I< E whenever E E 9and E c A, and (ii). Iv(E)-DI,fdpI 0. By (ii), there exists A in b a ( l l , 9 ) and a nonnegative number k such that
- k p ( F )5 A (F)5 k p ( F ) for every F in 9, and IIu -A
)I5 s/3. Then we have
- ~ / 3- k p ( F ) < A (F)< k p ( F ) + ~ / 3 for every F in 9. By applying Lemma 6.3.3(ii) for ~ ' = 2 ~ / there 3 , is a simple function f on ll such that
176
THEORY OF CHARGES
for every F in 9. Consequently, for any F in 9,
1
v (F)- D
jFf & 1
5
Iv(F) -A
(F)I+ IA (F)- D
IF 1 f dp
< ~ / +32 4 3 = E . This proves (iii). (iii)+(i). Let E >O. Let f be a simple function on SZ such that
for every F in 9.Let k = max {I f(w)l; w E R}. Take S = ~ / 2 k (Assume, . without loss of generality, that k > O . ) We show that if F E9and p(F) v(F)-D
f d p LIv(F)I- D
fdp
[ F I
Consequently, Iv(F)I < E . This proves (i).
0
6.3.5 Example. Exact Radon-Nikodym derivative may not exist, i.e. there may not exist a D-integrable function f (with respect to p ) such that
v(F)=D
I,
f dp
for every F in 9, in Theorem 6.3.4. Let R = {1,2,3, . . .} and 9the finite-cofinite field on SZ. Let v and defined on 9by
p
be
v(F) = 0, if F is finite, = 1,
p(F)=
c 21
keF
=1+
T,
1 1T , kGF
Note that
v 0. Then
Applying Holder’s inequality to the numbers l / t l and l / t z , i.e.,
we obtain
5 IIvI12:111vI12z.
Since this inequality is true for every P in 9, we obtain
1.”,
5
The case t l = 0 or tz = 0 is trivial.
ll~Il,”:f111~ll~f2. 0
The following theorem provides a sufficient condition for strong conp ) for p > 1 in terms of strong convergence in vergence in V,(Q 9, V l ( Q %, p ) .
7.4.2 Theorem. Let 1< r 5 00. Let (a,%, p ) be a probability charge space p ) satisfying the following two and v, v,, n L 1 a sequence in Vr(Sz,g, conditions. (i). v,, n 2 1 converges to v in V1(Sz,%, p ) . (ii). Supnzl llvnllr Then v,, n 2 1 converges to v in V,(Sz,9, p ) for any 15 p 1in terms of strong convergence in V1(R,9, p). 7.4.3 Theorem. Let (i2,9,p) be a probability charge space and v, v,, n 2 1 a sequence in Vl(R, 9, p ) having the followingproperties. (i). v,, n 2 1 converges to v in V,(R, 9,p ) . (ii). lv,I S A for every n 2 1 for some A in Vp(i2,9, p ) with 1 s p 1. By (ii), it follows that each v, eVp(R,5F, p ) . By (i), it follows that v,(F) = v(F) for every F in 9. It also follows that limn+m(v,l(F) = IvI(F) for every F in 9.(Use the fact that 1 ~ ~ 1n, 2 1 converges to 1.1 in Vl(R, 9, p ) . ) Consequently, IvI < A . Hence v E V,(R, 9, p ) also. Assume, without loss of generality, that each v, is positive. From the lattice properties of ba(R, 9), for any m, n 2 1, we have (A A mp)(v, A m p )S A - v,. See Theorem 1.5.4(29).Therefore, from this inequality, we obtain 0 5 v, -(v, A m p )s A -(A A mp).
Hence llv, - (v, A mp)IlpI l(A - (A ll v - v n l l p s l l v - ( v
A
mp)llp Thus for any m,n 2 1,
~ m l . ~ ) l l p + l l ( v ~ ~ m l ~ ) - ( v n ~ m ~ ~ ) I I~pm+ ~ l l )( v- v , nllp
5 IIv - (v A
mp
)]Ip
A
m p ) - (v, A mp )11p
+ Ilk - (A
A
m p )IIp
Observe that for each fixed m 2 1, (vn A mp),n 2 1 converges to (v A m p ) in V1(R,5F, p ) . This follows from the inequality II(v A m p )- (v, A m p ) l l l s ((v- v,(I1 for every n L 1 and (i). Further, (v, A m p )Im p for every n 2 1. Consequently, by Theorem 7.4.2, (v, A mp), n 2 1 converges to (v A mp) in V,(R, 9, p ) for every fixed m 2 1. Therefore, for every fixed m L 1, 0 s lim sup I(v- v,IIp 5 I1v - (v A mp)llp+ 0 +(\A - (A n-m
A
mp
)IIp
7. V,-SPACES
199
Now, by letting m + 00, by Theorem 7.2.15, we have OsIim sup IIv - vnllp < O + O . n-W
Hence limn,, ))v- vnllp = 0. This completes the proof.
0
The following theorem gives necessary and sufficient conditions for strong p ) for 1< p 0 such that Iv(A)I< E for every v in M
whenever A E 9 and IA I(A) 0, there exists a natural number N L 1 such that
12,
P (Ad
I
0, we can find n o 2 1 such that lp,,({k})l 77/8. We can find, for each j 2 1, a set Bi C U I ~Ai T, such that Ipnl(Bj)l< q / S . Since pnl is a bounded charge on 9and Bj, j 2 1 is a sequence of pairwise disjoint sets in .F,limjem ,unl(Bj)= 0. This contradiction establishes the claim. Let S1 = Ti,. Let p1 be the smallest element in S1. Then p 1 # 1. For, if p1= 1, then q / 8 Aj) 2 lFn,(A1)l2 77, by 5"(b). This contradiction shows that p1 # 1. By decomposing S1 -(PI} into a countable number of pairwise disjoint infinite sets, as above, we can find an infinite subset Sz c S1-(PI} such that finp,(UiES2 Ai)5 q / 8 . Hence the desired S1, p1 and S 2 are obtained satisfying (a)', (b)' and (c)'. If p 2 is the smallest element in Sz, then p 1 < p z . Continuing this way, we obtain the desired sequences having properties (a)', (b)' and (c)'. 7". Let A = Ui2lA , . For any fixed i 2 1, write
ujzl
2fi,l(ujo~1
i-1
u A , u u A,. By 5"(b), Ipnpi(APi)177. Note that max ( u;.:APi) : < mpi-l+lI m p ( ,by Sob). A = A, u
j=l
B
Consequently, by 5"(c),
Since
jzi+l
8.
NIKODYM AND VITALI-HAHN-SAKS
THEOREMS
209
by 6"(c)'. Consequently,
3
>q-q/8-~/8=zq.
-
Since P I < p 2 < * np,< npz< * .Therefore, pnp,,pnpz,. . . is an infinite pnPi(A) = 0. The above subsequence of p l , p z ,. . . . So, by (i), inequality is a contradiction to this limit. Hence (ii) is valid. (ii)+ (iii). Note that a,
m
0 ~r lirn sup n+m
~r 2
C lpn({k})I= limn-msup rn-mk-1 lim 1 kzl
( p , ({k})I
lim sup Sup {lp,,(A)l;A c R and A finite} = 0, n-rm
by (ii). Hence (iii) follows. (iii)+ (ii). Note that 0 5 lirn sup Sup {(pn(A)I; A c n and A finite} n+m
5 lirn sup n-m
C
Ipn( { k } ) (= 0,
by (iii).
k z l
Hence (ii) follows.
8.3.4 Remark. In the above lemma, (iii)+(i) is not true. Take any sequence pn, n 2 1 of distinct 0-1 valued charges on P(R) each of which vanishes on finite sets.
8.4 NIKODYM THEOREM We develop this theorem in the framework of Boolean algebras. The various notions connected with Boolean algebras are given in Section 1.4.
210
THEORY OF CHARGES
8.4.1 Definition. A Boolean algebra B is said to have Seever property if for every two sequences a,, n 2 1 and b,, n 2 1 in B satisfying the following conditions (i). a l s a z s a 3 * ~, ~ ~ (ii). bl? bz 2 b3 L * and (iii). a, b, for every m, n 2 1, there exists c in B such that a, s c 5 6, for every m, n 2 1.
-
Boolean cr-algebras and cr-fields of subsets of any set constitute an important class of Boolean algebras with Seever property. We give some examples.
8.4.2(i) Example. Let R = {1,2,3, . . .} and 9 the finite-cofinite field on $2.The Boolean algebra 9does not have Seever property. Take A, = { 2 , 4 , 6 , .. . ,2m},
m 21,
and
B , = R - { l , 3 , 5 ,..., 2n+1},
n ~ l .
There is no C in 9such that A, c C c B, for all m, n L 1. The following is an example of a Boolean algebra with Seever property which is not a Boolean cr-algebra.
8.4.2(ii) Example. First, we prove the following result. Let B1 and Bzbe two Boolean algebras and h onto homomorphism from B1 to Bz. If B1 is a Boolean algebra with Seever property, then Bz also has Seever property. Let a,, n 2 1 and b,, n L 1 be two sequences in BZsuch that a, 5 u,+l5 bm+l5 b , for all m, n 2 1. Let c,, n L 1 and d,, n L 1 be two sequences in Bl such that h(c,) = a n for every n L 1 and h(d,) = b, for every m L 1. Define e l = c1 A dl, f l = c1 v d l , f n t l = f , A (cnClv d,+l v e,) and e,+l = e , v ( c , + l ~ d , + l ~ f , ) ,n s l . Then e l 5 e z " * * " f 2 5 f l , h(e,)=a, and h ( f m )= b, for all m, n L 1. Since B1 has Seever property, there is an x in B1 such that e, s x sf, for all m, n L 1. Then a, Ih ( x )5 6 , for all m, n 2 1. This completes the proof. Now, we give the desired example. Let R = {1,2, . . .} and 9 the ideal of all finite subsets of 0. Since the natural homomorphism from P(R) to the quotient Boolean algebra P ( R ) / 9 is onto and "(0) has Seever property, by what we have proved above, P(R)/,9 has Seever property. We show that P(Q)/9 is not a Boolean a-algebra. Let B,, n 2 1 be a sequence of pairwise disjoint subsets of R each of which is infinite. Then Vnrl [B,] does not exist. This we prove as follows. Let [C]z[B,] for every n 2 1. Then we exhibit [D] in P ( R ) / 9 such that [C] > [D] 2 [B,] for every n 2 1.
8.
NIKODYM AND VITALI-HAHN-SAKS THEOREMS
211
Since B, - C is finite for every n 2 1, we can find a point x , in C n B , for every n L 1. Let A = {x,; n r 1)and D = C - A. This D serves the purpose. [B,] does not exist. Consequently,
v,
The most important result about Boolean algebras with Seever property is the following.
8.4.3 Theorem. Let B be a Boolean algebra with Seever property. Let p be any bounded charge on B. Let X = {b E B; lp 1(b)= 0). Then the quotient Boolean algebra B / N is complete.
Proof. Since B/X satisfies the countable chain condition, i.e. any family of pairwise disjoint elements in B / N is at most countable (because there cannot be more than n pairwise disjoint elements b in B such that Ip I(b)> l k l ( l ) / n ) ,it suffices to show that B/X is a Boolean c+-algebra.See Theorem 1.4.8. Let [a,], n 2 1 be a sequence of equivalence classes in B / N . We show that Vn,l [a,] exists in B/X. Without loss of generality, assume that a l r a 2 1 - * - . Let C = { b € B ; b r a , for every n r l } . Let r = Inf {Ipl(b);b E C}. Then there exists a sequence b,, n 2 1 in C such that r= Ip](b,).Without loss of generality, assume that bl 2 b22. * . Since B has Seever property, there exists c in B such that a, 5 c I b, for every m, n z 1. It is now obvious that V,,I [a,] = [ c ] . Thus B/X is a 0 Boolean c+-algebra.
-
Now, we are ready to prove Nikodym theorem for charges on complete Boolean algebras.
8.4.4 Theorem. Let B be a complete Boolean algebra and p,,, n 2 1 a sequence of bounded charges on B. Suppose for every b in B, Sup,,, Ipn(b)l< 00. Then Sup,,, llpnll< 00. Proof. Suppose Sup,,,, Ilp,ll= 00, i.e. Sup,,l SUPbeB Ip,(b)l = CO. First, we find a sequence c,, n L 1of pairwise disjoint elements in B and an increasing sequence mk,k r 1 of positive integers such that limk+mIpmk(ck)l= 00. For each a in B, define t, = Sup,,, SupbSa Ikn(b)l.Note that for each a in B, either t, = co or tl-, = co, where 1 is the unit element in B. For, if t, < 00 and tl-, < 00, then tl = Sup,,, SUpbeB Ip,(b)l < 00. Similarly, if t, = 03 for a in B, then either t b = co or t a - b = co for any b in B satisfying b 5 a. From the supposition that Sup,,,, Ilp,,ll=m, we can find m l z l and dl in B such that Ipml(dl)(>Supnrl Ip,(1)1+2. If tdl=co, t a k e e l = l - d l . We find that Iprn1(Cl)l=
I~rnl(1-dl)lz Ikrnl(dAl - I p m l ( l ) l
2 Ipm1(dl)l-SUPnz,
I/~n(1)1>2.
212
THEORY OF CHARGES
If tl-& = 00, take c1= dl. In this case, we find that lprnl(cl)l= Iprnl(dl)l>2. In any case, we have Ipml(cl)l> 2 and tl-cl = 00. Since tl-E1= co,we can find dZI1 - c1 and m2 > ml such that (prn2(d2)1 > Sup,,l Ip,(1-cl)l+3. If t&=co, takecz=(1-c1)-d2. Note that Iprnz(cz)l= Iprnz((1
-cl)-dZ)I LIprnz(dz)I-Ipm,(l
-cI)I
~ I ~ r n z ( d 2 ) l -I pSn ~( 1~- ~ 1 ) 1 > 3 . nzl
If t(l-cl)-dz = 00, take c z = dZ.Now, Ipm2(c2)I = Iprn2(d2)I > 3. In any case, we observe that m2 > ml, C I A cz = 0, I p m l ( C 1 ) 1 > 2, IKrnz(Cz)l> 3 and tl-(clvc2)= 00. Continuing this procedure, we obtain a sequence c,, n L 1 of pairwise disjoint elements in B and an increasing sequence mk, k L 1 of positive integers such that limk--tmI/Arnk(Ck)l = a. For each k L 1, we define A k on P(R), where R = {1,2,3, . . .}, by
for A c R . Note that each Ak is a bounded charge on P(R). Further, limk,m &(A) = 0 for every A c a. This follows from limk+mI&,,(Ck)l= and SUpkzl Iprnk(ViE~ cj)l < 00 for any A c R. By Phillips' lemma 8.3.3, we conclude that limk,m Ak({k}) = 0. But hk({k}) = 1 for every k 1. This contradiction proves the result. 0 Now, we generalize Theorem 8.4.4 to Boolean algebras with Seever property.
Theorem. Let B be a Boolean algebra with Seever property and p,, n 2 1 a sequence of bounded charges on B. Suppose for every b in 5, Supnzl Ipn(b)l 0. Since M is uniformly absolutely continuous with respect to u, there exists 8 > 0 such that p (A) < e/2 for every p in M whenever A E 9 and u ( A ) O such that for every S > 0, we can find A in 9 and n 2 1 such that $(A) < S , but lpn(Nl2 28. For S = 1, we find A1 in 9 and n l L 1 such that lpnl(A1)l22.5 and $(Al) < 1. Since pl,p 2 ,. . . ,pnlare absolutely continuous with respect to $, there exists S1> O such that Ip,(A)I < .5/2’ for j = 1,2, . . . , nl whenever A E and ~ $(A) nl. Also, Ipnl(B)I< .5/2’ whenever B E 9 and B c A2, since $(B) 5 $(A2) < 81. Since PI, pz, . . . ,pnz are absolutely continuous with respect to 4, there exists SZ>O such that 2 =~ 1,2, . . . , nz whenever A E 9and $(A) < S2.For S = SZ, Ipj(A)I< ~ / forj we find A3 in 9 and n 3 2 1 such that IpCLn,(A3)]>2~ and $(A3)nz. Also, lpnl(B)I< ~ / and and B c A3, since $(B) 5 $(A3) < SZ. Proceeding this way, we obtain a sequence Ak, k L 1 of sets in 9 and a sequence n l < nz < * * * of positive integers having the following properties. (a). Ipnk(Ak)l2 2.5, k 2 1. (b). lpni(B)I< for j = 1,2, . . . , k - 1, k 2 2 whenever B E 9 and B C Ak. We relabel pnl,p n z , .. . as p l , pz,. . . . This is admissible since the property (iv) is hereditary, i.e. any subsequence of p,, n 2 1 has property (iv) . Now, we construct a sequence Hk,k 2 1of pairwise disjoint sets in 9and find a sequence 0 = p o < p l
E
for every j L 0. This then gives a contradiction as property (iv) fails to hold for the sequence p l , ppltl, pp,+l,. . . , and the property (iv) is hereditary. First, we give H1 and p l . Let F1 = Al. If there is an integer i > 1 such that Ipi(F1nAi)l > ~ / 2 let , il be the smallest positive such integer and F2=F1-Ail. If there is an integer i > i l such that Ipi(FznAi)I>E/2, let i2 be the smallest positive such integer and F3 = Fz -Ai,. If this process does not stop at any finite stage, we get a sequence F1n Ail,F2nAi,, . . . of pairwise disjoint sets in 9such that ]pi*(FknAc)I > ~ / for 2 every k 2 1.This contradicts the validity
8.
NIKODYM A N D VITALI-HAHN-SAKS THEOREMS
223
of property (iv) for the sequence pi*, k z 1. So, let us assume that the above procedure stops at some finite stage. This means that there is a positive 2 every i >pl. Take H1 = Fpl.We integer p1 such that pi(Fpln Ai)5 ~ / for show that Ipl(H1)I2 2s - ~ / 2 . = IPl(Fp1-1 -AiplL1)I IPl(H1)I = Ip~(Fpi)l
= ICL1(Fp1-2-Ai,l-2-Ai.1-1)/
1~ 2 IP =
1(A1- Ail - Ai2 - * 1
-*
* *
-A
ipl-l
)I
(AllI - IP 1 (A1 Ail)I - IP 1 (A1 nAiJl * *
- Ipi(A1nAipl-,>l
- &/2’2- . . . - & / y P , - 1
2 2&- &/24
12E - & / 2 > & . Look at the sequence AI1)=Apl+l -FPlri r 1 and the sequence pI1)= ppl+i,i r l of charges. We claim that these sequences have properties similar to those of (a) and (b) with 2~ replaced by 28 - ~ / 2Note . that
IpL1)(AI1))I = IFpl+i(ApI+i-Apl+i nH1)1 2 Ippl+i(Apl+i)I - Ippl+i(Apl+in
z 2.5 - &/2 for every i r 1.
To prove (b), let B E 9 and B c A!’), i z 2. Then for j = 1,,2, . . . ,i - 1,
ICLjl)(~)I 5 &/2plCi I 3&/zi+l= 3 1 2 E F
1 2
= (2E - &/2)7.
Thus the sequences A!’), i 2 1 and p !I), i z 1 have the following properties. - ~ / 2for every i 1 1. (a)’. Ipi1)(A!1))122~ (b)’. Ipjl’(B)II(2.5 -&/2)(1/2’) for j = 1 , 2 , . . . ,i - 1, whenever B E 9 and B c A!” for i z 2. By using the argument given after (a) and (b) for the sequences figuring 2 ~ / 2 we ~ ,obtain an integer p 2 > p 1 , a in (a)’ and (b)’ and replacing ~ / by set H2 in 9 and sequences A!”, i z 1 and p?), i z 1 having properties ~ ( H2 ~~~) ~/ 2 -) ~ / 2 ’ . similar to (a)’ and (b)’. Further, lp!‘)(H2)I= I , U ~ ~ + 2 Observe that H2 c A\*)= APl+l-HI and so, HI n Hz = 0. Continuing this procedure, we get the desired sequences Hk, k 1 1 and pk, k r 0. Hence (iv)+ (v).
224
THEORY OF CHARGES
+
(v) (vi). It suffices to show that $ E, we can find m l > j l such that x,2jlIpl(Ej)I> E . We can also find P I > ml such that Ipl(Ej)l< ~ / 4 . Now, for pp17 we can find m2 >j,, such that Ipp,(Ej)l> E and p z > m2 such that lppl(Ej)l< ~ / 4 . Continuing this procedure we obtain two sequences ml, m 2 , .. . and pl, p z , . . . of positive integers satisfying the following conditions. (a). jl< m l< p l I j,, < m z < p z s jm< m3 F , i = 1,2,3, . , . . (~1.Cjzpi+,I ~ p i ( ~ j ) l < ~ / i4 =, 1,293, * (We ignore p1.) 3". Look at the sets Ej,j = j,,, j,, 1, j,, + 2, . . . ,mz and p,,. By segregating ) 0, we can find those Ei for which pp,(Ej)< 0 and those for which p p , ( E j 2 F1, FZ,. . . ,Fkl,a subcollection of these Ej's, such that either cT1, pPl(Fi)< -&/2 with each pPl(Fi)< 0 or pp,(Fi)> ~ / with 2 each ppl(Fi)2 0. We can achieve this because of (b) for i = 1. Now, look at the sets Ei, j =j,,, jm+ 1, jm+ 2, . . . , m3 and p p z . By the same technique as above, we can find Fkl+l,Fkl+Z,.. . ,Fk2, a subcollection k of these Ei's, such that either Ci2kl+lppz(Fi)< - ~ / 2with each ppz(Fi)C 0 k or cj&l+l pp2(Fi)> ~ / with 2 each ppz(Fi)P 0. Continuing this procedure, we get a sequence F1,Fz, . . ,Fk,, Fkl+l, Fkl+Z,. . . , Fk,, Fkz+l,Fk,+Z, . . . , Fka. . . of pairwise disjoint sets in 9such that for each n 2 0,
cizjl
cjzp,
cizp,
-
+
.
I
k"+l
with the understanding that ko = 0.
I
228
THEORY OF CHARGES
4".We show that property (ii) fails to hold for the sequence ppl,p p z , .. . and the sequence F1, FZ,. . . of pairwise disjoint sets from g.This would prove the implication (ii)j(i). For every n L 0, note that
z
> €12- €14= €14. This follows from the elementary inequality la + b I L la1 - 161. Since k, -+ 00 as n + CO, property (ii) fails to hold for the sequence ppl,p p 2 ,. .with respect to the sequence F1, FZ,. . . of pairwise disjoint sets in 9. (i)j(iii). This is obvious. Let (iii)j(i). Let A,, n L 1 be a sequence of pairwise disjoint sets in 9. E >O. Since (iii) is assumed to be true, we can find m o r 1 such that for every m L mo,
.
SUP SUP
1c
D C N ~ E Mn z m nED
p(An)J< E / 2 *
Now, if m L mo and p E M, we have
c
nzm
c
c
nED
nsE
Ip(An)I=)n z m F ( A n ) l + /n z m p(An)l 0. This contradiction shows that 9, is a maximal filter in 9. 9". We claim that if E = ( E ~ E, ~ .,. .) and S = (Sl, S2, . . .) are two sequences of 0's and 1's such that E # 8, then 9, and 9 6 are distinct. Suppose 9, =9 6 . Since E # S , there exists n 2 1 such that ( & I , E Z , . . . , E , ) # (61, SZ, . . . ,6,).
236
THEORY OF CHARGES
,,
So, A,,.,, n A,,.,, ..... = 0 E 9,a, contradiction. Thus the claim is valid. 10". We claim that for every n 2 1, { E E (0, l}Ko; A (A) 5 l / n for every A has at most n elements. (Recall that (0, l}Ko is the space of all in sE} sequences of 0's and 1's.) Suppose the above set has more than n elements. Pick up any n + 1 distinct elements E " ) , E " ) , . . . ,E ( n + l ) from this set. Since FEq 9,9. . . ,9,(n+1) are distinct, we can find Bi in Sc(:), i = 1,2, . . . , n + 1 such that B1, Bzr. . . ,B,,' are pairwise disjoint. Since A (Bi)5 l / n for every i = 1,2, . . .,n + 1,h(Urf: Bj)~ (+ ln) / n .But A (0)= 1.This contradiction establishes the claim. 11". Since
h (A) 2 l / n for every A E PE},
the set on the left is countable. Since (0, l}Nois uncountable, there exists q E {O,l}"o such that InfA,F,, A (A) = 0. 12". Let u on 9be defined by u ( A ) = 0, if A &9,,, A E 9,
=1, i f A E g V . u is a 0-1 valued charge on 9. 13". Observe that u LA. See Proposition 8.5.l(v). Hence the Lebesgue decomposition of u with respect to A is 0 v. Therefore, T ( u )= 0. 14". On the other hand, we show that J,fu>:. It suffices to show that
+
given P={E1, EZ,. . . , E m } in 9,there exists a finer partition P = (F1, Fa, . . . ,F,} in 9 such that I:=,f(Fi)v(Fi) 2;.For P = {El, Ez, . . . ,E m } in 9,there exists exactly one i, say i = 1, such that u(Ei)= 1, i.e. El E g,,. We can find B e F , , n B such that B c E 1 . Take P '= {B, EI-B, Ez, E3,. . * Em}. Thus 0 = T ( u )# J,fu 2 .; This contradiction shows that 9is superatomic. Now, we prove (ii)+ (i). This is carried out in the following steps. 1". We, first, collect some basic facts about derived sets in topology. Let X be any topological space and A c X. Set A' = A, A' (the derived set of A') = {x E A'; x is an accumulation point of A'}, if (Y is a limit ordinal, set A" = ADand for any ordinal a,set A"+'= (A")'. Then A", a 2 0 is a decreasing net of sets and each A" is a closed subset of A. For what follows, we assume that X is a scattered compact Hausdorff totally disconnected space. Then, there exists an ordinal (YO such that X"O is a non-empty finite set and Xn0+l = 0. This can be proved as follows. Let p be the least ordinal such that Xp = Xp+l. Then X p = 0.For, if Xp # 0 , then Xp+' is a proper subset of X p . (Since X is scattered, the 9
no1. The following theorem goes beyond finite cardinals. 11.3.1 Theorem. Let K be any infinite cardinal less than or equal to the cardinality of the continuum. Then there is a set R, a field 9 of subsets of R and a real charge p on 9such that the cardinality of the range R(p) of p is K. Proof. Let X be any subset of R having the following properties. (i). Cardinality of X = K. (ii). If x, y E X and a , p are rational numbers, then a x +by E X . Such a set X can be constructed as follows. Let B be any subset of the real line R with cardinality K. Let X = ( ( ~ 1 x +1 a 2 ~ 2+ *
* *
+(Y,x,;
X I ,~
2
. ., . ,X ,
E B,
al, a2,. . .,a , rational numbers and n 2 I},
Then the set X has the above properties (i) and (ii).
11.
257
RANGES OF CHARGES
uy=l
[ai,bi), Take R = R. Let 9 be the collection of all sets A of the form where [al, b l ) , [a2,b2),. . . , [anr6,) are pairwise disjoint intervals, a l 5 b l , a 2 s b Z,..., a , 5 b n , a l , a z , ...,a , E X , 61, b2,..., b , ~ X a n d n r l ,and their complements. 9is clearly a field on R. Define p on 9by n
@(A)=
n
1 (bi- a i ) , i=l
= -@(AC),
p
if A is of the form
u [a;, bi),
i=l
if A' is of the above form.
is a real charge on 9 and R ( p )= X. This shows that R(p) has cardinality
K.
0
11.3.2 Remarks. (i). One can construct a bounded charge p with cardinality of R(p) = tc in Theorem 11.3.1. Further, one could have I.L to be positive as well. (ii). If p is allowed to take infinite values, the construction of a positive charge p with cardinality of its range being a prescribed infinite cardinal number could be made much simpler.
11.4 CHARGES WITH CLOSED RANGE If p is a bounded charge on a field 9 of subsets of a set R, then its range R(p) need not be a closed subset of the real line R. See Theorem 11.3.1. In the following, we give an example of a bounded charge p on a a-field '% of subsets of a set R such that its range R(p) is not a closed set.
11.4.1 Example. Let R = {1,2,3, . . .} and '% = P(R), the class of all subsets of R. Let po be any probability charge on '% such that pO(A) = 0 for any finite subset A of R. For each n r 1, let p, on '% be the measure defined by p,(A)=O, if n EA , = 1,
if n E A and A c R .
Let p =CnrO(1/2,+')pn. Note that $ & R ( p but ) $ is an accumulation point of R ( p ) .Hence R ( p ) is not a closed set. In view of the above example, it is of interest to derive a set of sufficient conditions under which R ( p ) is a closed set. Sobczyk and Hammer Decomposition theorem (See Theorem 5.2.7.) provides a basis for further exploration in this direction. We need a definition, to begin with.
258
THEORY OF CHARGES
11.4.2 Definition. Let 9be a field of subsets of a set fl. A sequence p,,, n L 1 of 0-1 valued charges is said to be discrete if for any given positive integer n, there exists a set A in 8 such that @,(A)= 1 and pm(A)= 0 for every m # n. Let us state a lemma about discrete sequences of charges on fields.
11.4.3 Lemma. If p,,, n L 1 is a discrete sequence of 0-1 valued charges on a field 9 of subsets of a set fl, then there exists a sequence A,, n 2 1 of pairwise disjoint sets in 8 such that p,,(A,) = 1 and p,(A,,) = 0 for all m and n such that m # n. Proof. If B,, n L 1 is a sequence of sets from 9 such that p,(B,) = 1 and p,(B,) = 0 for m # n, then the sequence A,, n 2 1 defined by
u B,,
n-1
A1=B1, and
A, =B,-
n22
m=l
serves the purpose of the lemma. The notion of discreteness introduced above is weaker than the notion of infinite disjointness. See Remark 5.2.3(i). The following is an example amplifying this point. Let f l = { 1 , 2 , 3 , . . . , 00) and 8 the collection of all finite subsets of {1,2,3, . . .} and their complements. On the field 9,for each n 2 1, define pn by @,(A)= 1, if n E A,
=0, ifngA. This sequence p,, n 2 1 of distinct 0-1 valued charges is discrete but not infinitely disjoint. This is because for any countable partition {Fl, Fz, . . .} of R in 9, all but a finite number of sets among {Fl, Fz, . . .} are empty. The following is an instance when the range is a compact set.
11.4.4 Theorem. Let a,,, n L 1 be a sequence of real numbers such that be a a-field of subsets of a set fl and p,,, n 2 1 a discrete sequence of 0- 1 valued charges on 8.Let
Cnzl la,] 0 there exist C in % and G in 9such that G c C c F and p (F- G )< E . Then p is a measure on S. Christensen (1971) gave some conditions under which a charge becomes a measure. We present some of his results. Let 9 be a a-field of subsets of a set R. For v in ca(R, S),let h, be the map on .F defined by
h,(F) = v(F),
F E9.
Let % be the smallest a-field on 9with respect to which each of the maps h , is measurable for v in ca(R, 9).
Theorem A.2. Let p be a real charge on 9, If p is measurable with respect to the a-field % on 9, i.e.
h i ' (B) = {FE9; p(F) E B}E % for every Borel subset B of the real line R, then p is a measure on 9. Another result in the context of Polish spaces, i.e. complete separable metric spaces, can be described as follows. Let R be a Polish space and 9 its Borel a-field, i.e. the smallest a-field on R containing all open subsets of R. Let p be a probability charge on 9. Let R* be the collection of all closed subsets of R. The gist of the following result is that if p restricted to R* is a decent function on R", then p is a measure on 9. We now elaborate this statement. We can introduce a suitable metric d" on St" so that (a*,d " ) becomes a separable metric space. Let d be a metric on R compatible with its topology. Since R is separable, one can always choose d to be a precompact metric on R. Define a metric d* on R* by d * ( A , B) =Sup {max { d ( a ,B), d ( A , b)}},
A, B E R".
aeA beB
Then
(a*,d * ) is a separable metric space. Let 9"be the Borel a-field on
R*. Theorem A.3. If the map p restricted to R* is measurable with respect to 9*, then p is a measure on 9.
274
THEORY OF CHARGES
Rao (1971) gave a sufficient condition under which a charge becomes a measure. Let p be a positive real charge defined on a field 9 of subsets of a set SZ. A subfield goof 9is said to be p-pure if the following conditions are met. (i). p (AN)= 0 for some N L 1 wheneve; A,, n L 1 is a sequence in 90 satisfying Al 3 A2 3 A3 3 . . and A, = 0. (ii) p(A)=Inf(C,,,p(A,); { A f l , n Z 1 } c 9 o and U n ~ l A , ~ Aforl every A in 9. (This condition means that the Caratheodory measure induced by p on gocoincides with p on 9.)
-
n,,
Theorem A.4. If there exists a p-pure subfield 90 of 9, then p is a measure on 9. Rao (1971) stated that the converse of the above theorem is true. This is not correct, however, as the folIowing discussion demonstrates.
Theorem AS. Let 9 be a a-field of subsets Q f a set s2 and p a nonatomic probability measure on 9. Let 90be a p-pure subfield of 9 and 91the smallest u-field on SZ containing 90. Then p is nonatomic on 9 1 . (For the definition of a nonatomic measure, see Chapter 5.) Proof. Obviously, 9 1 c 9. We show that 9 and g1are p-equivalent, i.e. given A in 9 there exists B in slsuch that p (A A B) = 0. To begin with, given E > 0 we show that there exists a set B, in g1such that p (A A B) < E . Since 90is a p-pure subfield of 9, there exists a sequence En,n 2 1 in 90 such that UnZlE, 3 A and Cnzl p(E,) < p (A) + E . Clearly, UnZl E, E gl. Take B, =UnzlEn. Thus for each n 2 1, we can find B, in sl such that B,. Since A A (lim p ( A A B,) < 1/2". Take B = Iim B,) c lim sup,,oo (A A B,) and p (lim sup,+m (A A B,)) = 0 (Borel-Cantelli Lemma), it follows that p ( A A B) = 0. Finally, since 9 and 9 1 are pequivalent, p is nonatomic on S1. 0 Theorem A.6. Let 9be a a-field of subsets of a set s2 and p a nonatomic probability measure on 9. Let g o be a p-pure subfield of 9.Then p is strongly continuous on 90. Proof. This is a consequence of Theorem A5 above and Proposition 5.3.7. Theorem A.7. Let 9 be a a-field of subsets of a set SZ and p a nonatomic probability measure on 9. Let 90 be a p-pure subfield of 9. Then there exists a set A in 9of cardinality greater than or equal to the cardinality of the coniinuum c such that p (A) = 0.
Proof. By Theorem A6, p is a strongly continuous probability charge on 90. So, there exist two sets Bo and B1 in .F0 such that Bo n B 1 = 0 , O < p (Bo) < 1/1(2) and O0, then F = E + E = { ( x 1 + y 1 , x 2 + y 2 , . . .):(x1,x2,. . . ) E E and (y1,y2,. . .)EE} should contain a point of {(xl, x2, . . .) E C: xi = 0 for all but a finite number of i's}. See Oxtoby (1971) and Bhaskara Rao and Bhaskara Rao (1974). But F = E. This contradiction proves (a). In Theorem 11.5.2, the proof that D is not 7-measurable is essentially due to Sierpinski (1938).
CHAPTER 12 The main result of this chapter is from Maharam (1976). See also Weissacker (1982) and Talagrand (1981) for further related results.
APPENDIX 2
Selected Annotated Bibliography BOOKS To begin with, we give a list of books we have consulted at one time or the other in our study of finitely additive measures.
1. BIRKHOFF, G. “Lattice Theory” American Mathematical Society Colloquium Publications, New York, 1948. 2. DUBINS, L. E. and SAVAGE, L. J. “How to Gamble If You Must (Inequalities for Stochastic Processes)”. McGraw-Hill, London, 1965. 3. DUNFORD, N. and SCHWARTZ, J. T. “Linear Operators, Part I: General Theory”. Wiley-Interscience, London, 1954. 4. FUCHS, L. “Infinite Abelian Groups,” Vol. 1. Academic Press, London and New York, 1970. 5 . HALMOS, P. R. “Measure Theory”. Van Nostrand, London, 1950.
6. HALMOS, P. R. “Lectures on Boolean Algebras”. Van Nostrand, London, 1963. 7. KAMKE, E. “Theory of Sets”. Dover Publications, New York, 1950.
8. KELLEY, J. L. “General Topology”. Van Nostrand, London, 1955. 9. KURATOWSKI, K. “Topology”, Vol. 1. Academic Press, London and New York, 1966. 10. OXTOBY, J. C. “Measure and Category”. Springer-Verlag, New York, 1971.
11. PFANZAGL, J. and PIERLO, W. “Compact Systems of Sets”, Lecture Notes in Mathematics No. 16. Springer-Verlag, New York, 1966. 12. SCHAEFER, H. H. “Banach Lattices and Positive Operators”. SpringerVerlag, New York, 1974.
13. SIKORSKI, R. “BooIean Algebras”, Third Edition. Springer-Verlag, New York, 1969.
PAPERS We now give a list of research papers which we have come across in our quest to achieve a good understanding of the world of finitely additive measures. This list
APPENDIX 2
283
is by no means exhaustive on this subject. We provide a brief description of some of the salient features of some of the papers which we think are relevant to the main theme of this book. Most of the papers contain a lot more information than the cursory annotation we provide here. ALBANO, L. (1974). Teoremi di decompozione per funzioni finitamente additive in un reticolo relativamente complementato, Ricerche Mat. 23, 63-86. Lebesgue, Jordan, Yosida-Hewitt Decomposition theorems are discussed for charges taking values in a complete vector lattice. ALEKSANDROV, I. I. (1973).The decomposition of a finitely additive set function (in Russian), Comment. Math. Univ. Carolinae 14, 87-93. Using results on vector lattices, Lebesgue Decomposition theorem for charges is proved. See Section 6.2. ALIC, M. and KRONFELD, B. (1969). A remark on finitely additive measures, Glasnik Mat., Ser. III 4(24), 197-200. The problem of embedding a charge space into a measure space is considered. See also Fefferman (1968). ANDO, T. (1961). Convergent sequences of finitely additive measures, Pacific J. Math. 11, 395-404. Vitali-Hahn-Saks theorem for sequences of charges defined on u-fields is proved. See Chapter 8. ARMSTRONG, T. and PRIKRY, K. (1978). Residual measures, Illinois J. Math.
22, 64-78. ARMSTRONG, T. and PRIKRY, K. (1981).Liapounoff’s theorem for non-atomic finitely-additive, finite-dimensional vector-valued measures, Trans. Amer. Math. SOC.266,499-514. We came across this paper at the proof-reading stage of this book. Ranges of charges defined on fields of sets is the main theme of this paper. See Chapter 11. There is some overlap of results between this paper and that of Bhaskara Rao (1981). ARMSTRONG, T. and PRIKRY, K. (1982). On the semimetric of a Boolean algebra induced by a finitely additive probability measure, Pacific J. Math. 99,
249-263.
Let p be a probability charge on a field 9 of subsets of a set n and N, the ideal of all p-null sets. On the quotient Boolean algebra 9/NW, there is a natural metric d, defined by d,([A], [B]) = p (AAB) for [A], [B] in 9/N,. The d,) is studied in detail in this paper. completion of the metric space (9/NW, See also Bhaskara Rao and Bhaskara Rao (1977). AUSTIN, D. G. (1955).An isomorphism for finitely additive measures, Proc. Amer. Math. SOC.6, 205-208. An isomorphism theorem for charge spaces analogous to the classical Halmos and von Neumann (1942) theorem for measure spaces is proved. See also Buck and Buck (1947) for a similar result.
284
THEORY OF CHARGES
BANACH, S. (1948).On measures in independent fields, StudiaMath. 10,159-177. Let (R,.9,,,EL,), a E r be a collection of probability charge spaces in which each p, is a probability measure. Let 9 be the field on R generated by is,, CY E r}.A common extension of all these probability measures to 9 as a probability measure with a special property is sought. See also Marczewski (1951). BARONE, E. (1978). Sulle misure sernplicimenten additive non continue, Atti Sem. Mat. Fix Univ. Modena 27, 39-44. An example of a nonatomic charge which is not strongly nonatomic is given. BARONE, E. and BHASKARA RAO, K. P. S. (1981). Misure di probabilita finitamente additive e continue invarianti per transforrnazioni, Boll. Un. Mat. Ital. 18, 175-184. Existence of a nonatornic probability charge invariant with respect to a transformation is discussed. BARONE, E. and BHASKARA RAO, K. P. S. (1981). PoincarC recurrence theorem for finitely additive measures, Rendiconti di Matematica 1, 521-526. The classical PoincarC recurrence theorem in Ergodic theory is discussed in the context of a charge space. BARONE, E., GIANNONE, A. and SCOZZAFAVA, R. (1980). On some aspects of the theory and applications of finitely additive probability measures, Pubbl. Istit. Mat. A p p l . Fac. Univ. Stud. Roma Quaderno 16, 43-53. Sobczyk-Hammer Decomposition theorem for charges on w-fields is proved. See Section 5.2. BAUER, H. (1955). Darstellung additiver Funktionen auf Booleschen Algebren als Mengenfunktionen, Archiv der Math. 6, 215-222. Let B* be a Boolean algebra and B a subalgebra of B*. The notion of a positive bounded charge on B being a measure relative to B* is introduced and some of the results of Yosida and Hewitt (1952) and Hewitt (1953),are generalized. BELL, W. C. (1977). A decomposition of additive set functions, Pacific J. Math. 72,305-311. Every positive bounded charge p on a field 9 of subsets of a set R can be written as a sum of positive bounded charges p l and p zon 9with the following properties. (i) p 1 and hzare mutually singular. (ii) The linear functional induced by the Lebesgue Decomposition of charges with respect to p I has a refinement integral representation. See Chapter 9. BELL, W. C. (1979).Unbounded uniformly absolutely continuous sets of measures, Proc. Amer. Math. SOC.71, 58-62. A uniformly absolutely continuous set of charges can be decomposed into bounded and finite dimensional parts. See Section 8.6. BELL, W. C. (1979). Hellinger integrals and set function derivatives, Houston J. Math. 5, 465-481.
APPENDIX 2
285
Using the concept of a refinement integral (see Chapter 9), the author introduces the notion of derivative of a bounded charge on a field 9 of sets with respect to a real valued function on 9and studies some of its properties. BELL, W. C. (1981). Approximate Hahn decompositions, uniform absolute continuity and uniform integrability, J. Math. Anal. Appl., 80, 393-405. A sequence p,,, n 2 1 of bounded charges on a field B of subsets of a set R is said to be disjoint if IpnlA (pml= 0 for all n # m. A subset G of ba(R,B) is uniformly absolutely continuous if and-only if each disjoint sequence in (&)+ is norm convergent to zero, where (G)’ is the set of positive elements in 6 = {q E baW, 9); Iq I < Ipl for some p in G}. See Theorem 8.7.7 for a related result. BELL, W. C. and KEISLER, M. (1979). A characterization of the representable Lebesgue Decomposition Projections, Pacific J. Math. 84, 185-186. Representability of the linear functional induced by the Lebesgue Decomposition of charges with respect to a fixed charge is studied. BHASKARA RAO, K. P. S. (1981). Remarks on ranges of charges, to appear in Illinois J. Math. See Chapter 11 and Armstrong and Prikry (1981). See also Notes and Comments on Chapter 11. BHASKARA RAO, K. P. S. and AVERSA, V. (1982). On Tarski’s extension theorem for group valued charges, a pre-print. See Notes and Comments on Chapter 3. See also Carlson and Prikry (1982). BHASKARA RAO, K. P. S. and AVERSA, V. (1982). A remark on E. Green’s paper “Completeness of L,-spaces over finitely additive set functions”, to appear in Coll. Math. See Notes and Comments on Chapter 7. BHASKARA RAO, K. P. S. and BHASKARA RAO, M. (1973). Charges on Boolean algebras and almost discrete spaces, Mathematika 20, 214-223. A systematic study of nonatomic, strongly continuous and strongly nonatomic charges is made. Superatomic Boolean algebras are characterized. See Chapter 5. BHASKARA RAO, K. P. S. and BHASKARA RAO, M. (1974). A category analogue of the Hewitt-Savage zero-one law, Proc. Amer. Math. SOC.44, 497-499. See Notes and Comments on Chapter 11. BHASKARA RAO, K. P. S. and BHASKARA RAO, M. (1977). Topological properties of charge algebras, Rev. Roum. Math. Pures et A p p l . 22, 363-375. Let p be a positive bounded charge on a field 9 of subsets of a set R. p induces a natural semi-metric or pseudo-metric d, on 9by d,(A, B) = p (AAB) for A, B in 9. This paper studies some topological properties of the semi-metric space (9,d,). See also Armstrong and Prikry (1982). BHASKARA RAO K. P. S. and BHASKARA RAO, M. (1978). Existence of nonatomic charges, J. Austral. Math. SOC.25 (Series A), 1-6.
286
THEORY O F CHARGES
A set of necessary and sufficient conditions for the existence of a nonatomic charge on a given Boolean algebra is provided. See Chapter 5. BHASKARA RAO, K. P. S. and BHASKARA RAO, M. (1981).On the separating number of a finite family of charges, Math. Nuchr. 101, 215-217. Given any finite number of distinct charges on a field 9 of subsets of a set R, a partition of R in 9 with minimal number of sets is sought which separates the charges. BHASKARA RAO, K. P. S., BHASKARA RAO, M. and RAO, B. V. (1982). A note on ,u-pure sub-fields, a pre-print. Let p be a probability measure on a countably generated u-field of subsets of a set 0. The following are equivalent. (i) There exists a p-pure sub-field of 9. (ii) p is perfect. (iii) p is compact. For the notions of compactness and perfectness of measures, see Ryll-Nardzewski (1953). This result was anticipated by Frolik and Pachl (1973). See also Notes and Comments on Chapter ‘ I
L.
BHASKARA RAO, M. and HALEVY, A. (1977). On Leader’s V,-spaces of finitely additive measures, J. Reine Angew. Math. 2931294, 204-216. V,-spaces (Leader (1953)) are shown to be isometrically isomorphic to L,spaces of a measure space using the Stone Representation Theorem for Boolean algebras. See Notes and Comments on Chapter 7. BOCHNER, S. (1939).Additive set functions on groups, Ann. Math. 40,769-799. V,-spaces (1s p 5 a)in the setting of charges are introduced. Radon-Nikodym theorem for charges is also proved. See Chapters 7 and 6. See also Notes and Comments on Chapters 6 and 7. BOCHNER, S. (1940). FiniteIy additive integral, Ann. Math. 41,495-504. Representation of positive linear functionals on vector lattices is provided. BOCHNER, S. (1946). Finitely additive set functions and stochastic processes, Proc. Nut. Acad. Sci., U.S.A. 32,259-261. This paper introduces a notion called stochastic phenomenon. Let P be a probability measure on a u-field ? ofIsubsets of a set S and 9 a field of subsets of a set R. A real valued function f defined on 9 x S is called a stochastic phenomenon if f(ul=, Ei, - ) = f ( E , . ) a.e. [PI for every finite number of pairwise disjoint sets El, E2,.. . , E, in 9. A stochastic phenomenon can be regarded as a general type of stochastic process and it includes many known processes. BOCHNER, S. and PHILLIPS, R. S. (1941). Additive set functions and vector lattices, Ann. Math. 42, 316-324. This is a fundamental paper on vector lattices. Riesz Decomposition Theorem in the general setting of vector lattices is proved. See Section 1.5. Lebesgue Decomposition Theorem in the setting of charges is observed. See Section 6.2. BOeDAN, V. and OBERE, R. A. (1978). Topological rings of sets and the theory of vector measures, Dissert. Math. 154. Nikodym and Vitali-Hahn-Saks type of theorems for finitely additive vector
APPENDIX 2
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measures on rings of sets are presented along the lines initiated by Drewnowski (1972a, b, c). BROOKS, J. K. (1969). On the Vitali-Hahn-Saks and Nikodym theorems, Proc. Nut. Acad. Sci., U.S.A. 64,468-471. Simplified proofs of Vitali-Hahn-Saks and Nikodym theorems for measures on cr-fields are presented. BROOKS, J. K. (1972). Weak compactness in the space of vector measures, Bull. Amer. Math. SOC., 78, 284-287. A set of necessary and sufficient conditions are given for a subset of ba(R, 9‘) to be conditionally weakly compact in a general setting. BROOKS, J. K. (1973).Equicontinuity, Absolute continuity and weak compactness in Measure Theory, a paper in “Vector and Operator valued Measures and Applications”, (D. H. Tucker and H. B. Maynard, eds). pp. 51-61. Academic Press, London and New York. Some extensions of the result in Brooks (1972) are dealt with. BROOKS, J. K. (1974).Interchange of limit theorems for finitely additive measures, Rev. Roumaine Math. Pures et A p p l . 19, 731-744. Let 9 be a field of subsets of a set R and 9*the smallest u-field on R containing 9. Let K c ba(fl, 9*). Equivalence of uniform s-boundedness of K over 9*and uniform s-boundedness of K over 9is examined. See also Brooks and Dinculeanu (1974). BROOKS, J. K. and DINCULEANU, N. (1974). Strong additivity, absolute continuity and compactness in spaces of measures, J. Math. Anal. A p p l . 45, 156-1 7 5. The notion of strong additivity of a charge studied in this paper is the same as s-boundedness we have used in this book. See Definition 2.1.4. Uniform s-boundedness of a collection of bounded charges is characterized in terms of uniform absolute continuity. See Theorem 8.7.7 for another characterization of uniform absolute continuity of a sequence of charges. BROOKS, J. K. and JEWETT, R. S. (1970). On finitely additive vector measures, Proc. Nut. Acad. Sci., U.S.A.61, 1294-1298. Vitali-Hahn-Saks and Nikodym theorems for charges on cT-fields are proved. BUCK, R. C. (1946). The measure theoretic approach to density, Arner. J. Math. 68,560-580. Density charges are constructed from simple set functions defined on a particular class of subsets of { l , 2 , 3 , . . . }. See Section 2.1. BUCK, E. F. and BUCK, R. C. (1947). A note on finitely additive measures, Amer. J. Math. 69,413-420. Isomorphism of the charge spaces (R, 9, F ) and (R’, 9:,m*), where R’= { 1 , 2 , 3 , . , . }, 9; contains all arithmetic progressions and m” is a density-like charge on 9:,is investigated.
288
THEORY OF CHARGES
BUMBY, R. and ELLENTUCK, E. (1969). Finitely additive measures and the first digit problem, Fund. Math. 65, 33-42. A class S of probability charges on the power set of the set of all natural numbers is constructed such that for any p in S, p (P,) = log,, ( n + l),where P, is the set of all natural numbers whose first significant digit lies between 1 andn,lsns9. CANDELORO, D. and SACCHETTI, A. M. (1978). Su alcuni problemi relativi a misura scalari sub additive e applicazionial caso dell’additivita finita, Atti. Sem. Mat. Fis. Uniu. Modena 27,284-296. Connectedness of the range of a bounded charge is studied. CARLSON, T. and PRIKRY, K. (1982). Ranges of Signed Measures, a pre-print. Theorem A.8 is true for all abelian groups. See Notes and Comments on Chapter 3. This paper came to our notice at the proofreading stage of this book. CHENEY, C. A. and de KORVIN, A . (1976/77). The representation of linear operators on spaces of finitely additive set functions, Proc. Edinburgh Math. SOC.2(20), 233-242. An integral (Kolmogorov-Burkill type) representation of a continuous linear operator from V,(R, 9, p ) to a Banach space is provided. See also Edwards and Wayment (1974). CHERSI, F. (1978). Finitely additive invariant measures, Boll. Un. Mat. Ital. A(5) 15, 176-179. Existence of invariant probability charges is shown. See Section 2.1. CHRISTENSEN, J. P. R. (1971). Borel structures and a topological zero-one law, Math. Scand. 29, 245-255. A probability charge p on the Borel c+-field of a complete separable metric space X is a measure if p is measurable as a function on the space of all closed subsets of X equipped with a natural distance (metric) function. See Notes and Comments on Chapter 2. COBZAS, S. (1976). Hahn Decompositions of finitely additive measures, Arch. Math. 27, 620-621. Let 9 be a field of subsets of a set R. Let ba(R,9) and % ( R , F ) be as in Sections 2.2 and 4.7 respectively. ba(R, 9) is equipped with the total variation norm and %(a, 9)is equipped with the supremum norm. ba(R, 9)is the dual In this paper, it is proved that a p in b a ( R , 9 ) admits an exact of %(R, 9). Hahn decomposition if and only if p attains its norm on the unit ball of %‘(a, 9). DARST, R. B. (1961). A note on abstract integration, Trans. Amer. Math. SOC. 99,292-297. A real valued function on a set R is 9-continuous if and only iff is integrable where 9 is a field on R. See with respect to every bounded charge on 9, Section 4.7. See also Leader (1955).
APPENDIX 2
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DARST, R. B. (1962a). A decomposition of finitely additive set functions, J. Reine Angew. Math. 210, 31-37. Lebesgue Decomposition Theorem for bounded charges is proved. See Section 6.2. DARST, R. B. (1962b). A decomposition for complete normed abelian groups with applications to spaces of additive set functions, Trans. Amer. Math. SOC. 103,549-558. A Lebesgue type decomposition theorem is proved in a general setting. Validity of Lebesgue Decomposition Theorem for unbounded charges is examined. See Section 6.2. See also Notes and Comments on Chapter 6. DARST, R. B. (1963). The Lebesgue Decomposition, Duke Math. J. 30,553-556. An extension of a result in Darst (1962b) is established. DARST, R. B. (1966). A direct proof of Porcelli’s condition for weak convergence, Proc. Amer. Math. SOC.17, 1094-1096. See Section 8.7 and Notes and Comments on Chapter 8. DARST, R. B. (1967). On a theorem of Nikodym with applications to weak convergence and von Neumann algebras, Pacific J. Math. 23, 473-477. Nikodym theorem for sequences of charges on a a-field is proved. See Section 8.4. DARST, R. B. (1970a). The Vitali-Hahn-Saks and Nikodym theorems for additive set functions, Bull. Amer. Math. SOC. 76, 1297-1298. Vitali-Hahn-Saks and Nikodym theorems are proved for charges on a-fields. See Sections 8.4 and 8.8. DARST, R. B. (1970b). The Lebesgue Decomposition, Radon-Nikodym derivative, conditional expectation and martingale convergence for lattice of sets, Pacific J. Math. 35, 581-600. The Lebesgue Decomposition Theorem and the Radon-Nikodym theorem are considered in a general setting. DARST, R. B. and GREEN, E. (1968). On a Radon-Nikodym theorem for finitely additive set functions, Pacific J. Math. 27, 255-259. Radon-Nikodym theorem for finitely additive bounded complex valued functions on a field of sets is proved. See Fefferman (1967). See also Notes and Comments on Chapter 6. DIESTEL, J. and UHL, J. J. Jr. (1977). Vector measures, American Mathematical Society Math. Surveys 15, Providence. A sharper version of Phillips’ lemma due to Rosenthal is presented. DREWNOWSKI, L. (1972a). Topological rings of sets, continuous set functions, integration I, Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys. 20, 269-276. Rings equipped with a topology such that the operations A and fl become continuous are presented. Vitali-Hahn-Saks theorem for charges taking values in a topological group is proved.
290
THEORY OF CHARGES
DREWNOWSKI, L. (1972b). Topological rings of sets, continuous set functions, integration 11, Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys. 20, 277-286. This is a continuation of the paper of Drewnowski (1972a) in which extensions of s-bounded group-valued charges on a ring of sets to the a-ring generated by the ring are sought. DREWNOWSKI, L. (1972~).Topological rings of sets, continuous set functions, integration 111, Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys. 20, 439-445. Nikodym theorem for group-valued measures is proved. DREWNOWSKI, L. (1972d). Equivalence of Brooks-Jewett, Vitali-Hahn-Saks and Nikodym Theorems, Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys. 20, 725-731. See the following paper of Drewnowski (1973). DREWNOWSKI, L. (1973). Decomposition of set functions, Studia Math., 48, 23-48. This paper and the above paper give analogues of Vitali-Hahn-Saks and Nikodym theorems for sequences of strongly bounded charges defined on o-rings of sets taking values in a commutative Hausdorff topological group. DREWNOWSKI, L. (1973a). Uniform boundedness principle for finitely additive vector measures, Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys. 21, 115-118. Nikodym theorem for s-bounded charges on a o-ring of sets taking values in a normed group is proved. DOLGUSEV, A. N. (1981). Remark on finitely additive measures, Sibirsk. Mat. 2 . 2 2 , 105-120. DUBINS, L. E. (1969). An elementary proof of Bochner’s finitely additive RadonNikodym Theorem. Amer. Math. Monthly 76, 520-523. See Notes and Comments on Chapter 6. EDWARDS, J. R. and WAYMENT, S. G. (1971). Representations for transforma154 251-265. tions continuous in the BV norm, Trans. Amer. Math. SOC. An integral representation theorem for continuous linear functionals on V,(Q 9, p ) , where a =[0,1], can be deduced using v-integrals. EDWARDS, J. R. and WAYMENT, S . G. (1974). Extensions of the v-integral, Trans. Amer. Math. SOC.191, 165-184. An integral (with respect to a charge) representation of continuous linear p ) into a Banach space can be deduced. See also Cheney operators on V,(Q 9, and de Korvin (1976/77). FAIRES, B. F. (1970). On Vitali-Hahn-Saks-Nikodym type theorems, Ann. Insti. Fourier, Grenoble 26, No. 4, 99-114. Vitali-Hahn-Saks and Nikodym type theorems are studied in the setting of
APPENDIX 2
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Boolean algebras with interpolation property (which are same as Boolean algebras with Seever property) for Banach-space-valued s-bounded charges. See Seever (1968) and Chapter 8. FEFFERMAN, C. (1967). A Radon-Nikodym theorem for finitely additive set functions, Pacific J. Math. 23, 35-45. Radon-Nikodym theorem for bounded complex valued charges on a field of sets is proved. See also Darst and Green (1968). FEFFERMAN, C. (1968). L,-spaces over finitely additive measures, Pacific J. Math. 26, 265-271. The problem of embedding a charge space into a measure space is considered. See also AliC and Kronfeld (1969). de FINETTI, B. (1955). La Struttura delle Distribuzioni in un insieme astratto qualsiasi, Giorn. Ist. Ital. Attuari 18, 15-28. A decomposition theorem similar to the one given by Sobczyk and Hammer (1944) is proved. FROLIK, Z. and PACHL, J. (1973). Pure measures, Comment. Math. Uniu. Carolinae 14, 279-293. Properties of charges p which admit p-pure subfields of 9 are studied. Pure measures discussed here are different from pure charges studied in Chapter 10. This paper pointed out an error in M. M. Rao's (1971) paper. See Bhaskara Rao, Bhaskara Rao and Rao (1982) and also Notes and Comments on Chapter 2. GAIFMAN, H. (1964). Concerning measures on Boolean algebras, Pacific J. Math. 14,61-73. Existence of a strictly positive charge on a field 9 of subsets of a set R is related to some conditions in Set Theory. Most importantly, he exhibited a Boolean algebra satisfying countable chain condition having no strictly positive charge on it. See also Kelley (1959). GOULD, G. G. (1965). Integration over vector-valued measures, Proc. London Math. SOC.15, 193-225. Integration of scalar-valued functions with respect to vector-valued charges is developed. See Section 4.5. GRECO, G. H. (1981). The continuous measures defined on a Boolean algebra (Italian), A n n . Univ. Ferrara Ser. VII(N.S.)26, 213-218. A characterization of superatomic Boolean algebras B in terms of exact Hahn Decomposition of bounded charges on B is provided. GREEN, E. (1970/71). Completeness of L,-spaces over finitely additive set functions, Coll. Math. 22, 257-261. See Notes and Comments on Chapter 7. See also Bhaskara Rao and Aversa (1982).
292
THEORY OF CHARGES
GUY, D. L. (1961). Common extensions of finitely additive probability measures, Portugal. Math. 20, 1-5. A necessary and sufficient condition is given for the existence of a common extension of two probability charges defined on two different fields on the same set to any field containing these two fields as a probability charge. See Section 3.6. HALMOS, P. R. (1947). The set of values of a finite measure, Buff.Amer. Math. SOC. 53, 138-141. A simple proof of a result of Liapounoff on the range of a measure is given. HALMOS, P. R. (1948). The range of a vector measure, Bull. Amer. Math. Soc.
54,416-421. A simple proof of two results of Liapounoff on the range of a measure with values in a finite dimensional vector space is provided. HALMOS, P. R. and von NEUMANN, J. (1942). Operator methods in classical mechanics 11, Ann. Math. 43, 332-350. Isomorphism between two measure spaces is abstractly characterized. HATTA, L. and WAYMENT, S. G. (1973). A Radon-Nikodym theorem for the v-integral, J. Reine Angew. Math. 259, 137-146. An analogue of the classical Radon-Nikodym theorem is considered in the setting of v-integrals for charges. an der HEIDEN, U. (1978). On the representatation of linear functionals by finitely additive set functions, Arch. Math. 30, 210-214. Necessary and sufficient conditions for the existence of a charge p for a given linear functional on a Stonean lattice of functions to be expressed as an integral with respect to p are derived. HEWITT, E. (1951). A problem concerning finitely additive measures, Mat. Tidsskr. B 81-94. The structure of all bounded charges on the field 9 on Q = [0, 1) generated by all intervals of the type [a, b ) with 0 5 a I b 5 1 is determined. See Section
10.4. HEWITT, E. (1953). A note on measures on Boolean algebras, Duke Math. J. 20,
25 3-25 6. Distinction between measures on fields and measures on Boolean algebras is pointed out. See Section 10.5. HILDEBRANDT, T. H. (1934). On bounded linear functional operations, Trans. 36,868-875. Amer. Math. SOC. The dual of the Banach space of all 9-continuous functions is shown to be ba(Q, 9-),where 9is a field on Q. See Section 4.7. HILDEBRANDT, T. H. (1938). Linear operations of functions of bounded variation, Bull. Amer. Math. SOC.44, 75.
APPENDIX 2
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Integral representation of continuous linear functionals on a subspace of is given, where R = [0, 11. ba(R, 9) HILDEBRANDT, T. H. (1940). On unconditional convergence in normed vector spaces, Bull. Amer. Math. SOC.46, 959-962. Properties of unconditional convergence in normed linear spaces are used to define some simple measures on P(R), where R = {1,2,3, . . . }. HILDEBRANDT, T. H. (1958). On a theorem in the space el of absolutely convergent sequences with applications to completely additive set functions, Math. Research Center Report No. 62 Madison, Wisconsin. HODGES, J. L. Jr. and HORN, A. (1948). On Maharam’s conditions for measure, Trans. Amer. Math. SOC.64, 594-595. One of the conditions in the set of necessary and sufficient conditions given by Maharam (1947) for a Boolean c+-algebrato admit a strictly positive bounded measure is shown to be redundant. HORN, A. and TARSKI, A. (1948). Measures on Boolean algebras, Trans. Amer. Math. SOC.64, 467497. Extension of set functions defined on a collection Y? of subsets of a set R to a field 9 on R containing %? as charges are sought. See Chapter 3. See also Notes and Comments on Chapter 3. HUFF, R. E. (1973). The Yosida-Hewitt Decomposition as an Ergodic theorem, a paper in “Vector and Operator Valued Measures And Applications”, (D. H. Tucker and H. B. Maynard, eds), pp. 133-139. Academic Press, London and New York. The Yosida-Hewitt (1952) Decomposition of a charge as a sum of a pure charge and a measure is obtained using an ergodic theorem for commutative semigroup of idempotent linear operators on a Banach space. This approach covers both the scalar valued and vector valued charges. JECH, T. and PRIKRY, K. (1979). On projections of finitely additive measures, Proc. Amer. Math. SOC.74, 161-165. There exists a translation invariant charge p on P(R), where R = {1,2,3, . . . } and a function f from R to R such that p = pf-’ and p (A)5 ; iff is one-to-one on A c R . JORSBOE, 0. G. (1966). Set transformations and Invariant measures, A Survey, Math. Inst. Aarhus Universitet Various Publications Series, No. 3, Aarhus, Denmark. Invariant charges are constructed using Banach limits. See Section 2.1. KEISLER, M. (1979). Integral representation for elements of the dual of ba(9, Z), Pacific J. Math. 83, 177-183. If 9is a superatomic Boolean algebra, then every continuous linear functional on ba(R, 5)has a refinement integral representation. See Chapter 9. See also Notes and Comments on Chapter 9.
294
THEORY OF CHARGES
KELLEY, J. L. (1959). Measures on Boolean algebras, Pacific J. Math. 9, 11651177. Necessary and suficient conditions for a Boolean algebra to admit a strictly positive charge are given. KELLEY, J. L. and SRINIVASAN, T. P. (1970/71). Pre-measures on lattices of sets, Math. Ann. 190, 233-241. Necessary and sufficient conditions are given for a positive bounded charge defined on a lattice of sets closed under countable intersections admits an extension as a measure to the cr-field generated by the lattice. KELLEY, J. L., NAYAK, M. K. and SRINIVASAN, T. P. (1973). Pre-measures on lattice of sets 11. “Proceedings of a Symposium on Vector and Operator valued measures and Applications” held at University of Utah, August 7-12, 1972, (D. H. Tucker and H. B. Maynard, eds) Academic Press, London and New York. Some improvements of the results of Kelley and Srinivasan (1970/71) are presented. KHURANA, S. S. (1978).A note on Radon-Nikodym theorem for finitely additive measures, Pacific J. Math. 74, 103-104. Radon-Nikodym theorem for charges is proved using the corresponding result for measures. The argument is essentially that of Dunford and Schwartz (1954), p. 315. KINGMAN, J. F. C. (1967). Additive set functions and the theory of probability, Proc. Camb. Phil. SOC.63, 767-775. A certain notion dense subset of a set fl in the context of a field of subsets of fl is introduced and its ramifications are studied. KISYNSKI, J. (1968). Remark on strongly additive set functions, Fund. Math. 63, 3 2 7-332. Smiley’s (1944) result on the extension of a strongly additive set function defined on a lattice of sets containing the null set to the ring generated by the lattice is reproved. See Section 3.5. LADUBA, I. (1972). Sur quelques gCnQalisations de thkorbmes de Nikodym et de Vitali-Hahn-Saks, Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys. 20,447-456. Some generalizations of Nikodym and Vitali-Hahn-Saks theorems are presented for charges on a-fields taking values in a specified space of functions. LEADER, S. (1953). The theory of L,-spaces for finitely additive set functions, Ann. Math. 58, 528-543. A systematic study of V,-spaces is presented. See Chapter 7. See also Notes and Comments on Chapter 7. LEADER, S. (1955). On universally measurable functions, Proc. Amer. Math. SOC. 6,232-234.
APPENDIX 2
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A real valued function f on a set R is 9-continuous if and only iff is integrable with respect to every bounded charge on 9, where 9 is a field on R. See Section 4.7. See also Darst (1961). LEMBCKE, J. (1970). Konservative Abbildungen und Fortsetzung regularer Masse, 2. Wahrscheinlichkeitstheorie und Verw. Gebiete 15, 57-96. A certain order relation on the set of all real measures on a ring of sets is introduced and the maximal elements in this order are identified. LEMBCKE, J. (1972). Gemeinsame Urbilder endlich additiver Inhalte, Math. Ann. 198,239-258. LIPECKI, Z. (1971). On strongly additive set functions, Coll. Math. 22, 255-256. Another proof of a result of Smiley (1944) is presented. See Section 3.5. LIPECKI, Z. (1974). Extensions of additive set functions with values in a topological group, Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys. 22, 19-27. Extensions of group-valued charges are discussed. LIPECKI, Z. (1982). On unique extensions of positive additive set functions, a pre-print. LIPECKI, Z. (1982). Maximal-valued extensions of positive operators, a pre-print. LIPECKI, Z. (1982). Conditional and simultaneous extensions of group-valued quasi-measures, a pre-print. LIPECKI, Z., PLACHKY, D. and THOMSEN, W. (1979). Extensions of positive operators and extreme points I, Coll. Math. 42, 279-284. The result of Plachky (1976) concerning extreme points of a certain convex subsets of ba(R, F ) is generalized. Extensions of results of Jlos and Marczewski (1949) are derived in a Functional Analytic setting. LLOYD, S. P. (1963). On finitely additive set functions, Proc. Amer. Math. SOC. 14,701-704. Pure charges on Boolean algebras are characterized in terms of measures on the Stone space of the Boolean algebras. See Section 10.5. LOMNICKI, Z. and ULAM, S. (1934). Sur la thCorie de la mesure dans les espaces combinatoires et son application au calcul des probabilitks I. Variables indkpendantes, Fund. Math. 23, 237-278.
ZOS, J. and MARCZEWSKI, E. (1949). Extensions of measures, Fund. Math. 36, 267-276. The problem of extending a charge from a subfield of a field 9 of subsets of a set R to 9 as a charge is tackled. See Section 3.3. LUXEMBURG, W. A. J. (1963/64). On finitely additive measures in Boolean algebras, J. Reine Angew. Math. 213, 165-173. A special class of Boolean algebras in which every charge is a measure when restricted to some suitable ideal is studied. MAHARAM, D. (1947). An algebraic characterization of Measure algebras, Ann. Math. 48, 154-167.
296
THEORY OF CHARGES
Necessary and sufficient conditions are given for a Boolean a-algebra to admit a strictly positive bounded measure. See also Hodges and Horn (1948). MAHARAM, D. (1958). On a theorem of von Neumann, Proc. Amer. Math. SOC.
9,987-994. Lifting exists in complete measure spaces. See Chapter 12. MAHARAM, D. (1972).Consistent extensions of linear functionals and of probability measures, “Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (University of California, Berkeley, 1970/71)”, Vol. 2, Probability theory, p. 127-147, Univ. California Press, Berkeley. Let Fa,Q E r be a colIection of fields on a set R and 9 the smallest field on R containing this collection. For each a in r, let p, be a bounded charge on Fa.The existence of a charge on 9agreeing with pa on gafor every a E r is discussed. A simple case of this problem is studied in Section 3.6. MAHARAM, D. (1976). Finitely additive measures on the integers, Sankhya, Series A 38,44-59. Lifting fails to exist in the setting of charge spaces. See Chapter 12. MAHARAM, D. (1977). “Category, Boolean algebras and measures, General Topology and its relation to modern analysis and algebra”, pp. 124-135. Springer-Verlag, Berlin. MARCZEWSKI, E. (1947). Sur les mesures ti deux valeurs et les idCaux premiers dans les corps d’ensembles, Ann. SOC.Polon. Math. 19, 232-233. MARCZEWSKI, E. (1947). Two-valued measures and prime ideals in fields of sets, SOC.Sci. Lett. Varsovie C. R . Cl. ZZZ.Sci. Math. Phys. 40, 11-17. Let 9be the smallest field on [0,1] containing all sub-intervals of [0,1]. There is no non-trivial two-valued measure on 9. MARCZEWSKI, E. ~-(1947).IndCpendance d’ensembles et prolongement de mesures (Rksultats et Problkmes), Coll. Math. 1, 122-132. MARCZEWSKI, E. (1948). Ensembles d’indkpendants et leurs applications a la thkorie de la mesure, Fund.Math. 25, 13-28. MARCZEWSKI, E. (1951). Measures in almost independent fields, Fund.Math.,
38,217-229. This paper and the two papers above deal with the following problem in all its facets. Let 9-, a E r be a collection of fields on a set 0 and 9 a field on R containing all 9,’s. Let pa be a probability charge on Fa for each a in r. Is there a common extension p (with a special property) of all pa’s to 9 as a probability charge? This problem is linked with the notion of almost-independence of the fields This problem was also studied by Banach (1948) in the setting of probability measures. MARCZEWSKI, E. (1953). On Compact measures, Fund.Math. 40, 113-124. See Notes and Comments on Chapter 2. MAYNARD, H. B. (1972). A Radon-Nikodym theorem for operator valued measures, Trans. Amer. Math. SOC.173, 449-463.
APPENDIX 2
297
MAYNARD, H. B. (1979). A Radon-Nikodym theorem for finitely additive bounded measures, Pacific J. Math. 83, 401-413. Necessary and sufficient conditions for the existence of exact Radon-Nikodym derivative in the setting of charges are presented. See Section 6.3. See also Notes and Comments on Chapter 6. MOLTO, A. (1981a). On the Vitali-Hahn-Saks theorem, Proc. Roy. SOC.Edinburgh, Sec. A 90, 163-173. Boolean rings with property (f) are introduced. These include Boolean algebras with Seever property strictly. See Seever (1968) and Definition 8.4.1. Let G be a commutative Hausdorff topological group. It is proved that if wn, n 2 1 is a sequence of G-valued s-bounded charges defined on a Boolean ring with property (f), pointwise convergent and En, n 2 1 is a sequence of pairwise disjoint elements in the ring, then limp+- pn(Ep)= 0 uniformly in n. See also Faires (1976). MOLTO, A. (198 lb). On Uniform boundedness properties in exhaustive additive set function spaces, Proc. Roy. SOC.Edinburgh, Sec. A 90, 175-184. Uniform boundedness of a family of s-bounded G-valued charges defined on a Boolean ring having the property (f) is discussed. See Molto (1981a). NAYAK, M. K. and SRINIVASAN, T. P. (1975). Scalar and Vector-valued premeasures, Proc. Amer. Math. SOC.48, 391-396. Let 9 be a lattice of subsets of a set fl and 9*the smallest a-field on R containing 9. Conditions under which a charge on 9 taking values either in R or in a Banach space is extendable as a measure to 9*are presented. NAYAK, M. K. and SRINIVASAN, T. P. (1976). Vector-valued inner-measures, “Lecture Notes in Mathematics”, Vol. 541, pp. 107-1 16. Springer-Verlag, Berlin. Extension of a vector valued charge defined on a lattice of sets to the a-field generated by the lattice as a measure is discussed. See also Nayak and Srinivasan (1975). NUNKE, R. J. and SAVAGE, L. (1952). On the set of values of a nonatomic, finitely additive, finite measure, Proc. Amer. Math. SOC.3, 217-218. A nonatomic charge whose range is not convex is exhibited. See Section 11.4. OLEJeEK, V. (1977). Darboux properties of finitely additive measures on a 8-ring, Math. Slovaca 27, 195-201. An example of a nonatomic charge defined on a 6-ring which is not strongly nonatomic is given. See Definition 5.1.5, Theorem 5.1.6 and Remarks 5.1.7. OLEJCEK, V. (1981). Ultrafilters and Darboux property of finitely additive measure, Math. Slovaca 31, 263-276. The notion of an ultrafilter-atom is introduced in the setting of a charge space and some of its properties are studied. PACHL, J. (1972). An elementary proof of a Radon-Nikodym theorem for finitely additive set functions, Proc. Amer. Math. SOC.32, 225-228. See Notes and Comments on Chapter 6.
298
THEORY O F CHARGES
PACHL, J. (1972). On projective limits of probability spaces, Comment. Math. Univ. Carolinae 13, 685-691. Let p be a non-atomic probability measure on a a-field 9 of subsets of a set R. If there exists a p-pure sub-field of 9, then there is a set A in 9such that p ( A )= 0 and the cardinality of A is at least that of the continuum. See Notes and Comments on Chapter 2. PACHL, J. (1975). Every weakly compact probability is compact, Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys. 23,401-405. Let p be a probability measure on a a-field 9 of subsets of a set 0. If there is a p-pure sub-field of 9,then p is compact. See Ryll-Nardzewski (1953) for the notion of a compact measure. PETTIS, B. J. (1951). On the extension of measures, Ann. Math. 54, 186-197. Various extensions of set functions are dealt with. See Section 3.5. PHILLIPS, R. S. (1940). On linear transformations, Trans. Amer. Math. SOC.48, 5 16-54 1. Lemma 3.3 of this paper is Phillips’ lemma. See Section 8.3. PHILLIPS, R. S. (1940a). A decomposition of additive set functions, Bull. Arner. Math. SOC.46, 274-277. Let 9 be a cr-field of subsets of a set SZ and K an infinite cardinal number can be written as a not greater than the cardinal of R. Every p in ba(R, sum p l+ p z uniquely with pl,p z in ba(R, and pz vanishing on every set of cardinal %K in 9. PIERCE, R. S. (1970). Existence and uniqueness theorems for extensions of zero-dimensional compact metric spaces, Trans. Arner. Math. SOC.148, 1-21. Some comments on countable superatomic Boolean algebras are made. See Chapter 5. PLACHKY, D. (1971). Decomposition of Additive Set Functions, “Transactions of the Sixth Prague Conference on Information theory, Statistical Decision Functions, Random Processes”, pp. 715-719. Publishing House of the Czechoslovak Academy of Sciences, Prague. A general decomposition theorem is proved from which the Yosida-Hewitt Decomposition and the Lebesgue Decomposition of bounded charges follow as corollaries. PLACHKY, D. (1976).Extremal and monogenic additive set functions, Proc. Amer. Math. SOC.54, 193-196. Let 9 be a field of subsets of a set SZ and 9oa sub-field of 9. Let v be a The extreme points of the convex set of all probability probability charge on go. charges p on 9which agree with v on goare characterized. PLACHKY, D. (1980). Darboux property of measures and contents, Math. Slouaca 30, pp. 243-246. Let 9,, and g1be two cT-fields on a set such that 9,,c Let p o be a positive bounded charge on Po.Then po is strongly continuous if and only if every
APPENDIX 2
positive bounded charge strongly continuous.
p
defined on g1whose restriction to gois
299 po
is
PORCELLI, P. (1958a). On weak convergence in the space of functions of bounded variation, Math. Research Center Reports No. 39, Madison, Wisconsin. See Porcelli (1960). PORCELLI, P. (1958b). On weak convergence in the space of functions of bounded variation 11, Math. Research Center Reports, No. 68, Madison, Wisconsin. See Porcelli (1960). PORCELLI, P. (1960). Two embedding theorems with applications to weak convergence and compactness in spaces of additive type functions, J. Math. Mech. 9, 273-292. Weak convergence in ba(@ .F) is characterized using Porcelli (1958a and 1958b). See Section 8.7. See also Notes and Comments on Chapter 8. PORCELLI, P. (1966). Adjoint spaces of Abstract L,-spaces, Port. Math. 25, 105-122. V,-spaces are studied from another angle. See Chapter 7. See also Leader (1953). PTAK, V. (1969). Simultaneous extension of two functionals, Czechoslovak Math. J. 3, 553-569. The results of this paper are relevant to the problem studied by Maharam (1972). PYM, J. S. and VASUDEVA, H. L. (1975).An algebra of finitely additive measures, Studia Math. 54, 29-40. Maximal ideals in the algebra b a ( Q .F) are determined, where Cl is a discrete semigroup which is a totally ordered set with multiplication as max. RAMACHANDRAN, D. (1972). A note on finitely additive set functions, Proc. Amer. Math. SOC.31, 314-315. A counterexample is presented to a conjecture of Yosida and Hewitt (1952) concerning the correspondence between charges on a Boolean algebra and the measures on the Stone space of the Boolean algebra. RANGA RAO, R. (1958). A note on finitely additive measures, Sunkhya 19, 27-28. Another proof of the Yosida-Hewitt (1952) Decomposition of a charge as a sum of a pure charge and a measure is presented. See Chapter 10. RAO, M. M. (1971). Projective limits of Probability spaces, J. Multivariate Anal. 1, 28-57. Some conditions are given for a charge to be a measure. See Notes and Comments on Chapter 2. RICKART, C. E. (1943). Decomposition of Additive Set Functions, Duke Math. J. 10,653-665. Generalizations of a result of Phillips (1940a) are presented.
3 00
THEORY OF CHARGES
RIEFFEL, M. A. (1968). The Radon-Nikodym theorem for the Bochner integral, Trans. Amer. Math. SOC.131, 466-487. Hahn Decomposition Theorem for measures on a-fields is presented using Banach space methods. RYLL-NARDZEWSKI, C. (1953). On quasi-compact measures, Fund. Math. 40, 125-130. Perfect and compact measures are discussed. SASTRY, A. S. and SASTRY, K. P. R. (1977). Measure extensions of set functions over lattices of sets, J. Indian Math, SOC. 41, 317-330. Extension of vector-valued set functions from a lattice of sets to the ring generated by the lattice is examined. SCOZZAFAVA, R. (1978). On finitely additive probability measures, “Transactions of the Eighth Prague Conference on Information theory, Statistical Decision functions, Random Processes, (Prague 1978)”, Vol. C, pp 175-180. Reidel, Dordrecht. Let p be a strongly continuous probability charge on the power set P(n)of an infinite set a. Given 0 < a < 1, there exists a sequence F,, n 2 1of pairwise disjoint subsets of fl such that a = p(Unrl F,) = Cnrl p(F.). SCOZZAFAVA, R. (1979). Complete additivity, on suitable sequences of sets, of a simply additive and strongly nonatomic probability measure (Italian), Boll. Un. Mat. Ztal. B 5 , 16, 639-648. Sobczyk-Hammer Decomposition theorem for nonconcentrated charges p, i.e. p ( { o } )= 0 for every o in R, on the power set P(0)of R is proved. SEEVER, G. L. (1968). Measures on F-spaces, Trans. Amer. Math. SOC.133, 267-280. Nikodym and Vitali-Hahn-Saks theorems are presented for Boolean algebras having Seever property. See Sections 8.4 and 8.8. See also Notes and Comments on Chapter 8. SIERPINSKI, W. (1938). Fonctions additives non complhtement additives et fonctions non mesurables, Fund. Math. 30, 96-99. A non-Lebesgue measurable function on the unit interval [0, 11is constructed where fl ={l, 2,3,. . .}. See Notes and Comments based on a charge on P(L?), on Chapter 11. SINCLAIR, G. E. (1974). A finitely additive generalization of the FichtenholzLichtenstein theorem, Trans. Amer. Math. SOC.193,359-374. An analogue of Fubini’s theorem is established in the setting of charges. SMILEY, M. F. (1944). An extension of metric distributive lattices with an application in general analysis, Trans. Amer. Math. SOC. 56,435-447. Every strongly additive set function defined on a lattice of sets containing the empty set can be extended in a unique manner as a charge on the smallest ring containing this lattice. See Section 3.5.
APPENDIX 2
301
SOBCZYK, A. and HAMMER, P. C. (1944). A decomposition of additive set functions, Duke Math. J. 11, 839-846. Sobczyk-Hammer Decomposition Theorem is proved. See Section 5.2. See also Notes and Comments on Chapter 5. SOBCZYK, A. and HAMMER, P. C. (1944). The ranges of additive set functions, Duke Math. J. 11, 847-851. Some results on the ranges of charges are obtained. See Chapter 11. See also Notes and Comments on Chapter 11. SRINIVASAN, T. P. (1955). On extensions of measures, J. Indian Math. SOC., (N.S.)19, 31-60. Extension of measures is discussed using inner measures. STRATIGOS, P. D. (1980). Extensions of additive set functions, Serdica 6 , 197201. Extension of regular bounded charges on fields of sets generated by u-topological spaces is discussed. SUCHESTON, L. (1967). Banach limits, Amer. Math. Monthly 74, 308-311. Existence of Banach limits is shown using an old-fashioned version of the Hahn-Banach theorem. See Section 2.1. TALAGRAND, M. (1981). Non existence de relbvement pour certaines mesures finiement additives et retract& de PN, Math. Ann. 256, 63-66. Under continuum hypothesis, the author constructs a separable subset of P N - N which is not a retract of PN, where N is the set of all natural numbers with discrete topology and /3 N its Stone-Cech compactification. This example is used to show non-existence of a lifting in the setting of charges. See Maharam (1976) and Chapter 12. TARSKI, A. (1930). Une contribution 42-50.
la thkorie de la mesure, Fund. Math. 15,
TARSKI, A. (1938). Algebraische Fassung des Massproblems, Fund. Math. 31, 47-66. TARSKI, A. (1939). Ideale in Vollstandigen Mengenkorpern I, Fund. Math. 32, 45-63. Weak and strong accessibility of cardinals are discussed and existence of measures on some quotient Boolean algebras is considered. TARSKI, A. (1945). Ideale in Vollstandigen Mengenkorpern 11, Fund. Math. 33, 51-65. There exists a 0- 1 valued charge on a Boolean algebra B vanishing on all atoms of B if and only if B contains a countable set of disjoint elements. THOMSEN, W. (1978). On a Fubini-type theorem and its application in game theory, Math. Operationsforsch. Statist. Ser. Statist. 9,419-423. Sinclair's (1974) analogue of Fubini's theorem for measures in the setting of charges is generalized.
302
THEORY OF CHARGES
THOMSEN, W. (1979). The common domain of uniqueness of the products of finitely additive probability measures, “Transactions of the Eighth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, (Prague, 1978)”, Vol. C, pp. 31 1-316, Reidel, Dordrecht. Let B(X) be the Banach space of all bounded real valued functions defined on a set X, F a subset of B(X), B(X, F) the closed subspace of B(X) generated by F and B*(X) the dual of B(X). Let p be a real valued function on F such that there exists a probability charge b on P ( X ) such that p (f)= f d& for all f in F. Let U(p) = { f B(X); ~ pl(f) =p2(f) for all pi in BT(X) with pi/F = p } which is the domain uniqueness of p. It is proved that n , U ( p ) = B ( X , F ) . An extension to product spaces is also considered. TOPSOE, F. (1978). On construction of measures, “Proceedings of the Conference on Topology and Measure I (Zinnowitz 1974)”, Part 2, pp. 343-381, ErnstMaritz-Arndt Univ., Griefswald. A general result is proved in Section 8 of this paper from which the result of Smiley (1944) on the extension of strongly additive functions on a lattice of sets containing the null set to the ring generated by the lattice as a charge follows. TOPSOE, F. (1979). Approximate pavings and construction of measures, Coil. Math., 42, 377-385. A condition under which a positive bounded charge on a field of sets becomes a measure is given. See Notes and Comments on Chapter 2. TRAYNOR, T. (1972). Decomposition of Group-valued Additive Set Functions, Ann. Inst. Fourier, Grenoble 22, Part 3, 131-140. Lebesgue-type decomposition theorem is obtained for group-valued charges. TRAYNOR, T. (1972). A general Hewitt-Yosida Decomposition, Can. J. Math. 24, 1164-1169. Yosida-Hewitt (1952) Decomposition of a group-valued charge is presented using Caratheodory process. TUCKER, D. H. and WAYMENT, S. G. (1970). Absolute continuity and the Radon-Nikodym theorem, J. Reine Angew. Math. 244, 1-19. A general discussion about Radon-Nikodym Theorem in various settings is presented. TULIPANI, S. (1979). On continuous and invariant measures for a transformation (Italian), Rend. Mat. 12, 249-256. Let R be a set and T a map from R to R. Existence of a nonatomic, T-invariant probability charge on the power set P ( R ) of R is discussed.
UHL, J. J. (1967). Orlicz spaces of finitely additive set functions, Studia Math. 29, 19-58. Spaces of set functions more general than the V,-spaces (Leader (1953)) are studied. VOROB’EV, N. N. (1962). Consistent families of measures and their extensions, Theory Prob. Appl. 7 , 147-162.
303
APPENDIX 2
This paper treats the problem described in the annotation of Maharam’s (1972) paper in the setting of probability measures on cr-fields. Some combinatorial methods are used to solve the problem. WAJDA, L. (1972). Remarks on infinite products of finitely additive measures, Coll. Math. 25, 269-271. Product charge of a sequence of probability charge spaces is shown to exist. WALKER, H. D. (1975). Uniformly additive families of measures, Bull. Math. SOC.Sci. Math. R. S. Roumanie (N.S.)18,217-222. If K is uniformly Let 9 be a field of subsets of a set 0 and K c ba(0, s-bounded and pointwise bounded, then K is a bounded subset of ba(Cl, 9). For some more results in this direction, see Section 8.5. See also Brooks (1974).
a.
WEBER, H. (1982). Unabhangige Topologien, Zerlegung von Ringtopologien, Math. 2. 180, 379-393. WEBER, H. (1982). Vergleich monotoner Ringtopologien und absolute Stetigkeit von Inhalten, Comment. Math. Univ. St. Pauli 31,49-60. WEBER, H. (1982). Die atomare Struktur topologischer Boolescher Ringe und s-beschrankter Inhalte, a pre-print. WEBER, H. (1982). Der Verband der s-beschrankter monotoner Ringtopologien und Zerlegung s-beschrankter Inhalte, a preprint. WEBER, H. and VOLKMER, H. (1982). Der Wertebereich atomloser Inhalte, a pre-print. WEISSACKER, H. U. (1982). The non-existence of liftings for arithmetical density, a pre-print. The argument presented in Maharam’s (1976) paper is clearly explained. See Chapter 12. WILHELM, M. (1976). Existence of additive functionals on semi-groups and the von Neumann minimax theorem Coll. Math. 35,267-274. A general result which may be considered as a common generalization of a result on charges due to Kelley (1959) and of the von Neumann minimax theorem in Game theory is presented. WOODBURY, M. A. (1950). A decomposition theorem for finitely additive set functions, Abstract presented in Bull. Amer. Math. SOC.56, 171. A forerunner of Yosida-Hewitt (1952) Decomposition Theorem was announced. YASUMOTO, M. (1979). Finitely additive measures on N, Proc. Japan Acad. 55, Ser. A, 81-84. An improved version of a theorem of Jech and Prikry (1979) is established.
304
THEORY OF CHARGES
YOSIDA, K. (1941). Vector lattices and additive set functions, Proc. Imp. Acad. Tokyo 17,228-232. ba(R, 9)is studied from the point of view as a vector lattice. YOSIDA, K. and HEWITT, E. (1952). Finitely additive measures, Trans. Arner. Math. SOC.72,46-66. Yosida-Hewitt Decomposition of a charge into a pure charge and a measure is presented. See Chapter 10.
APPENDIX 3
Some Set Theoretic Nomenclature
1. Empty set or null set is denoted by 0. 2. The symbol R is used to denote an “abstract space” or “whole space” or “master set” which is a nonempty set of elements. The members of R are denoted generically by w . The sets in a collection of sets we consider are usually subsets of R. 3 . Membership. If w is a member of a set El we use the notation ME. If a set E is a member of a collection of sets d , we use the symbol M . 4. Inclusion. For any two sets E and F, EcF indicates that E is a subset of F, i.e. every member of E is a member of F. 5 . Union. If {Ea;a E r} is a nonempty collection of sets, we denote the E, and is defined to be the the set {w ;w E E, union of these sets by UaEr for some a in r}. 6 . Intersection. If {E, ; a E r}is a nonempty collection of sets, the intersection of these sets is denoted by neGrEa and is defined to be the set {w ; w i E, for every a in r}. 7. Difference. If E and F are any two sets, the difference of E and F is denoted by E-F and is defined to be the set {w ;w E E and w & F}. 8 . Complement. If E is any subset of R, the complement of E is denoted and is defined to be the set R - E. by 9. Symmetric difference. If E and F are any two sets, the symmetric difference of E and F is denoted by E A F and is defined to be the set (E-F)u (F-E).
Index of Symbols and Function Spaces
Function Spaces =The space of ba(C q
all bounded charges defined on the field
n. 43 space of all bounded charges defined on the u-field 9l of subsets of R vanishing on the cr-ideal9 in 9l. 140 =The space of all essentially bounded real valued functions defined on R. 90 =The space of all bounded measures defined on the field 9of subsets of R. 50 =The space of all bounded measures defined on the field 9 of subsets of R, when 9 is viewed as a Boolean algebra. 248 133 =The space of all 9-continuous functions defined on R. =The space of all real valued functions defined on R. 88 (This space is topologized in such a way that convergence in this space is precisely equivalent to hazy convergence, see p. 94.) = Thespace of all equivalence classes of B(R, 9, p ) formed under the equivalence relation f - g iff = g a.e. [ p ] . 90 =The space of all TI-measurable functions defined on R 121, 178 such that Ifl” is D-integrable, 1s p < CD. =The space of all equivalence classes of L,(R, 9, p) formed under the equivalence relation f g iff = g a.e. [P I. 178 = The space of all essentially bounded TI-measurable functions defined on 0. 122,178 =The space of all essentially bounded measurable functions defined on 0, where 9l is a u-field on R and 9 is a cr-ideal in ‘3. 137 =The space of all equivalence classes of L,(R, 9) formed under the equivalence relation f - g if f - g is a null function. 138 =The space of all bounded pure charges defined on the field 9 of subsets of a. 248 = The space of all simple functions. 101 =The space of all simple charges defined on the field 9 of subsets of R. 188 9of subsets of
ba(R, %,9)
B(R, 9, p) caW, .%
G(R,9) Y(R9 q C(R, 9, p)
= The
-
a,
INDEX OF SYMBOLS A N D FUNCT!ON SPACES
Sim(R, 9, p)
V,(n, 9, p)
(a,R P I
=The space of all D-integrable simple functions defined on R. =The space of all bounded charges A on 9 absolutely continuous with respect to p and satisfying IlA 1, < co. : Charge space, i.e. p is a charge on the field 9of subsets of a.
307 132 185 87
Operations on Boolean Algebras B Bl9
: Boolean Algebra : Quotient Boolean algebra
18 20
Operations on Charges
AB WP
=The collection of all bounded charges on 9 absolutely continuous with respect to A = a l v l + a2v2+ * ' +a,v,, where a l , a2,. . . , a , are real numbers and vl, v2, . . . , v, are 0 - 1 valued charges on 9. : A,(A)=A(AflB), A E ~BE9fixed. , = maxlsis, p (Fi), P = {Fl,F2, . . . ,F,} is a partition of R in
9. = Positive variation o f j . =Negative variation of-p.
=Total variation of p . = Outer charge induced by p. = Semi-variation
of
p.
: See Definition 3.2.6. : See Definition 3.2.6. : See Definition 2.5.2. : See Definition 2.5.2. : p is absolutely continuous with respect to u. : p is weakly absolutely continuous with respect to v. : p is strongly absolutely continuous with respect to u. : p and Y are singular. : p and v are strongly singular.
216 180 145 45 45 45 86 206 66 66 52 52 159 159 159 164 164
Operations on Functions f' fIf I f vg fAg f = g a.e. f s g a.e.
=Positive part off. =Negative part of f. = Modulus of f. = Maximum of f and g. = Minimum off and g. : See Definition 4.2.4. : See Definition 4.2.4.
11 11 11 11 11 88 88
308
INDEX OF SYMBOLS AND FUNCTION SPACES
I*
= Indicator function of the set A.
O(f, F)
12 134
: See the proof of Theorem 4.7.3.
Operations on Sets =A
=A',the complement of A. = Closure of A. =Interior of A. =Number of points in the set. =The class of all subsets of R. =The collection of all finite partitions of R in 9. = Thecollectionof all finitepartitionsof F i n F f o r F i n F . : Equivalence relation on a set. : Partial order on a set. : Relation directing a set. : The cardinality of the continuum.
6 6 15 15 41 3 15 15 14 13 13 192
Operations in Vector Lattices : Lattice supremum of x and y. : Lattice infimum of x and y. : Positive part of x. : Negative part of x. : Modulus of x. : x and y are orthogonal. : The orthogonal complement of S.
X"Y X)Y
X X
-
I
Ix xl Y
S'
24 24 24 24 24 24 29
Miscellaneous Symbols ,1
II II *
Il*IIP,
1s p
Dlf d& Slf d&
JfP a=6
{O, 1F"
39 : The space of all bounded sequences of real numbers. 33 : Norm on a linear space. : Normson2p(R,9, p)-spacesoronV,(R, 9, &)-spaces.121, 122, : See Definition 4.4.11. : See Definition 4.5.5. : Refinement Integral of f with respect to p. : a and 6 are numbers satisfying la -61 5 1. : The space of all sequences of 0's and 1's.
178,180,183 104 116 23 1 270 17
INDEX A Accumulation point, 15 of a sequence of charges, 265 Additive-class, 2 Additivity uniform, 226 uniform countable, 204 Antisymmetric relation, 13 Atom of a Boolean algebra, 22 of a charge (p-atom), 141 of a field, 7 Axiom of choice, 14 B Bake category theorem, 17, 267 property of, 17 o-field, 17 Banach lattice, 34 limit, 39 space, 33 Base topological, 16 filter, 134 Boolean algebra, 18 atomic, 22 complete, 19 nonatomic, 22 pairwise disjoint elements in a, 21 quotient, 20 Boolean algebras atomic, 22 homomorphism between, 19 isomorphic, 19 isomorphism between, 19
Stone representation theorem for, 20 Boolean a-algebra, 19 Borel-Cantelli lemma, 274 Bore1 a-field on R,12 C Cantor set, 17, 265 Caratheodory extension theorem, 81 measure, 274 Cardinal number, 14 Cartesian product space, 13 Cauchy-Schwartz inequality, 123 Cauchy sequence, 16 weak, 33 Chain, 13 Chain condition countable, 21, 211 Charge, 35 atomic, 213 bounded, 35 convex function with respect to a, 238 density, 41 finitely many valued, 249 general invariant, 41 infinitely many valued, 245 modular, 36, 60 negative variation of a, 45 nonatomic, 141 0-a valued, 35 outer, 86 positive, 35 positive bounded, 35 positive real partial, 64 positive variation of a, 45 probability, 35
310 Charge (cont.) pure, 240 range of a, 249 real, 35 real partial, 64 s-bounded, 41 shift-invariant, 39 simple, 188 strongly continuous, 142 strongly nonatomic, 142 total variation of a, 45 unbounded, 42 Charge space, 87 probability, 179 complete, 265 Chebychev’s inequality, 127 Class additive-, 2 compact, 49,245 equivalence, 14 monocompact, 273 Clopen set, 16 Closed under complementation, 5, 6 under countable intersections, 3 under countable unions, 2 under differences, 3 under finite disjoint unions, 5 under finite intersections, 3 under finite unions, 3 under proper differences, 3 under symmetric differences, 3 Closure of a set, 15 Cofinite set, 3 Compact class, 49, 245 Compact topological space, 16 Condition countable chain, 21, 211 Continuous function, 16 9-,133 Continuity absolute, 99, 159 strong absolute, 159 uniform absolute, 127, 204 weak absolute, 159 Convergence hazy, 99 in measure, 92 of a net, 15 Convergence theorem
INDEX
dominated, 88 Lebesgue dominated, 131 Convex function with respect to a charge, 238 Cover open, 16 sub-, 16
D D-integral, 96 Decomposition E-Hahn, 56 exact Hahn, 57 Decomposition theorem general Jordan, 52 Hahn, 56 Jordan, 52 Lebesgue, 168 Riesz, 29 Sobczyk-Hammer, 146 Yosida-Hewitt, 240, 241 Decomposition theorem for measures on cr- fields Hahn, 165 Jordan, 56 Dense-in-itself set, 16, 251 Dense set, 17 Density charge, 41 Derived set, 236 Determining sequence, 104 Directed set, 13 Dominated convergence theorem, 88 Lebesgue, 131 Dual space, 33 Dual of V,-space, 193
E E-Hahn decomposition, 56 Egyptian fraction theorem, 280 Equivalence class, 14 Equivalence relation, 14 Essential boundedness, 89 Exact Hahn decomposition, 57 Exhaustion, principle of, 143 Extension theorem Caratheodory, 81 Extremely disconnected topological space, 278
311
INDEX
F 9-continuous function, 133 F,-set, 15 Field, 2 atomic, 8 discrete, 3 finite-cofinite, 49 p-pure sub-, 274 nonatomic, 8 superatomic, 151 Filter base, 134 in a Boolean algebra, 19 in a field, 10 Finite-cofinite field, 49 Finite dimensional set, 216 Finite intersection property, 16 Finite partition, 8, 14 Finitely disjoint sequence of charges, 144 Finitely many valued charge, 249 First category set, 17 Function continuous, 16 %-continuous, 133 indicator, 12 @-measurable, 91 measurable, 12 modular, 61 null, 88 simple, 90 smooth, 91 strongly additive, 61 T1-measurable, 101 T2-measurable, 101 Functional induced by a real valued set function, 59 linear, 31 Functions equal almost everywhere, 88 G G8-set, 15 General invariant charge, 41 General Jordan decomposition theorem, 53 Generator, 4
H Hahn-Banach theorem, 32
Hahn decomposition E - , 56 exact, 57 Hahn decomposition theorem, 56 for measures, 165 Hamel basis, 32 HausdorfT topological space, 16 Hazy convergence, 92 Holder’s inequality, 122 Homomorphism between Boolean algebras, 19
I Ideal in a Boolean algebra, 19 in a field, 10 m-,137 Image of a set under a map, 16 Indicator function, 12 Inequality Cauchy-Schwartz, 123 Chebychev’s, 127 Holder’s, 122 Minkowski’s, 124 Infinitely disjoint sequence of charges, 145,258 Infinitely many valued charge, 249 Integral D-, 96 lower, 116 refinement, 231 S-, 116 upper, 116 Interior of a set, 15 Invariant charge general, 41 shift-, 39 Isolated point, 15 Isomorphic Boolean algebras, 19 Isomorphism between Boolean algebras, 19 J Jordan decomposition theorem, 52 for measures on (+-fields,56
K Kolmogorov’s Zero-One law, 265
312
INDEX
L L,-space, 121, 178 Lattice Banach, 34 boundedly complete vector, 29 modulus of an element in a vector, 24 negative part of an element in a vector, 24 normal sub-, 28 normed vector, 34 of sets, 1 orthogonal complement of a subset of a vector, 29 orthogonal elements in a vector, 24 positive part of an element in a vector, 24 sub-, 28 vector, 24 Lebesgue decomposition theorem, 168 dominated convergence theorem, 131 measurable set, 264 measure, 49 Lifting, 268 Limit Banach, 39 infimum, 11 supremum, 11 Linear functional, 31 Linear order, 13 Linear space complete pseudo-normed, 33 normed, 33 Linearly independent set, 32 Linearly ordered set, 13 Lower integral, 116 Lower sum, 115
M p a t o m , 141 p-measurable function, 91 p-null set, 87 @-puresub-field, 274 Maximal filter in a Boolean algebra, 19 in a field, 10 Maximal ideal in a Boolean algebra, 19 in a field, 10
Measurable function, 12 p-7 91 Ti-, 101 T2-, 101 Measurable set Lebesgue* 264 7 - 9 264 Measure, 47 bounded, 47 Caratheodory, 274 convergence in, 92 Lebesgue, 49 nonatomic, 141 positive, 47 product probability, 265 real, 41 Metric pre-compact, 273 Metric space, 16 pseudo-, 16 Minkowski’s 124 Modular charge, 36, 60 Modular function, 61 Modulus of an element in a vector latt i e , 24 Monocompact class, 273
N Negative part of an element in a vector lattice, 24 Negative variation of a charge, 45 Net, 15 convergence of a, 15 sub-, 15 weakly convergent, 33 Nikodym theorem, 204 Nonatomic charge, 141 Nonatomic Boolean algebra, 22 Nonatomic field, 8 Nonatomic measure, 141 Norm, 33 pseudo-, 33 Norm bounded set, 33 Normal sub-lattice, 28 Normed linear space, 33 Normed vector lattice, 34 Nowhere dense set, 17 Null function, 88
313
INDEX
Number cardinal, 14 ordinal, 14
0 0-a valued charge, 35
Open cover, 15 Open set, 15 Order linear, 13 partial, 13 Ordered set linearly, 13 partially, 13 well-, 13 Ordered vector space, 23 Ordinal number, 14 Orthogonal complement of a subset of a vector lattice, 29 Orthogonal elements in a vector lattice, 24 Outer charge, 87 Oxtoby’s category analogue of Kolmogorov’s zero-one law, 265
P Pairwise disjoint elements in a Boolean algebra, 21 Partial order, 13 Partially ordered set, 13 Partition finite, 8, 14 refinement of a, 15 Perfect set, 16 Phillips’ lemma, 206 Polish space, 273 Positive bounded charge, 35 Positive charge, 35 Positive measure, 47 Positive part of an element in a vector lattice, 24 Positive real partial charge, 64 Positive variation of a charge, 45 Power set, 3 Pre-compact metric, 273 Principle of exhaustion, 143 Probability charge, 35 Probability charge space, 179 Product probability measure, 265
Property of Baire, 17, 264 Pseudo-metric space, 16 complete, 16 completion of a, 17 Pseudo-norm, 33 Pure charge, 240
Q Quotient Boolean algebra, 20
R Radon-Nikodym theorem, 174,191 Range of a charge, 249 Real charge, 35 Real measure, 47 Real partial charge, 64 Refinement integral, 231 Refinement of a partition, 15 Reflexive relation, 13 Relation antisymmetric, 13 equivalence, 14 reflexive, 13 symmetric, 13 transitive, 13 Relative topology, 16 Riesz decomposition theorem, 29 Riesz representation theorem, 136 Ring, 2
S s-bounded charge, 41 S-integral, 116 Scattered set, 16,236 Seever property, 210 Semi-field, 2 Semi-ring, 1 Semi-variation, 206 Sequence Cauchy, 16 determining, 104 weak Cauchy, 33 Sequence of charges accumulation point of a, 265 discrete, 258 finitely disjoint, 144 infinitely disjoint, 145, 258 Set Cantor, 17,265 clopen, 16
314
INDEX
Set (cont.) closed, 15 closure of a, 15 cofinite, 3 dense, 17 dense-in-itself, 16, 251 derived, 236 Fu-,15 finite dimensional, 216 first category, 17 Gs-, 15 image of a, 16 interior of a, 15 Lebesgue measurable, 264 linearly independent, 32 linearly ordered, 13 p-null, 87 norm bounded, 33 nowhere dense, 17 open, 15 partially ordered, 13 perfect, 16 scattered, 16, 236 7-measurable, 264 tail, 265 weakly closed, 33 well-ordered, 13 with the property of Baire, 17, 264 a-additivity across a sequence of sets, 253 a-class, 2 a-field, 2 Baire, 17 Borel, 12, 17 discrete, 3 a-ideal, 137 a-ring, 2 Shift-invariant charge, 39 Simple charge, 188 Simple function, 90 Singularity, 164 strong, 164 Smooth function, 91 Sobczyk-Hammer decomposition theorem, 146 Space Banach, 33 Cartesian product, 13 charge, 87 compact topological, 16
complete charge, 264 complete pseudo-metric, 16 completion of a pseudo-metric, 17 dual, 33 extremely disconnected topological, 278 Hausdorff topological, 16 Lp-, 121, 178 metric, 16 normed linear, 33 ordered vector, 23 Polish, 273 probability charge, 179 pseudo-metric, 16 Stone, 21 topological, 15 totally disconnected topological, 16 VP-, 185 vector, 31 weakly complete, 33 Stone representation theorem for Boolean algebras, 20 Stone space, 21 Strong absolute continuity, 159 Strong singularity, 164 Strong topology, 33 Strongly additive function, 61 Strongly continuous charge, 142 Strongly nonatomic charge, 142 Subcover, 16 Sub-field, p-pure, 247 Sublattice, 28 normal, 28 Subnet, 15 Sum lower, 115 upper, 115 Superatomic field, 151 Symmetric relation, 13
T TI-measurable function, 101 T,-measurable function, 101 7-measurable set, 264 Tail set, 265 Theorem Baire, 17, 267 Caratheodory extension, 81
315
INDEX
Theorem (cont.) dominated convergence, 88 Egyptian fraction, 280 general Jordan decomposition, 53 Hahn-Banach, 32 Hahn decomposition, 56, 165 Jordan decomposition, 52 Lebesgue decomposition, 168 Lebesgue dominated convergence, 131 Nikodym, 204 Radon-Nikodym, 174, 191 Riesz decomposition, 29 Riesz representation, 136 Sobczyk-Hammer decomposition, 146 Stone representation, 20 Vitali-Hahn-Saks, 204 Yosida-Hewitt decomposition, 240, 24 1 Topological space, 15 compact, 16 extremely disconnected, 278 Hausdorff, 16 totally disconnected, 16 Topology relative, 16 strong, 33 weak, 33 weak*, 158 Total variation of a charge, 45 Totally disconnected topological space, 16 Transfinite induction, 14 Transitive relation, 13 Tree, 150 U Uniform absolute continuity, 127, 204 Uniform additivity, 226 Uniform countable additivity, 204 Upper integral, 116
Upper sum, 115 Urysohn’s lemma, 17
V V,-space, 185 Vector lattice, 24 boundedly complete, 29 modulus of an element in a, 24 negative part of an element in a, 24 normed, 34 orthogonal complement of a subset of a, 29 orthogonal elements in a, 24 positive part of an element in a, 24 . Vector space ordered, 23 over the real line R, 31 over the field of rational numbers, 31 Vitali-Hahn-Saks theorem, 204 W Weak absolute continuity, 159 Weak Cauchy sequence, 33 Weak topology, 33 Weak* topology, 158 Weakly closed set, 33 Weakly complete space, 34 Weakly convergent net, 33 Well-ordered set, 13 Well-ordering, 13
Y Yosida-Hewitt decomposition theorem, 240,241 Z Zero-one law Kolmogorov’s, 265 Oxtoby’s category analogue of Kolmogorov’s, 265 Zorn’s lemma, 14
Pure and Applied Mathematics A Series of Monographs and Textbooks Editors
Samuel Eilenberg and Hyman Base
Columbia University, New Y o r k
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