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Limit measures on second countable locally compact regular spaces Lin, Jung-Fang 1969

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LIMIT MEASURES ON SECOND COUNTABLE LOCALLY COMPACT REGULAR SPACES by JUNG-PANG LIN Sc., Tamkang College of Arts and Sciences, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE;-i n the Department of MATHEMATICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C o lumbia, I a g r e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u rposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada i i ABSTRACT A. Appert proved i n [2] that every sequence of strong measures on a separable weakly l o c a l l y compact metri-zable space has a subsequence converging i n the sense of A to a strong measure. We extend th i s r e s u l t to a second countable l o c a l l y compact regular space. A. Appert also proved that on a weakly l o c a l l y compact metrizable space, a sequence of strong measures converges i n the sense of A to a strong measure i f f i t converges i n the sense of A^ to that strong measure. In [ 3 ] , D. J. H. Garling extended this r e s u l t to a sequence of monotone set functions on a weakly l o c a l l y compact Hausdorff space under certain conditions. We show that this r e s u l t s t i l l holds on a l o c a l l y compact regular space with the same conditions. i i i TABLE OP CONTENTS Page SECTION 1 INTRODUCTION 1 SECTION 2 PRELIMINARIES 3 SECTION 3 PROPERTIES OP SECOND COUNTABLE LOCALLY COMPACT REGULAR SPACES 5 SECTION 4 MEASURE THEORY 10 SECTION 5 CONVERGENCE OP SEQUENCES OP STRONG MEASURES 22 SECTION 6 RELATIONSHIP BETWEEN CONVERGENCE IN THE SENSE A AND ^ 31 BIBLIOGRAPHY '59' ACKNOWLEDGEMENT I would like to express my thanks to Dr. P. S. Bullen for Introducing me to this topic and the guidance given during the preparation of this thesis. I also like to thank the University of British Columbia for the financial support. - 1 -SECTION 1 INTRODUCTION Some results established by A. Appert [1], [2] for metrizable spaces were extended by D. J. H. Garling [3] to more general spaces. We w i l l generalize, make more precise and complete their results. In sections 2 and 3 we give the terminology and notations of point set topology and some properties of second countable locally complact regular space. An example shows that second countable locally compact regular spaces need not be metrizable and i t is to such spaces that we ex-tend Theorem 5 of [2]. In section k we l i s t without proof some results from measure theory. We also define Borel # , Baire # and strong measures and give a characterization of Baire measures. The essential part of section 5 obtains the results of [2] in this more general setting. The definitions of convergence of sequences of strong measures are extended to general topological space. As we note in section 6, these definitions can also be extended to monotone set functions. In [2] i t is shown that our two types of convergence of strong measures are equivalent on a weakly locally compact separable metrizable space. D. J. H. Garling, [3], generalized - 2 -th i s r e s u l t to a weakly l o c a l l y compact Hausdorff space under c e r t a i n condi t ions . In sect ion 6 we show that the r e s u l t s t i l l holds on a l o c a l l y compact regular space under the same condi t ions . - 3 -SECTION 2 PRELIMINARIES We assume here the basic properties of set theory and use [5] as standard reference except f o r some notations and d e f i n i t i o n s given below. Throughout this thesis we w i l l use the contraction " i f f " f o r the phrase " i f and only i f " . The following conventions w i l l be used. 1. 0 denotes the empty set. 2. ft denotes the extended r e a l l i n e . 3. N = (1, 2, 3, • • •} • 4. A ~ B = { x : x e A and x £ B) . 5* ^ - x i ^ i = i = X2' '''» Xn^ * 6. ( x i } i e i = £ xi , i € 1} , where I i s an index set. Let f be a function and A be a set. 7. f[A] i s the d i r e c t f-image of A , i . e . f[A] = Cy : y = f(x) f o r some x € A] . 8. f - 1 [ A ] i s the counter f-image of A , i . e . f _ 1 i A ] = {x : y = f(x) f o r some y € A) . Let <$- he .a, f a m i l y of sets with 0 e G •k-9 . ira = n A . AeG 1 0 . aG = U A . AeG 11. G ~ = { B : B = a G ~ A for some AeG} . 12 . G = {B : B = u A. for some subsequence G' = {A.}. „ a i 6 N 1 1 lew of G} . 13- Ge = [B : B = n A. for some subsequence G' = fA.}. „ 6 i€ N 1 1 i € N of G} • 14 . G is a covering of A i f f A c aG . 15 . G is a a - f i e l d i f f G~ c G and G c G • 0" 16 . Borel G = T { Q : Q is a a-field and G c ()}, i.e. the smallest cr-field containing G . Let X be a topological space and A c X. 17. p(A) = a l l subsets of A . 18 . A denotes the closure of A . 1 9 . A° denotes the interior of A . f 2 0 . A denotes the boundary of A. 21. CA = X ~ A . 2 2 . Baire 9 = TT{^' : 3?' is a a-field and 5 fl (} 6 c 3 ' ) , i.e. the smallest a- f i e l d containing 3 (1 (J6 , - 5 -SECTION 3 PROPERTIES OF SECOND COUNTABLE  LOCALLY COMPACT REGULAR SPACE In this section X w i l l denote a topological space with Q i t s family of open sets. 3-1 D e f i n i t i o n : X i s l o c a l l y compact i f f o r each x e X there i s am open r e l a t i v e l y compact set containing x . 3 . 2 Lemma; If X i s l o c a l l y compact, then there i s a base f o r the topology consisting of open r e l a t i v e l y compact sets. Proof: Let S = (0 : x e X and 0 v i s an open r e l a t i v e l y compact set containing x} . Let IS be the family of a l l f i n i t e intersections of members of S . Then the in t e r s e c t i o n of two members of 6 i s again a member of 8 . By [5, p. 47 Theorem 11] , B i s a base f o r the topology f o r X . Furthermore, since any f i n i t e i n t e r -section of open r e l a t i v e l y compact subsets i s an open r e l a -t i v e l y compact subset, (B consists of open r e l a t i v e l y compact subsets. - 6 -We note that i f X i s second countable l o c a l l y compact, then X has a countable tease consisting of open r e l a t i v e l y compact subsets of X . For convenience, we adopt the following d e f i n i t i o n . 3.3 D e f i n i t i o n : A topological space i s weakly l o c a l l y compact i f f each point has at least one compact neighborhood. I t i s clear that every l o c a l l y compact space i s weakly l o c a l l y compact. But we can see from the following example that the converse i s not true. 3.4 Example; Let ft be the r e a l l i n e and U be the usual topology of ft . Let X = R U {z} where z i s an addi-t i o n a l point. Let Q = { G : G c X , G = : 0 or G = u l ) { z } where u € U] Then the closure of any neighborhood of z i s the whole space, which i s not compact. Hence X i s not l o c a l l y compact. But i t i s e a s i l y seen that X i s weakly l o c a l l y compact. 3«5 Theorem: I f X i s weakly l o c a l l y compact space whi.ch._is - 7 -either Hausdorff or regular, then the family of closed compact neighborhoods of each point i s a base for i t s neighborhood system. Proof: [5, p. 146 Theorem 17] In p a r t i c u l a r i t follows that every weakly l o c a l l y compact Hausdorff space i s regular. 3.6 Corollary: A weakly l o c a l l y compact Hausdorff space i s a l o c a l l y compact regular space. 3.7 D e f i n i t i o n : A topological space X i s c a l l e d p e r f e c t l y normal i f f X i s normal and every closed set i s a Q^-set. Consequently, i n a p e r f e c t l y normal space, every open set i s an 3-^-set. We know, by [5, p. 113 Lemma 1], that each regular Lindelbf space i s normal. But each second countable space i s Lindelof. Hence every second countable regular space i s normal but i n f a c t more i s true. 3.8 Theorem: Every second countable regular space i s p e r f e c t l y normal. - 8 -Proof: We have seen X i s normal, i t remains to prove that i f P i s a closed subset of X then P i s a . Qg-set. For every x £ F , by r e g u l a r i t y of X , there exists G e Q such that x e G cr G c CF . So the family X X X £ = [G„ : x e CF and there exists G e Q such that X X x e G c G c CF) i s an open covering of CF . Since X X X i s second countable, there i s a countable subfamily £' of e such that CF = U G i . e . CF i s an 3 -set and G x n a X so F i s a Qg-set. The following lemma w i l l be used l a t e r . 3-9 Lemma: If X i s normal, F i s closed, G i s open and F c G , then there exi s t A e 3 fl Q& and B e Q H 3 ^ such that F c B c A c G . Proof: By Urysohn's lemma, there i s a continuous function f : X - R, such that 0 < f(x) < 1 , f[ F ] = 0 and f[CG] = 1. 08 Let B = f _ 1 [ [ 0 , i ) ] = U ( f ^ t t O , ^ " ^ 3 ] ] } e Q n > a 00 A = f _ 1 [ [ o , i - ] ] = n [ ^ [ [ o i + h]} e 3 n Q . * . n=2 n Then P c B c A c G 3.10 Lemma: If X Is a second countable regular space, then Baire 3 = Borel 3 . Proof: By Theorem 3»8, every closed set Is , so 3 = 3 n Q6 . Hence Balre 3 = Borel 3 . We mention here that a second countable l o c a l l y compact regular space need not be metrizabie. Since i t need not be Hausdorff. However, by [5], i t i s metrizabie i f i t i s Hausdorff. Consider the following example. 3.11 Example: Let X = (1, 2, 3) , Q = (0, (1), (2,3), {1,2,3}}. Then X i s regular but not Hausdorff. Clearly X i s second countable and l o c a l l y compact. Since each metrizabie space i s Hausdorff, so X i s not metrizabie. - 10 -SECTION 4 MEASURE THEORY In t h i s section X w i l l be a given set. 4.1 D e f i n i t i o n : If B e P(X) , a set function g on B i s a function on B to ft . 4.2 D e f i n i t i o n : A non-negative set function u on P(X) i s ca l l e d a measure on X i f f (i ) = 0 and ( i i ) |i(A) <. T, |i(B. ) , whenever A c U B. c X . ieN 1 ieN 4.3 D e f i n i t i o n : M(X) = : u i s a measure on X} . 4.4 D e f i n i t i o n : If |Jt € M(X) , A c X , then A i s u-measurable i f f f o r every T c X , |i(T) = |i(T PI A) + u(T ~ A) . 4.5 D e f i n i t i o n : If u e M(X) , then 17[ = {A : A c X and A i s n-measurable. - 11 -In the following theorems, we l i s t some basic properties of measurable sets. 4 . 6 Theorem: If ( A n ^ n e N i s a d i s j o i n t subfamily of TH , then U( U A ) = Z n(A ) . neN neN 4 . 7 Theorem: I f U n ) n e N i s a subfamily of ^ and A n c A n + 1 f o r every n e N , then |i( U A ) = lim n(A ) . neN n n-» n 4 . 8 Theorem: Let n e M(X) . (a) I f A,B e %l , then A ~ B e % . <b> I f [ V n e N C \ • t h e n n ^ A n e \ t ' (c) X e n and 0 e ^ . 4 . 9 D e f i n i t i o n : I f u e M ( X ) , A'c X , then the contraction of p. by A , n A , i s the measure on X defined by U A(B) = |i(A n B) f o r every B c X . - 12 -4.10 Lemma: If A c X , then A e 7H i f f any of the following conditions holds. (a) For every T c X , |i(T) < co , p.(T) > u(THA) + u(T ~ A). (b) For B c X and T c X , ^ ( T ) = (Xg(T fl A) + L L ^ T ~ A) . (c) For every B c X , |i(B) < • , Hg(X) = MgfA) + ^ ( X ~ A). 4.11 Corollary: Let [X e M(X) and A c X . Then A e to i f f f o r every B e x and |i(B) < «> , A e 7I\ . 4.12 Theorem: If g i s a non-negative set function on # , p7 e M and g(P') = 0 , then the set function v defined on P(X) by Y(A) = i n f f t : t = S g(H ), (H } c » and A c U Hn} neN neN i s a measure. We c a l l v the measure generated by g on B . 4.13 Lemma: If v i s the measure generated by g on ~A , then 1 - 13 -(a) f o r every A c X , v ( A ) < «> and e > 0 , there exists a B e tt such that A c B and Y ( A ) + e > Y ( B ) . such that A c B and y(A) = Y ( B ) . 4.14 D e f i n i t i o n : If Y i s the measure generated by g on tt , then Y i s c a l l e d a regular measure i f f (a) Y ( H ) = g ( H ) f o r every H e tt . (b) K c ^ y . 4.15 D e f i n i t i o n : A family of sets & i s ca l l e d a semi-ring i f f f o r A e tt and B e M , (a) A n B' € M • a (b) f o r every A .c X , Y(A) < * , there exists a B' € M We write g = Y 13* i f Y ( H ) = g ( H ) f o r every K e tt n (b) A ~ B = U H, where (H. ) k=l k k i s a f i n i t e d i s j o i n t subfamily of W . 4.16 D e f i n i t i o n : A set function g i s said to be additive on a family of sets tt i f f f o r every countable d i s j o i n t subfamily - 14 -f B } ,T of M with U B e tt , g( U B ) = E g ( B ) . n n e N neN neN n neN n Here we have two theorems which characterize regular measures; Theorem 4.18 i s known as the extension theorem. 4.17 Theorem: If g i s a non-negative set function additive on a semi-ring M , then the measure y generated by g on & i s a regular measure. Proof: Suppose that H^, E^ e W and c Hg • Then, n since H i s a semi-ring, H 2 = ^ U (Hg ~ H 1) = E± U U H^ where f H i ^ i - i i s a f i n i t e d i s j o i n t subfamily of W . n Since g i s additive on M , g(Ho) = g(H n U U H.') = d 1 i = l 1 n g(H 1) + X g(H/) . So that g(H 2) > g(H 1) . Suppose H e M , H c U E. and (E.}. „ c JJ . ieN 1 1 1 € N Putting M1 = E fl E 1 , N 2 = H ~ M1 , then there i s a f i n i t e k? _ ^2 d i s j o i n t subfamily C N ^ J J ^ of » such that Ng = U N 2 1 Putting M2 = E 2 0 N 2 , N 3 = H - M1 ~ M2 , k2 k2 M„ = E 9 n ( U N U , ) = U (E„ fl N Q. ) and * ^ 1=1 d X 1=1 ^ ^ then - 15 -k 2 k2 k3 k o N 3 = ^ N 2 ± ~ M 2 = ^ ( N 2 i ~ M , ) = N 3 ± where f ^ } * ^ i s a f i n i t e d i s j o i n t subfamily of M . Let M ^ = fl N^ . Continue i n this way, we have a d i s j o i n t family ( } ^ e j g and M n = E n 0 N n = ^ ( E ^ 0 N n l) where i \ ± ) ^ 2 i s a f i n i t e d i s j o i n t subfamily of W . Then H = M 1 U N 2 = M 1 U ((N 2 0 E 2 ) U (N"2 ~ Eg)) = K U M 2 U ( H ~ ~ E g ) = M ^ U Mg U ( H ~ M ^ ~ Mg) = U M 2 U N 3 . k n Consequently, H = U M = U U ( E 0 N . ) U ( H 0 E , ) and neN n n>2 1=1 n n i 1 we have kn g(H) = g ( H fl E ) + E S g(E n N ) -1 n>2 1=1 n n i kn = g ( H fl E ) + S g ( E fl U N ) 1 n>2 n 1=1 n i < g ( E T ) + E g ( E ) = T, g(E ) , n>2 n neN n This implies g ( H ) <_ Y ( H ) . But i t i s clear that g ( H ) I y ( H ) . So g ( H ) = Y ( H ) f o r every H e » . Let T c X , Y ( T ) < » and e > 0 , then there exists (B } M c H such that T c U B and n neN „ n neN - 16 -Y ( T ) + e > S g(B ) = E g((B fl H) U (B ~ H>) . neN n neN n n m m Since B i s a semi-ring, B ~ H = U B . where (B .}. , i 1 — i s a f i n i t e d i s j o i n t subfamily of B . Then •'in Y ( T ) + e > S g((B n H) U ( U B.)) neN n 1=1 n i In = S g(B n H) + S S g(B ) neN n neN 1=1 n i = Y ( T n H) + Y ( T ~ H) . Hence H c to. . 4.18 Theorem: I f Y i s the measure generated by g on B , then Y i s a regular measure i f f f o r every semi-ring B ' , B c B ' c to, and i f g' = y|#' then the measure generated by g' on B ' i s exactly the measure y • Proof: Let Y be a regular measure, B be a semi-ring, B e B' c to, and g' = y\W . Let Y ' be the measure generated by g' on B' , g" = Y | ^ 6 and y" be the measure generated by g" on B^g . Since g" i s additive on B^6 c to, and B^6 i s a semi-ring, by Theorem 4 .17 , Y " - 17 -i s a regular measure. Clearly y" <_ y' <_'y . Let T c X . I f Y"(T) = « , then Y " ( T ) = Y ' ( T ) = Y ( T ) . If Y " ( T ) < «, by Lemma 4.13(b), there exists S c X , S e such that T c S and Y " ( T ) = Y " ( S ) . Since Y " i s regular, g" = y"\rb , then Y"(T) = Y"(S) = g"(S) = Y(S) > Y ( T ) . Hence Y"(T) = Y ( T ) , i . e . y" = y ' = Y • The converse follows immediately from Theorem 4.17* The proof i s complete. 4.19 Lemma: 0 Let u e M ( X ) and (A } „ be a family of subsets ~ v 7 • n neN J of X such that A n + 1 c A n f o r every n e N . I f , f o r every T c. X , LI(T) < «• , ' ° An+1' + M(T ~ A J = U(T 0 An+1 U T ~ V ' then fl A e 7ft . neN n ^ Proof: Suppose that T c X , LI(T) < « and v = LI t , then the above condition reduces to v(A _ ) + v ( X ~ A ) = v(A - U X ~ A ) . v n+1' v n' v n+1 n 7 Let T ' = T f l , by the given condition again, H ( T ' n A n + 1 ) + U ( T'.~ A n) = u ( T ' fl A n + 1 U T ' ~ A j . - 18 -i . e . n(T n A n + 1 ) + n(T H ( A n . 1 ~ A j ) - n(T 0 ( A n + 1 U ( A ^ ~ A J ) ) i . e . v ( A n + 1 ) + v ( A n . 1 ~ A J = v ( A n + 1 U ( A ^ ~ A J ) < v f A ^ ) . Then v ( A n _ 1 ~ A j < v ^ ) ~ v ( A n + 1 ) . So S v(A ~ A ) < v(A ) + v(A ) < » . n>2 n 1 n 1 d Let e > 0 , there exis ts m e N such that > v ( A n - l ~ A n^ < e ' n=m Let A = D A , then neN n v(A) + v(X - A) < v ( A m + 1 ) + v(X ~ A m ) + v (A m - A) < v(A ) + v(X ~ A ) + £ v(A ~ A _) — v m+1' v nr > v n n+ l ; n—m < v ( A m + 1 ) + v(X ~ A m ) + « = v ( A m + 1 U X ~ A J + « <_ v(X) + e . By Lemma 4.10(c), A e fl? 4.20 D e f i n i t i o n : I f B e P(X) , A 1 , A e P(X) , then A ± and A g are sa id to be separated by B i f f there ex i s t and R"2 i n B such that A c n B c H 2 and n H 2 = gS . Let X be a topo log i ca l space. - 19 -4.21 D e f i n i t i o n : If n e M(X) , LI i s ca l l e d a Borel 3 measure i f f 3 c TH • 4.22 D e f i n i t i o n : If u e M(X) , u i s ca l l e d a Baire 3 measure i f f 3 n Q 6 c ^ . The following theorem i s an important r e s u l t f o r Ba ir e 3 measure. 4.23 Theorem: If X i s normal,' ti e M(X), then LI i s " a Baire, 3 measure i f f LI(A u B) = n ( A ) + \x{B) whenever A and B are separated by 3 • Proof: Suppose that LI i s a Baire 3 measure, then 3 n Qfi c fli . Let A c X , B c X and A, B be separated by 3 . Choose Fi>F2 e 3 s u c h t h a t A C P l * B C P2 a n d P 1 fl P 2 = J2T . By Lemma 3»9, there exists A' e 3 (1 Q 6 such that A c F 1 c A' C CF 2 . Since A' e fll , |i(A U B) ='n(A U B fl A') + |i(A U B ~ A') = |i(A) + u(B) . Conversely, assume the condition holds and A e 3 fl Q 6 . Then A = n' G n where ( G n } n g N <= Q . Let - 2 0 -G ' = n G . , then G ' => G ' _ , G ' e Q f o r every n e N and n i < n 1 n — n+1 n * J A = n G ' . By normality of X , there exists a sequence neN (U } M of open sets such that A c t f - c U - C U c G ' . L n neN c n+1 n+1 n n Hence ^n+2_ a n d CU are separated by 3- . Suppose that T e x and |i(T) < » . Since T D U n + 1 and T n CU n are also separated by 3\ , by hypothesis, ^ T 0 Un+1 } + w ( T 0 CV = ^ ( ( T n u n + l ) U ( T 0 C U n } ) = ii((T n u n + 1 ) u ( T ~ U R ) ) . By Lemma 4.19, n U e #L , i . e . A e # L . Hence n i s a neN " I - 1 *x Baire 3 measure. ¥e extend the d e f i n i t i o n of strong measure of [ 2 ] to a general topological space. 4.24 Defintion: If M. i s a measure on X and if i s a family of subsets of X , then u i s M-regular i f f Jt e fl? and f o r every A c X , n(A) = i n f ( t : t = n ( E ) , A C E and E e ») . 4 . 2 5 D e f i n i t i o n : A Q-regular measure w i l l be ca l l e d a strong - "21 -measure. We note here that not a l l measures are strong measures. Consider the fo l lowing example. 4.26 Example: Let X be an i n f i n i t e set and Q = { E : E c X , E i s e i ther empty or X ~ E i s f in i te} . For every E c X , define n(E) = number of points of E . Then, c l e a r l y , n i s a measure. For G e Q , G ^ 0 , then |i(G) = » . Then i f E c X and E i s f i n i t e se t , u(E) < » . So LI (E) ^ inf{u(G) , E c G , G € Q} . i . e . LI i s not a strong measure. - 22 -SECTION 5 CONVERGENCES OF SEQUENCES OF STRONG MEASURES Let X be a topological space,-.with the family of open sets Q . 5.1 D e f i n i t i o n : A sequence ^n^ngN °^ strong measures on X i s said to converge i n the sense of A to a strong measure u i f f f o r every r e l a t i v e l y compact subset E of X .' \i(E°) < lim u (E) < lim" ^  (E) < . For convience, we write u -• M-(A) i f ( ^ J ^ ^ M converges i n the sense of A to yx • 5.2 D e f i n i t i o n : If f^n^ngjj a n d ^ a r e strong measures on X , then \i - n(A*) i f f f o r every r e l a t i v e l y compact subset E of X , u(E f) =0 implies lim H n (E) = |i(E) . n-,» 5.3 Lemma: If f^n^ngN a n d ^ a r e strong measures on X , then |i - u(A) implies u n -* n(A*) . - 25 -Proof: Let E be a r e l a t i v e l y compact subset of X . o f* Since E = E U E and p. i s a strong measure, L i ( E ) = u ( E ° ) + L i ( E F ) = H ( E ° ) . So H ( E ) = |i(E) = L l ( E ° ) . By hypothesis, H(E) m n ( E ° ) < lim M N ( E ) < Urn" U ( E ) < = u ( E ) . Hence L I ( E ) = lim n ( E ) . n-»«> n 5-4 D e f i n i t i o n : A family ( E ^ } l € l of subsets of X i s ca l l e d a to p o l o g i c a l l y monotone parametrized family i f f f o r i ^ j either E . c E ° or E . c E ? . 1 3 3 i We write E . c E . - i f E. c E ° . Since i t j 1 2 X ~ E , c X ~ E ° and X ~ E. e (X ~ E . ) ° , then i f { E , } . T 1 1 i v i y ' 1 i l e i i s a t o p o l o g i c a l l y monotone parametrized family on X , so i s (X ~ E i ^ e i • L e t E c X and P c X , then we have (i) E c E i f f E = E ° i f f E i s closed and open, ( i i ) E c. p i f f E° c E c P° c P , ' ( i i i ) E c P i f f E f n F f = pf and E c P . 5.5 D e f i n i t i o n : A property which holds f o r a l l but an at most countable number of element § of an index set I i s said - 2h -to hold f o r nearly a l l % e I . If a property P holds f o r nearly a l l 5 e I , we write P f o r n.a. § e I . 5.6 D e f i n i t i o n : If f^n^ngN a n d ^ a r e strong measures on X , then u n u(A^) I f f f o r every t o p o l o g l c a l l y monotone para-metrized family [ E ^ ) o f r e l a t i v e l y compact subsets of X , lim ^ ( E , ) = Ut(E ) f o r n.a. i e l . n-*<» The following theorem i s a more general form of Theorem 5 of [2] . 5*7 Theorem: Every sequence of strong measures on a second countable l o c a l l y compact regular space has a subsequence converging i n the sense of A to a strong measure. Proof: Suppose that t ^ ^ g N ^ s a s e c l u e r i e e °f strong measures on X , a second countable l o c a l l y compact regular space. We can choose a countable family 0 of open r e l a t i v e -l y compact sets as a base of X . We may assume without loss of generality that i s closed with respect to f i n i t e union and i n t e r s e c t i o n . By the Cantor "diagonal process", we can extract a subsequence fn, •} • \t °f (M M). m such that f o r every K J jeJM 1 l e N B e © , lim p., .(B) exists ( f i n i t e or i n f i n i t e ) . Let j-=o K J Y(B) = lim L L . ( B ) f o r every B e © . Now define f o r every E c X , M-(E) = i n f f t : t = S Y(B, ), (B, ). „ c B, E c U B ) ... [A] ieN 1 1 1 € N ieN 1 Then, by Theorem 4 . 1 2 , we have that p i s a measure. Let E c X , p(E) < «. and e > 0 . By Lemma 4 . 1 3(a), there exists G e © a such that E c G and p(E) + e > p(G) . If n(E) = eo , then f o r every G e Q and E c G , we have p(G) = oo . Hence, f o r every E c X , p.(E) = i n f [ t : t = p.(G) , G e Q and E c G} . We now prove Q c fl? by proving p. i s Baire 3 . By Lemma 3 ' 1 0 , Borel 3 = Baire 3 . Since Q c Borel 3 , so i f p i s a Baire 3 measure, then Q cr Borel 3 c 7H . Let H-^Hg c X > H i a n d H 2 b e separated by 3 . Then there ex i s t F-^Fg € 5 s u c h t h a t H l c F i » H 2 C P 2 and n P 2 = 0 . Since X i s normal, there exist G 1 , G 2 e $ s u c h fchat F l C G l ' P 2 C G 2 a n d G l 0 G 2 = ^ ' We may assume without loss of generality that v(E^ U Hg) < oo Let e > 0 , by [A], there exists a countable subfamily fB, }. „ of B such that H. U L c U B. and ^ 1 1 € N 1 2 ieN 1 L I ( H , U H P ) + e > S Y ( B , ) (1) 1 d ieN 1 Por every i e N , B^ i s an open r e l a t i v e l y compact subset of X . H^ (1 B^ i s a closed subset of the compact set B^ , H^ D B^ i s compact. Since G^ i s open, G, = U B! where fB'}. <z B . Now, since l l e N l l lew H.. n B. c H_ n B. c H_ c p c Gn , there i s a f i n i t e sub-1 i 1 l 1 1 1' covering, say (B^}!?_^ > °£ ^ j p i e N such that n n H n B . c H n B . c U B.' c G n putting G. . = U B . ' , then 1 1 1 1 i = 1 i i l i 1 = 1 l e B f o r every i e N . S i m i l a r l y , there i s e B f o r every i e N such that H g f l B 1 c H 2 f l B± c a c G g . Let H. = G . . fl B , , N. = G 0. fl B. , then M. , N . e 8 , l l i i ' i 2 i I * i ' l ' M i ° N i = ^ a n d M i U N i C B i ' P o r e v e r v j e N , u k i s a strong measure, so that M k.(M 1 U N ±) = p.kj(M.) + u k j(N.) < n k J ( B i ) . Letting j -we gave Y ( M ± ) + Y ^ ) <_ Y ( B ± ) . Then S Y ( M , ) + S Y(N, ) < S Y ( B , ) < (i(Hn U H„) + e by ( l ) ieN 1 ieN 1 ieN 1 1 * We claim that H e U M. and H 0 c U N. . Since 1 ieN 1 d ieN 1 - 2 7 -U M. = U (G n B. ) 3 U (H fl B. O B ) ieN 1 i€N 1 1 1 ieN 1 1 = u (H n B. ) = E1 n ( U B ) = H . ieN x 1 x ieN 1 1 S i m i l a r l y , H c U N. . By [A], n(H ) < S Y ( M . ) and d ieN 1 1 ieN -tt(H2) < S Y ( N ± ) . So, we get n ^ ) + |i(H 2) < yx(E1 U Hg) + e Since € i s a r b i t r a r y , we have n t H j + |i(H 2) < \s(E1 U H 2) . But we always have v(E^) + }i(Hg) >_ yx(E1 U Hg) . Consequenly, |a(H 1 U H 2) = u C ^ ) + |i(H 2) . By Theorem 4 . 2 3 , U i s a Baire 3 measure. Hence Q c to . F i n a l l y , we show that f^kj^jeN c o n v e r g e s to |i i n the sense of A . Suppose that E i s a r e l a t i v e l y compact subset of X . Since E° i s open, there exists o J (G.}. M c B such that E = U G. . Put E. = U G. , l lew ieN 1 0 1=1 1 then E.' c E _ , E. e ft f o r every j e N , and E° = U E.. •J J + 1 J jeN J Since [E } . „ c fl? , by Theorem 4 . 7 n(E°) = lim |i(E.) . j je i N M j-»» ^ Let T) >_ 0 and u(E°) >• TI . Choose a e N such that w ( E A ) > TI • Assume without loss of generality that n(E) < °° . Given e > 0 , by [A], choose (Qj^ieN C 1 5 s u c n that E c U Q. and T, y(Q.. ) < n(E) + e . Since E i s compact, i€N 1 ieN 1 • ,m there ,is a f i n i t e subfamily [ Q ^ ) , ^ of ( Q ± } l e N such that m — m ' Ec: U Q . So, we have E c E G c E c E c U Q. and 1=1 x l 1=1 1 1 m M k j(E a) < Mk.(E) < u k.(E) < ^ k ^ ± l ) • Le t t i n g J - «. , we get m Y(E ) < lim Wfc1(E) < lim U k 1(E) < E Y(Q 1 - t) < e + u(E) Since Y(E a) > |i(E a) > r\ , we get TI <_ lim U K 1 ( E ) 1 lim U K 1 ( E ) < u(E) + e . Hence u(E ) <_ lim U k 1(E) <. lim U k 1 ( E ) <_ |i(E) . i . e . [m •) J >T converges i n the sense of A to the strong measure L kj jeN & & . The proof i s complete. The following c o r o l l a r y i s the Theorem 5 of [ 2 ] . 5.8 Corollary: Every sequence of strong measures on a separable weakly l o c a l l y compact metrizabie space has a subsequence converging i n the sense of A to a strong measure. Proof: By Lemma 3-6, a separable weakly l o c a l l y compact metrizable space i s second countable and regular so this c o r o l l a r y follows immediately from Theorem 5 . 7 -Theorem 5 . 7 depends e s s e n t i a l l y on the f a c t that the space i s second countable as the following example taken from [ 3 ] shows. 5 . 9 Example: Let X = (s : s = [s^, s^, s^, ...} i s a subsequence of the integers} and Q = P(X) , i . e . the discrete topology. Then X i s a uncountable l o c a l l y compact and p e r f e c t l y normal space. For s = (s^, s^, s^, ...} e X and every n e N , l e t U (s) = l i f n = s f o r some j e; N ui (s) =0 otherwise. n v ' If E c X , l e t u (E) = S u (s) f o r every n e N . n seE n . Then t^n^ngjj i s a sequence of strong measures. But [ H n } n e N has no subsequence which converges i n the sense of A to a strong measure. I f CM-. 4 } • M i s a subsequence of [ un }neN ' l e t 1 = U j ] j e N e X ' T h e n V.Al) = 0 i f j i s odd u. Al) = 1 i f j i s even. - 30 -So lira Y±. ..(l) = 0 and lim n .(£) = 1 . For every strong measure I-t , since I i s open and closed, = = • Hence .} can not converge i n the sense of A to a strong measure. SECTION 6 RETATIONSHIP BETWEEN CONVERGENCE IN THE  SENSE OF A AND ^ Let X be a topological space with Q i t s family of open sets. 6.1 Definition: A set function M- on X is called a monotone set  function i f f the following two conditions are satisfied. (a) n(0) = 0 . (b) If E,P e P(X) and E c F , then u(E) <. n(F) . Evidently, every monotone set function is non-negative . The following lemma is taken from [3]• 6.2 Lemma: For every monotone set function p. on X and every topologically monotone parametrized family f E i ^ e j of subsets of X , \s(E±) = n(E°) for n.a. i e I . Proof: Let 1^ = ( i : i e I and there exists j e I , j / i such that E, = E.} . Then I. c I . - 3.2 ~ I f 1 e 1-^ , then there exists j e I , j / i such that = . Since t E i ^ i € i i s a t o p o l o g i c a l l y monotone parametrized family, we have either. E i c E^ or E. c E± . Then either E^ c E° = E° or E^ c E° = E° . So i n either case, E? = E^ = E^ . Let I 2 = I ~ I 1 . Define i <_ j i f f E± c E^ . It i s e a s i l y seen that the r e l a t i o n "<_" i s a t o t a l order on Ig . Define f ( i ) = n(E°) f o r every i e Ig . Then f i s monotonically increasing on Ig . Let L = ( i : i e L and f ( i ) < i n f f ( j ) ) . J j > i , j € l 2 Since the set of discontinuous points i s at most countable, I^ i s at most countable. If i e Ig and li(E°) 4 v(E±) , then, f o r every j e Ig , j > i implies E^ c E^ . Since ^±^±ej  i s a t o p o l o g i c a l l y monotone parametrized family, either E^ c E° or E. .<=• E° . But E.. c E° i s impossible, so we have E± c E 0 . Then f (j) = u(E°) > \x{E±) > \x(E°) f o r every j > i . We get f ( i ) < i n f f ( j ) and hence i e I, . j > i , j e l g 5 Consequently, n(E°) = v(E±) f o r n.a. i e l . - 33 -Let (p. } >T and p. be monotone set functions on 1 n neN X . We write | i - u(A) i f the condition of D e f i n i t i o n 5.1 holds. S i m i l a r l y , we write P-n - p(A*) and p. p,(A1) . Then Theorems 6 and 7 of [2] can be extended i n the following manner. 6.3 Theorem: Let (J be a strong measure and ^ n ^ n e N b e a sequence of monotone set functions on a l o c a l l y compact regular space X . Suppose that either (i ) each open r e l a t i v e l y compact set i s an 3 set or ( i i ) u(K) < » , f o r every compact subset K of X . Then P-N - p(&) i f f „ P N - p( ^ ) Proof: Let n n - p(A) and (E^^3 i ej be a top o l o g i c a l l y monotone parametrized family of r e l a t i v e l y compact subsets of X . Then, f o r every i e I , P ( E ° ) < lim P n ( E . ) < lim" p R ( E ) < p ( E . ) . n-»<» By Lemma 6.2, p ( E ° ) = P(E" 1) f o r n.a. i e I . Then p ( E 1 ) = n (E° ) = p ( E ± ) for n.a. i e I . Hence p ( E . ) = lim P n ( E . ) f o r n.a. i e I . i . e . P N - P-(A,) . n-*<» - 3 A -Next, we assume V-n'~* ) and E i s a relatively-compact subset of X . (a) If | i ( E ) < oo Given 6 > 0 , there exists an open set G such that E c G and |i(G) <_ | i ( E ) + e . By Theorem 3 .5, f o r every x e G , there i s a closed compact neighborhood V of x X such that x e V c G . Choose 0 e Q such that X X x e 0„ c V c G . Since 0 v i s a closed subset of V , so X X X X 0 i s compact and 0 i s an open r e l a t i v e l y compact set X X containing x . Then G = U 0 and [0.1 ~ i s an open xeG x x xeu covering of E . So, there i s a f i n i t e subcovering, say k k ( 0 . , of (0 } _ such that E c U 0. c G . Let 1 1 — 1 X X €VJ _^ 1 k H = U 0. , then H i s an open r e l a t i v e l y compact subset of 1=1 1  _ _ k k _ G and E c H c H = U 0. = U 0. <= G . Since X . i s regular, 1=1 1 1=1 1 H i s a compact regular space, by [5, p. I k l Theorem 10], H i s normal. By Urysohn's lemma, there exists a continuous function f on H to [0,1] such that f(x) = 0 f o r x € E and f(x) = 1 f o r x e H ~ H . Define E t = f _ 1 [ [ 0 , t ] ] f o r t € [0,1) . Then { E J ^ J - Q ^ i s a to p o l o g i c a l l y mono-tone parametrized family of closed compact sets and E ^ c H c H. By hypothesis, there i s t e (0,1) such that H(E.) = lim I-L ( E . ) . Since E c E , c H , n(E.) < n(H) and - 35'.-M-n(E) <.,Lln(E^.)' f o r every n e N , we have lim |i ( E ) 1 lim M- ( E T ) = n ( E T ) < n ( H ) <. n ( E ) + e . . . . (2) n-»os n-»°° Since .|i(i~) < =° , | i ( E ~ E ° ) < » . For the given e , there exists an open set U such that E ~ E ° c tj and M-(U) <. n ( E ~ E ° ) + e . Let M = E ~ U , then M c E ° and M i s closed compact. Since p i s a strong measure and M U U =5 ( E ~ U) U U = E = ( E ~ E ° ) U E ° , we have H ( M ) + n(u) > H ( M u u) > n(f ~ E ° ) + U ( E ° ) > n(u) - c + H ( E ° ) . Then u(M) >_ n.0E°) + e . Since E i s normal by Urysohn's lemma, there exists a continuous function g on E to [0,1] such that g(x) = 0 f o r x e M and g(x) = 1 f o r x € E ~ E ° . Define F f e = g _ 1 [ [ 0 , t ] ] f o r t e [0,1) . Then M c F c E ° and F^ i s a closed compact set, f o r every t e [0,1) . So . {F^.} e^j-Q i s a top o l o g i c a l l y monotone parametrized family of closed compact sets. By hypothesis, there i s t e (0,1) such that n(F t ) = lim n (p ) n-»» Then lim LI (E) >. lim LI (F, ) = |i(F.). > LI(M) > a(E°) - € n-»« ' n-»°=> i . e . lim M E ) > ^ ( E°) " e • (3) n—ao Combine (2) and (3), we get | i ( E ° ) - e 1.11m P n(E) <. lim u ( E ) < u ( E ) + c n-»oo n-*oo Since e i s a r b i t r a r y , L I ( E ° ) < lim n (E) < H i " ia (E) < L I ( E ) (B) n-*eo n-»« (b) I f Lt(f) = ~ . Clearly, lim LI ( E ) < L I ( E ) ( 4 ) n-*» n By assumption ( i ) , E ° = U D . ' where ( L V / L a t I S a sequence ieN 1 1 1 6 W : i of closed sets. Let D . = U D ' . f o r every i e N . Then 1 j=l J ^ i ^ i e N ^ s a n i n c r e a s i n S sequence of closed sets. By Theorem 4 . 7 , n(E°) = lim L I(D ) . If n ( E ° ) < co . Then, f o r e > 0 , there exists m e N such that n ( D M ) >. n ( E ° ) - e . Since E i s normal, by s i m i l a r argument as above, we have lim n (E) > U ( E ° ) - e ( 5 ) . n-»oo I f n ( E ° ) = co . Then, given a > 0 , there i s I e N such that ^ C 1 ^ ) >. a • Then, arguing as above, we have lim LI ( E ) >_ a . Hence n-*oo | i ( E ° ) = lim | i N ( E ) . . ( 6 ) n-»oo Combine ( 4 ) , ( 5 ) and ( 6 ) , since c i s a r b i t r a r y , - 37 U(E Q) < lim n n(E) < lim ^ ( E ) = u(E) (C) By (B) and (C), f i n a l l y we have l-KEj < lim U (E) < lim u (E) < n(E) n-*os n-*<» i . e . 6 . 4 Corollary: Let fu } n JneN and | i be strong measures on a separable weakly l o c a l l y compact metrizabie space, then U n - i f f ^ n - n(A 1) . Proof: By Theorem 3 * 8 , such a space i s p e r f e c t l y normal and so 6 . 3 ( i ) i s s a t i s f i e d and r e s u l t i s implied by Theorem We state the theorem of [3] i n the following c o r o l l a r y . 6 . 5 Corollary: Let H be a strong measure on a weakly l o c a l l y compact Hausdorff space X . Suppose that either ( i ) each open r e l a t i v e l y compact set i s an 3^-set or ( i i ) P-(K) < co f e r every_compact K . Then i f Ct-O-,.^  i s a'sequence of-monotone set functions 6 . 3 . - 3$, -on x , n - P (A) i f f n -» H ( A 1 ) . n v 7 n v l 7 Proof: By Corollary 3-6, X i s l o c a l l y compact and regular. Hence this c o r o l l a r y follows d i r e c t l y from Theorem 6.3« - 39 -BIBLIOGRAPHY [1] A. Appert: Mesures; et families topologiquement mono-tones . Comptes Rendus 260 (1965), 6493-95. [2] A. Appert: Mesures limites dans les espaces distancies  separables et localement compacts. Proc. London Math. Soc. (3) 18 (1968), 266-81. [3] D« J. H. Garling: A note on l i m i t measures on l o c a l l y  compact spaces. Proc. London Math. Soc. (3) 18 (1968), 282-84. [4] M. E. Munroe:. Introduction to measure and integration. Addison-Wesley, Cambridge, Mass., 1953-[5] J. Kelley: General Topology, Van Nostrand, New York, (1957). 

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