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Limits of inverse systems of measures Mallory, Donald James 1968

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LIMITS OP INVERSE SYSTEMS OF MEASURES b y DONALD J . MALLORY B . A . S c , U n i v e r s i t y of B r i t i s h C o l u m b i a , 1959 M . A . , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1962 A THESIS SUBMITTED IN PARTIAL FULFILMENT OP 'THE REQUIREMENTS FOR THE DEGREE.OF DOCTOR OF PHILOSOPHY i n the Department o f Mathematics We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1968 I n 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 t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e 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 a n d S t u d y . I f u r t h e r a g r e e t h a t 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 r p o s e s m a y b e g r a n t e d b y t h e H e a d o f m y D e p a r t m e n t o r b y h . i l 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 n o t b e a l l o w e d w i t h o u t m y w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a D a t e QU^^f 2 9 /? C % i i ABSTRACT S u p e r v i s o r : D r . M. S i o n I n t h i s paper we are c o n c e r n e d w i t h t h e p r o b l e m of f i n d i n g ' l i m i t s ' of i n v e r s e (or p r o j e c t i v e ) systems o f measure spaces ( f o r a d e f i n i t i o n o f these see e . g . C h o k s i : I n v e r s e L i m i t s o f Measure, S p a c e s , P r o c . London M a t h . S o c . 8, 1958). Our b a s i c l i m i t measure , ja, i s p l a c e d on the C a r t e s i a n product of the spaces instead of on the inverse l i m i t set, L . As a r e s u l t we o b t a i n an e x i s t e n c e theorem f o r t h i s measure w i t h fewer c o n d i t i o n s on the system t h a n are u s u a l l y n e e d e d . We a l s o i n v e s t i g a t e the e x i s t e n c e of a l i m i t measure on L b y r e s t r i c t i n g our measure ja t o L . T h i s e n a b l e s us t o g e n e r a l i z e known r e s u l t s and t o e x p l a i n some o f the d i f f i c u l t i e s e n c o u n t e r e d b y the s t a n d a r d i n v e r s e l i m i t measure . I n p a r t i c u l a r we show t h a t the p r o d u c t t o p o l o g y may be t o o f i n e t o a l l o w the l i m i t measure t o have good t o p o l o g i c a l p r o p e r t i e s ' ( e . g . t o be R a d 6 n ) . A n o t h e r t o p o l o g y w h i c h i s r e l a t e d t o the p r o d u c t s t r u c t u r e i s i n t r o d u c e d and we show t h a t l i m i t measures w h i c h are Radon w . r . t . t h i s t o p o l o g y can be o b t a i n e d f o r a wide c l a s s o f i n v e r s e systems o f measure s p a c e s . i i i CONTENTS Page INTRODUCTION 1 CHAPTER .1. SET THEORETIC PRELIMINARIES. . . . . . 4 1. Se t O p e r a t i o n s 4 2. C a r t e s i a n P r o d u c t s 8 3. ^-compact C l a s s e s 10 4. P r o o f s " •' 12 CHAPTER I I . MEASURE THEORETIC PRELIMINARIES . . . 21 1. C a r a t h e o d o r y Measure 21 2. A p p r o x i m a t i o n and G e n e r a t i o n o f Measures .• 24 3. P r o o f s i. 29 CHAPTER I I I . INVERSE SYSTEMS OP MEASURES 37 1. Concepts o f I n v e r s e Systems . . . '• 37 2. G e n e r a t i o n o f a Tr - l imit Outer M e a s u r e . . . . . 42 3. P r o o f s 46 CHAPTER I V . APPROXIMATION PROPERTIES OF LIMIT MEASURES ON PRODUCT SPACES. . . . . . . 51 1. The t o p o l o g i c a l S i t u a t i o n . . . . . . 52 2. The N o n - t o p o l o g i c a l . S i t u a t i o n . . . . . . . . 57 3. . P r o o f s 61 CHAPTER V . LIMIT MEASURES ON THE INVERSE LIMIT SET. . . . 73 , 1. D e f i n i t i o n s and N o t a t i o n . . . . . . . . . . . 73 2. E x i s t e n c e o f an I n v e r s e L i m i t Measure . . . . . 75 3« A p p r o x i m a t i o n P r o p e r t i e s of I n v e r s e L i m i t Measures . . . . . . . . . . . . . . 80 4. . P r o o f s . . . . . . . . . . . . . . . . . . . . 85 i v Page APPENDIX . . . . . 98 BIBLIOGRAPHY i ' 114 ACKNOWLEDGEMENTS The a u t h o r would l i k e t o thank h i s s u p e r v i s o r , . D r . M a u r i c e S i o n , f o r h i s h e l p and guidance t h r o u g h o u t the w r i t i n g o f t h i s t h e s i s and the a u t h o r ' s graduate s t u d i e s . INTRODUCTION 1 A n i n v e r s e (or p r o j e c t i v e ) system o f spaces c o n s i s t s o f a c o l l e c t i o n o f spaces X ^ , i e l , where I i s d i r e c t e d (by <), and f u n c t i o n s p . . : X . - X . , d e f i n e d f o r a l l i , j e l w i t h i < j , such t h a t f o r i<j-<k, p i k = P ± j 0 P j k ' . The system i s c a l l e d an i n v e r s e system of measures i f , i n a d d i t i o n , t h e r e e x i s t s f o r e v e r y i e l a measure d e f i n e d on a a - f i e l d ( or a - r i n g ) <5\ o f s u b s e t s o f X ^ , such t h a t whenever i < j and A e c ^ , we have and p-]DA] e a. U j C P i j C A - ] ) = ^ ( A ) . Such systems are u s e d i n many a r e a s of m a t h e m a t i c s , f o r -. example i n problems c o n n e c t e d w i t h s t o c h a s t i c p r o c e s s e s , m a r t i n g a l e s , d i s i n t e g r a t i o n o f measures , e t c . One o f the f i r s t ( i m p l i c i t ) uses was made b y K o l m o g o r o f f [ 8 ] , t o o b t a i n p r o b a b i l i t i e s on i n f i n i t e C a r t e s i a n p r o d u c t s p a c e s . The concept was l a t e r s t u d i e d e x p l i c i t l y b y B o c h n e r , who c a l l e d such systems s t o c h a s t i c f a m i l i e s (see [ 2 ] ) . S i n c e t h e n , i n v e r s e systems o f measure spaces have been the s u b j e c t o f a number o f i n v e s t i g a t i o n s (see e . g . C h o k s i [4],- M e t e v i e r [10], Meyer En], R a o u l t [12], S c h e f f e r [13])... The f u n d a m e n t a l p r o b l e m i n a l l o f these i n v e s t i g a t i o n s i s t h a t o f f i n d i n g a " l i m i t " f o r an i n v e r s e system of measure spaces. . A l l p r e v i o u s workers i n t h i s f i e l d have c o n c e n t r a t e d on g e t t i n g an a p p r o p r i a t e ' l i m i t 1 measure on the i n v e r s e l i m i t se t L (see d e f i n i t i o n 1.2, C h . I I l ) . Such an a p p r o a c h p r e s e n t s some s e r i o u s d i f f i c u l t i e s , e . g . L may be empty. In t h i s p a p e r , we a v o i d dependence on L, and hence many of these d i f f i c u l t i e s , b y c o n s t r u c t i n g a • l i m i t ' measure {Ion the C a r t e s i a n p r o d u c t i f o f the X ^ ' s . As a r e s u l t , we are a b l e t o get e x i s t e n c e theorems w i t h c o n s i d e r a b l y fewer c o n d i t i o n s on the systems.- ( i n a f o r t h -coming paper [14], C.L. S c h e f f e r a l s o gets away f r o m dependence on L b y w o r k i n g on an a b s t r a c t r e p r e s e n t a t i o n s p a c e . H i s methods, however , seem t o be v e r y d i f f e r e n t f r o m o u r s . ) S i n c e L c X , we i n v e s t i g a t e the more s t a n d a r d i n v e r s e l i m i t measure f r o m the p o i n t o f v iew of r e s t r i c t i n g j l t o L. T h i s e n a b l e s us not o n l y t o ex tend known r e s u l t s but a l s o t o g i v e a b e t t e r i n d i c a t i o n 1 o f the r e a s o n s f o r some o f the d i f f i c u l t i e s c o n n e c t e d ' w i t h the s t a n d a r d i n v e r s e l i m i t measures . I n p a r t i c u l a r , we show (see example 6 i n the a p p e n d i x ) t h a t the p r o d u c t t o p o l o g y may be t o o f i n e f o r any c o n t i n u o u s l i m i t measure t o be Radon w i t h r e s p e c t t o i t . T h i s l e a d s us t o the -i n t r o d u c t i o n of another t o p o l o g y w h i c h seems t o be more a p p r o p r i a t e f o r problems i n t h i s a r e a . . C h a p t e r s I and I I p r e s e n t r e s p e c t i v e l y some o f the r e s u l t s i n s e t t h e o r y and measure t h e o r y w h i c h we r e q u i r e i n the l a t e r c h a p t e r s . In C h a p t e r I I I we i n t r o d u c e and d i s c u s s the concept 3 of i n v e r s e systems of measure spaces, and prove a fundamental e x i s t e n c e theorem f o r the l i m i t measure jj . In Chapter IV we examine the t o p o l o g i c a l p r o p e r t i e s of jj (and r e l a t e d measures; on the product space X. Chapter V i s devoted t o an i n v e s t i g a t i o n of the e x i s t e n c e and p r o p e r t i e s of i n v e r s e l i m i t measures on L obtained by r e s t r i c t i n g {j t o L. The appendix c o n s i s t s of examples, which i l l u s t r a t e p o i n t s made i n the t e x t . The p r o o f s f o r each chapter are c o l l e c t e d i n a separate s e c t i o n at the end of the chapter. CHAPTER I SET THEORETIC PRELIMINARIES In t h i s chapter we give some set t h e o r e t i c d e f i n i t i o n s and n o t a t i o n , and c o l l e c t r e s u l t s which w i l l be needed f o r f u t u r e r e f e r e n c e . We d i s c u s s f a m i l i e s of s e t s c l o s e d under b a s i c s e t o p e r a t i o n s , product spaces and f a m i l i e s w i t h a f i n i t e i n t e r s e c t i o n p r o p e r t y . These f a m i l i e s w i l l p l a y an important r o l e i n our measure theory. Our n o t a t i o n i s f a i r l y standard except f o r the p a r t which i s used f o r f a m i l i e s c l o s e d under s e t o p e r a t i o n s j t h a t p a r t i s m o d i f i e d so as t o i n d i c a t e the c a r d i n a l i t y of the s u b f a m i l i e s i n v o l v e d . 1. Set Operations 1.1 D e f i n i t i o n s and N o t a t i o n For any r e l a t i o n R . 1 R [A] = {y : f o r some x e A, (x,y) £ / ? } . _ .2 0 denotes the empty s e t . . 3 denotes the s e t of n a t u r a l numbers 0 , 1 , 2 . . . , and hence the f i r s t i n f i n i t e o r d i n a l . .4 &a i s the c a r d i n a l c o r r e s p o n d i n g t o the o r d i n a l a under the standard o r d e r i n g of i n f i n i t e c a r d i n a l s . • .5 Card A denotes the c a r d i n a l i t y of the s e t A. _ . 6 A ~ B = { x : x e A , x i B ) . .7 A A B = (A ~ B ) W ( B ~ A ) . For any f a m i l y U of s e t s , . 8 U U = U a , 5 9 nu= n a , 10 Ga(u) = {A : A=\jW, v'crV, Card < S^} (Note t h a t the n o t a t i o n V i s commonly used f o r the f a m i l y we c a l l o^{U).) 11 a(*0 = (A ': A F U M ' , M'e y) = UCo- (#") : a an o r d i n a l } 12 6 a(V) = {A : A=nv', y'crV, Card-?/' < <*a} (Note t h a t the n o t a t i o n ^ 6 i s commonly used f o r the • f a m i l y we c a l l 6 ^ (w).) 13 6(V) =.CA : A=ntf', ^'c^} = L K 6 (#0 : a an o r d i n a l } . When i t i s c l e a r from the context t h a t a l l the elements of M are b e i n g c o n s i d e r e d as subsets of a space X, we l e t : 14 e(?/) = (A : A<=X and X ~ A e #},. 15 7 U = {A : A c x and A A H e U f o r a l l H e U], Note t h a t f o r any f a m i l y U, U U i s the s m a l l e s t space X w i t h t h e . p r o p e r t y t h a t a l l the elements of U are subsets of X. Thus, u n l e s s the space X i s . s p e c i f i e d i n advance we s h a l l use U U f o r X. The c l a s s e s 7^, J>^ w i l l p l a y r o l e s s i m i l a r t o those p l a y e d by c l o s e d s e t s and open s e t s r e s p e c t i v e l y i n t o p o l o g i c a l spaces (see sec. 3). We now c o l l e c t a few elementary lemmas f o r f u t u r e use. 6 1.2 Lemma. For any f a m i l y of se t s ^ and o r d i n a l a , .2 a'(a(*0) = aCtfh • 5 S a ( 6 a ( ^ ) ) = 6 a ( v ) , " .4 6(6(«r)) = 6(V). 1.3 Lemma. For any f a m i l y of se t s flf and o r d i n a l a , .2 aQ( 6(^)) <= 6(a 0(^)) J .3 V ^ W ) cr o - a ( , 6 0 ( ^ ) ) , .4 6 Q ( a ( y ) ) c= a ( 6 Q ( ^ ) ) . 1.4 Remark. The i n c l u s i o n s i n lemma 1.3 fliay be s t r i c t as can be seen by t a k i n g f o r W the s e t of compact i n t e r v a l s of the r e a l l i n e . Then b^M) = -U hence a Q ( 6 1 ( ^ ) ) = GQ(U) ., However 6 1 ( a Q ( y ) ) c o n t a i n s other s e t s , f o r example the Cantor s e t . 1.5 Lemma. For any f a m i l y of s e t s M and o r d i n a l a , .1 6 a(a QW) = a 0 ( 6 a ( a 0 ( V ) ) ) , .2 6(cr 0(y)) = CTQUUQM)), " .4 a(6 Q(v)) = 6 0 ( a ( 6 0 ( y ) ) ) . 1.6 Lemma. Let be a f a m i l y of se t s such t h a t f o r every A, B e y, A A B e y and there e x i s t s a f i n i t e d i s j o i n t f a m i l y w i t h A ~ B = U j . Then 7 .1 f o r each D e ° Q ( ^ 0 there, e x i s t s a f i n i t e d i s j o i n t f a m i l y 7 CU such t h a t D = U J. .2 ° Q ( ^ ) I S a r i n g . 1.7 Lemma. I f M i s a f a m i l y of subsets of a space X and f o r some o r d i n a l a, %C = 6 ( * 0 > then .1 c{U) = a a ( c ( y ) ) , 1.8 Lemma. I f V i s a f a m i l y of subsets of a space X and f o r some o r d i n a l a, ft" = a (v), then . 1 c(v) = • 6 a(c(v), 1 1.9 Lemma. I f V i s a f a m i l y of subsets of a space X and a i s an o r d i n a l then . 1 ^V <c .2 c • 3 c-.4 •V c • \ ( * ) ' 8 2 . C a r t e s i a n P r o d u c t s 2.1 D e f i n i t i o n s . .1 ( X , l ) i s a system o f spaces i f f t o e v e r y i e l t h e r e c o r r e s p o n d s a space X^, i . e . X i s a s e t v a l u e d f u n c t i o n on I . G i v e n such a system ( X , l ) : .2 II X. = (x : x i s a f u n c t i o n on I such t h a t f o r e v e r y i e l i e l 1 . x . e X . } . . 3 F o r e v e r y i e l , rr^ denotes t h e c a n o n i c a l p r o j e c t i o n onto X., i . e . n i s t h e f u n c t i o n on II X. such t h a t f o r e v e r y 1 1 i e l 1 x e II X., TT. (x) = x. . , i e l 1 1 ± . 4 F o r e v e r y a c n X., i e l 1 J a = { i e l : TTj.r.a] / X ± } . . .5 y i s a system o f f a m i l i e s o f s e t s w . r . t . ( X , l ) i f f ^  i s a f u n c t i o n on I such t h a t , f o r e v e r y i e l , ft\ i s a f a m i l y o f s u b s e t s o f X^. F o r such a system, %C : .6 C y l ( ^ ) = {a : f o r some i e l and some Aeft\, a = TT~ 1 [A]}. .7 R e c t U ) = 6 0 ( C y l ( y ) ) . Note t h a t R e c t ( ^ ) agrees' w i t h t h e u s u a l d e f i n i t i o n o f r e c t a n g l e s f r o m f a m i l i e s i f f f o r . e v e r y i ' e l , i s c l o s e d under f i n i t e i n t e r s e c t i o n s . We now c o l l e c t some- e l e m e n t a r y lemmas f o r f u t u r e • r e f e r e n c e . 9 2.2 Lemma. Let ( X , l ) be a system of spaces and f o r each i e l , l e t 0 / A. <= X;.. I f a = n A. then . 1 x- i e l 1 • . 1 TT^ [a] = A^ f o r ev e r y i e l and .2 a = n I ^ 1 [ i ^ E a ] ] : i e l ) 2 . 3 Lemma. Let % be a system of f a m i l i e s of s e t s w.r.t. ( X , l ) , and l e t 0 / a e R e c t ( ^ ) . Then .1 J i s f i n i t e . a .2 a = f U t>^ ETT. [a]] : j e J 3. U J U " . 3 I f 0 e R e c t ( v ) and a ^  3 = 0 then, f o r some j e J ^ n J g , TT, Ca ] ^ T T [f3] = 0. . 2.4 Lemma. L e t V be a system of f a m i l i e s of se t s w.r.t. ( X , l ) . Let 0 / a Rect(V) and T = U C J a : aec7]. Then f o r any xe Uc7, ' n [ T T - 1 ECTT ( X ) } ] « a jeT J J f o r some aec7. 2 . 5 Lemma. Let V be a system of f a m i l i e s of s e t s w.r.t. (X , l ) such t h a t f o r ev e r y i e l , = c(u^). Then .1 f o r ev e r y a, $eRect(&") there e x i s t s a f i n i t e d i s j o i n t f a m i l y ^ c R e c t ( v ) such t h a t a~3=Uj. .2 o-Q(Rect (&0 ) i s a r i n g . . 3 f o r every O £ O Q (Rect(ft ')) there e x i s t s a f i n i t e d i s j o i n t • f a m i l y J c Rect(^) w i t h a = U J. 10 3. ft-compact C l a s s e s . Here we d e f i n e c e r t a i n f a m i l i e s of se t s which resemble the f a m i l y of. c l o s e d compact se t s i n t o p o l o g i c a l spaces, and . study t h e i r p r o p e r t i e s . They w i l l l a t e r be used i n extending f i n i t e l y a d d i t i v e set f u n c t i o n s t o measures. These f a m i l i e s (those we c a l l ^-compact) were f i r s t used f o r such purposes by Marczewski [ 9 ] and have been used by many workers s i n c e (Choksi. [ 4 ] , M e t e v i e r [jo], Meyer [11] e t c . ) . We w i l l a l s o c o n s i d e r f a m i l i e s which w i l l a c t as c l o s e d s e t s and open s e t s . These are obtained i n the same way as c l o s e d and open s e t s i n forming k-spaces (see K e l l e y [ 7 ] ) . 3.1 D e f i n i t i o n s . . For any f a m i l y of s e t s <3 and i n f i n i t e c a r d i n a l K : .1 C- i s ft-compact i f f 0e<3 and f o r ev e r y Q)i*=- C- w i t h Card C-'^ ^  and r i d ' = 0 there e x i s t s a f i n i t e sub-f a m i l y ^ c c . ' such t h a t HJ = 0. .2 An ft-covering of a set A by elements of (3 i s a sub-f a m i l y &'<: & such t h a t Card o'^ K and A c U c ' . 3.2 Remark. The f a m i l y of c l o s e d compact s e t s i n a t o p o l o g -i c a l space i s ft-compact f o r every ft Furthermore a f a m i l y which i s ft-compact f o r every K may be c o n s i d e r e d as a f a m i l y of c l o s e d compact se t s i n an a p p r o p r i a t e t o p o l o g y (see lemma 3-7 below). We now c o n s i d e r f a m i l i e s r e s u l t i n g from v a r i o u s . op e r a t i o n s on ft-compact s e t s and whether these f a m i l i e s 11 are themselves K-compact. 3 . 3 Lemma. For any o r d i n a l a, i f (3 i s N -compact then 6 a +^(C-) i s N -compact. 3.4 Lemma. Let C-be K-compact. Then: . 1 every s u b f a m i l y of <3 i s ^-compact. .2 f o r ev e r y Ae<3, C A A C : Ce<3} i s K-compact, . 3 ^(c-) i s ^-compact. Note t h a t o^{c) i s not i n g e n e r a l ^-compact as t h i s i s not. t r u e f o r compact s e t s i n the r e a l l i n e . 3.5 Lemma. Let (X , l ) be a system of spaces, C-be a system of f a m i l i e s of s e t s w . r . t . ( X , l ) such t h a t f o r each i e l , i s K-compact, and l e t . $ = (A : A = E C , f o r some f u n c t i o n C on I i e l 1 w i t h C\e f o r ev e r y i e l } , 3- = '{A : A = n F. f o r some f u n c t i o n F on I i e l 1 w i t h F. e 7 f o r each i e l } . 1 & i Then ' .1 Rect(c-) i s K-compact, .2 <$ i s ^-compact, . 3 * <=^ . We next check some c o v e r i n g p r o p e r t i e s which p a r a l l e l the t o p o l o g i c a l case. 3.6 Lemma. Let c3 be an ft-compact f a m i l y o f subsets of a space X. Then . 1 every ft-covering of X by elements of c_(<3) can be . reduced t o a f i n i t e c o v e r i n g . .2 i f C e <3, then every ft-covering of c by elements of J' can be reduced t o a f i n i t e s u b -covering. The next lemma shows t h a t i f we have a f a m i l y which i s ft-compact f o r every N, we may c o n s i d e r i t as a f a m i l y of c l o s e d compacts. 3.7 Lemma. Let (3 be a f a m i l y of subsets of a space X, c3 be ft-compact f o r every ft and (3= 6(<3) = QQ (C3). Then i s a to p o l o g y i n which every element of <3 i s a c l o s e d compact s e t . 4. Proofs Proof of 1.2 Immediate from the d e f i n i t i o n s and the f a c t t h a t Card A < N and Card B < K i m p l i e s Card AxB<& whenever N i s an i n f i n i t e c a r d i n a l . Proof of 1.3. 1 Let A, B e 6 (#). Then f o r some I , J , w i t h Card I <K , Card J<K , we can w r i t e a a A = PI A.,'. B = fl B . , i e l 1 j e J J where A.e M f o r e v e r y i e l and B .e U f o r e v e r y j e J . Then A ^ B = n A . A O B . = n n ( A , A B . ) = n ( A . ^ B . ) i e l 1 j e J J i e l j e J 1 J ( i , j ) e I x J 1 J Since Card IxJ<K , A o B e 6 a ( c r 0 ( v ) ) . S i m i l a r l y i f Card I. < ^ and it CX . A, = n A ( i , k ) e 6 (V) f o r k=0, 1. . .n, and ,-we l e t S = I Q X * « * X I , then n n n UA, = U 0 A ( i , k ) = n U A ( j F k ) e 6 ( a n ( V ) ) , k=0 * k=0 i e l jeS k=0 K a -s i n c e Card S < K . a Proof of 1 . 3 . 2 » 1 . 3 . 3 , 1 . 3 .4 S i m i l a r t o above. Proof of 1.5.1 C l e a r l y S a ( a 0 W ) <=• a r 0 ( 6 a ( a 0 ( y ) ) ) . By lemma 1 .3 . 1, f o r any f a m i l y «£, CT0(6aU)) C 6 a ( c r 0 ( ^ ^ so t h a t by l e t t i n g J'.= a Q ( v ) , tf0Ua(°0(^))) c 6 a ( a 0 ( a 0 ( ^ ) ) ) = _ 6 a ( c J 0 ( V ) ) . ~ Proof of 1 . 5 . 2 , 1 . 5 . 3 , 1 . 5 . 4 S i m i l a r t o above. Proof of 1.6.1 Let D e a Q ( ^ ) . Then f o r some new n D = u D , m=0 ffi where D me %c f or m=0, 1 n. .. -I f n=0, 1.6.1 i s c l e a r l y t r u e . Suppose i t h o l d s f o r n-k -1 and l e t n=k. Then k -1 U D = U/? m=0 m 14 f o r some f i n i t e d i s j o i n t f a m i l y Be U. Thus •D = D K . v U B = D k \j U { B ^ D K : Be/?}, and these s e t s are d i s j o i n t . Since BcU and D^e U, f o r every BeB, B ~ \ = U ^ where 3 ^ i s a f i n i t e d i s j o i n t s u b f a m i l y of M. Thus D = D k v U {L : Le=fg f o r some BeB], and these s e t s form a f i n i t e d i s j o i n t s u b f a m i l y of U. Proof of 1.6.2 C T Q ( ^ ) i s c l e a r l y c l o s e d under f i n i t e unions and i n t e r s e c t i o n s . Let A, BeoQ(u). Then t h e r e e x i s t f i n i t e d i s j o i n t f a m i l i e s fi, 3 s u c h t h a t A =UA B =U^. Then A - B = U^~U^= U ( D ~ U J") = U n (D ~ F ) . For D, Fe#", D ~ F e r J Q ( y ) , so t h a t f o r ev e r y De^, n (D ~ F) e cr n (y) F e J u and thus A ~ B = U n (D ~ F) e a n ( y ) . De£ F e J U Hence on(u) i s a r i n g . Proof of 1 . 7 . 1 Immediate from De Morgan's Rules, Proof of 1 . 7 . 2 Let Fe 6 ( 7 , ) . Then f o r some I w i t h Card a # F = n F . i e l 1 where F.e 7 „ , f o r every i e l . I f KeU} then F n H = fl (P.^H) e 6 (tf) = K, i e l 1 a hence FeJ"^. Proof of 1 . 7 . 3 Immediate from 1 . 7 . 2 and 1 . 7 . 1 . Proof of 1 . 8 . 1 S i m i l a r t o 1 . 7 . 1 . Proof of 1 . 8 . 2 S i m i l a r t o 1 . 7 . 2 . Proof of 1 . 8 . 3 Immediate from 1 . 8 . 2 and 1 . 8 . 3 . Proof of 1 . 9 . 1 Let Aea {u) and Fe7.., then there e x i s t s a ft ^ ' c y w i t h Card %C*^ K such t h a t A = Uu'. Hence a P A A = F A U H = U F A H. EeU Rett Since F A H e V f o r every HeM', F A A = UPrxHeo ( y ) . He^ a Proof of 1 . 9 . 2 S i m i l a r t o the pr o o f of 1 . 9 . 1 . Proof of 1 . 9 . 3 , 1 . 9 . 4 Since = 1 . 9 - 3 and 1 . 9 . 4 f o l l o w from 1 . 9 . 1 and 1 . 9 . 2 r e s p e c t i v e l y . . Proof of 2.2 immediate from the d e f i n i t i o n . Proof of 2.3. 1 Let a = flH ^ where H me Cyl(V) f o r m^n, and l e t J = u J R . Then J i s f i n i t e s i n c e i s f i n i t e m=0 m m^ f o r each m^n (hy d e f i n i t i o n of Cyl(ftO). C l e a r l y J c hence J i s f i n i t e . J a a Proof of 2.3.2 With H^, m=0j1...n as above, f o r every i e l l e t Then a = II B. and the r e s u l t f o l l o w s from 2.2.2. i e l 1 Proof of 2.3.3 Suppose no such j e x i s t s . Then f o r every i e l choose x^e [a] A TT.. If? ]. Then {x} c n TT71 [{x,}] <= n TT-:1 [rr, [a]] i e l x x j e J ' 3 J ° a hence xea. S i m i l a r l y xef?, so t h a t a n 3 ^  0. Proof of 2.4 Let x e U a. Then there e x i s t s aec7-with xea. Then n T T " 1 [TT ( X) ] c n T T " 1 [ T T . ( X ) ] c a . jeT J J j e J ^ J J Proof of 2.5.1 For a, PeRect(^), l e t new and — ( J Q , J / | , . . « j } • Let -1 B = TT" LX, ~ TT [3]], and f o r 0 <• m ^ n l e t 1 m-1 B = TT^'LX. ~ TT , [ p ] ] n 0 TT ' [TT [0]]. m Jm Jm Jm 1=0 J l J l Then a ~ 3 = U ( a n B ) m=0 • m and t h e s e s e t s a r e d i s j o i n t s i n c e B^n B^ = J2f, lf£k. . P r o o f of 2 . 5 . 2 , 2 . 5 . 3 By 2 . 5 . 1 , V s a t i s f i e s the h y p o t h e s e s of lemma 1.6, hence 2 . 5 . 2 and 2 . 5 . 3 h o l d . P r o o f of 3 . 3 L e t 67 c 6 ,,(<3) w i t h Card c7 £ c< . F o r each —- a+1 a Ae#, t h e r e e x i s t s B cc, such t h a t Card /?. ^ ^ and A A a A = n 13 L e t = U £. . A e c 7 A Then C.:'<=<3, Card c-' £ N • K = N , a a a ' and = Wa. Thus i f T\<7 = 0, t h e n f i s ' = 0 and t h e r e e x i s t s a f i n i t e f a m i l y 3'c e / such t h a t f l = £f. F o r e v e r y Fe.7 t h e r e e x i s t s A„e a w i t h A„«:F so t h a t D A p c H J = 0. * _ FeJ P r o o f of 3 , 4 . 1 Immediate from t h e d e f i n i t i o n s . P r o o f of 3 . 4 . 2 Immediate from the d e f i n i t i o n s . P r o o f o f 3 . 4 . 3 (A s i m i l a r f a c t i s p r o v e n i n t h e same way by Meyer [11].) L e t <7<=oQ(c>) be such t h a t Card (Z ^ and f o r any f i n i t e f a m i l y <z'<=- CI, Y\dl ^ 0. F o r each Aec7 l e t A = U 13^ where 13^ i s a f i n i t e f a m i l y in-(3. L e t U be an u l t r a f i l t e r such t h a t # c U. Then f o r e v e r y Aec7 t h e r e e x i s t B A £ S U C H T H A T B A £ U * T I L E N (B A : Aec7} c C a r c l { B A : Aec7} ^ ^ a , and f o r any f i n i t e f a m i l y <7c{B A: Aec7}. , - J ^ 0 . Hence pf ^  T1{B A: Aec7} c R a . We prove lemma 3 . 5 i n the order 3 . 5 . 2 , 3 . 5 . 1 , 3 . 5 . 3 . Proof of 3 . 5 . 2 Let C7c<S, Card c7 ^  K and P I #V 0 f o r any f i n i t e s u b f a m i l y G 7 ' ' C C7. For every i e l , l e t B , = 0 TT L A ] . x Aetf 1Then ^ 0 f o r any i e l , s i n c e otherwise f o r some A Q , A ^ , . . » A e 67, m=0 ft ^ so t h a t fl A = 0. Thus m=0 m j2f ^ n B. = na. i e i Proof of 3 . 5 . 1 For each i e l l e t = C± v {X.}. Then Ci/ i s compact and R e c t ( c ) c { n C. : C. e a / f o r every i e l } , i e l 1 1 1 which i s ^-compact by 3 . 5 - 2 . Hence by 3 . 4 . 1 Rect(<3) i s ^-compact. 1 9 Proof of 3.5.3 Let Ee<$, De£ and E = il C. where C '.e a. i e l f o r e very i e l , and D = HP. where F. e 3„ f o r every i e l . i e l 1 1 i Then E A D = II (C.n P. ), i e l and s i n c e C. A P . S <3. f o r every i e l , EnDe#, hence De«7„ . 1 !L 3_ (3 Proof of 3 . 6 . 1 Let <2 be an ft-covering of X by elements of c_(<3). Then' {X ~ A : Aet7] <= &, Card[X ~ A : Aec7] £ N , and Il {X ~ A : Aec7] = 0 . Thus there e x i s t s a f i n i t e f a m i l y J c {x ~ A : Aed] w i t h f l 3 = 0, hence the complements of the elements of 7 form .a f i n i t e s u b f a m i l y of G and cover X. Proof of 3 . 6 . 2 Let 61 be an ft-covering of C by elements of Then {C A (X ~ A) : Aec7] <= <3, Card{C A (X ~ A ) : Aec7] ^ N and f l {C A (X ~ A) : Aec7] = 0. Hence there e x i s t s a f i n i t e s u b f a m i l y ^ of {C A (X ~ A ) : Aea] such t h a t V\7 = 0, so t h a t i f f o r each Fe.7 we choose A„e a such t h a t P = C A ( X ~ A F ) , then C <= U A „ Fe7 F 2 0 Proof of 3 . 7 By lemmas 1 . 7 . 3 and 1 . 8 . 3 , i s c l o s e d under a r b i t r a r y unions and f i n i t e i n t e r s e c t i o n s . Since 0 e C- we have 0 e hence XeJ? and thus 1 & i s a topology. For every Ce£% we have C e ^ j hence X ~ CeJ^. By lemma 3 . 6 . 2 every c o v e r i n g of C by elements of can be reduced to a f i n i t e c o v e r i n g . Thus f o r every Ce<3, C i s c l o s e d and compact. 2 1 C H A P T E R I I M E A S U R E T H E O R E T I C P R E L I M I N A R I E S I n t h i s c h a p t e r w e d e v e l o p t h e m e a s u r e t h e o r e t i c r e s u l t s w h i c h w e w i l l u s e i n o u r s t u d y o f i n v e r s e l i m i t s o f m e a s u r e s . T h e s e c o n s i s t o f e x t e n s i o n t h e o r e m s f o r f i n i t e l y a d d i t i v e a n d f i n i t e l y s u b a d d i t i v e s e t f u n c t i o n s , a n d t h e o r e m s c o n c e r n i n g c o n s t r u c t i o n o f R a d o n a n d s i m i l a r m e a s u r e s . We w i l l u s e C a r a t h e o d o r y m e a s u r e s t h r o u g h o u t a n d w e w i l l r e l y h e a v i l y o n t h e s t a n d a r d C a r a t h e o d o r y e x t e n s i o n t h e o r e m ( t h e o r e m 1.3). 1 c C a r a t h e o d o r y M e a s u r e s , . 1 . 1 D e f i n i t i o n s . .1 u i s a C a r a t h e o d o r y m e a s u r e • o n X I f f u i s r a f u n c t i o n o n t h e f a m i l y o f s u b s e t s o f X s u c h t h a t \d(0) = 0 a n d 0 s ^ A ) < 2 U(B ) £ oo n e LU w h e n e v e r A c u B . . 2 A I s ^ - m e a s u r a b l e I f f | j i s a C a r a t h e o d o r y m e a s u r e o n a s p a c e X , AaX, a n d f o r e v e r y T e x , | i ( T ) = n ( T z i A ) + U ( T ~ A ) . . 3 %l = (A : A. i s l a - m e a s u r a b l e } . . 4 A C a r a t h e o d o r y m e a s u r e (a o n a s p a c e X I s c a r r i e d b y A c X i f f |j(x ~ A ) = 0 . .5 A C a r a t h e o d o r y m e a s u r e |_i o n a s p a c e X I s p s e u d o - c a r r i e d b y A « = X i f f IJ(B) = 0 w h e n e v e r B e W a n d B e x ~ A . 2 2 .6 I f (a i s a Caratheodory measure on X, the r e s t r i c t i o n of [i t o A c X ; a | A, i s the measure v on X d e f i n e d toy v ( B ) - n ( B n A ) f o r a l l B c X . .7 (a i s an outer measure on X i f f a i s a Caratheodory measure on X and f o r every A <= x there e x i s t s B c X such t h a t Be^ , A c B and I^(A) = U(B). . 8 [j i s the Caratheodory measure on X generated toy g and U i f f V i s a f a m i l y of subsets of X, g(H) £ 0 f o r every EeW, and f o r every A<=X, n(A) - i n f { Z . S ( H ) : af'<= CardO')^ UL., and He?/' u ACUV'} . 9 [j i s a s e m i f i n i t e outer measure on X i f f u. i s an outer measure on X and f o r every A c X |i(A)=sup{|j(B) 1 B c A j J J C B ) < ~ } . 1 . 2 Remark. I f M i s c l o s e d under countable unions, g i s countable s u b a d d i t i v e on U and \j. i s the Caratheodory measure generated toy g and U, then f o r every A«= X, ia(A) . = inf{g(H) 2 EeU, A c H ] . We now c o n s i d e r what p r o p e r t i e s a measure generated as In d e f i n i t i o n 1 . 1 . 8 must have. These are c o n t a i n e d i n the f o l l o w i n g w e l l known theorem which i s e s s e n t i a l l y due t o Caratheodory. 1 . 3 Theorem. For any non-negative set f u n c t i o n g and f a m i l y U of subsets of a space X, the Caratheodory measure u, 23 generated by g and V has the f o l l o w i n g p r o p e r t i e s . .1 (a i s a Caratheodory measure, .2 i f A<=A 7 f o r some A''e o^U) ( i n p a r t i c u l a r i f H ( A ) < °°), then f o r ev e r y £ > 0 there e x i s t s Bea^ft") such t h a t A C B and H ( B ) ^ U ( A ) +e and hence there e x i s t s B:'e 6^(a ( V ) ) such t h a t A C B - ' and |_i(A) = ^ ' ( B ) . . 3 i f ^  i s a r i n g and g i s f i n i t e l y a d d i t i v e on M then V c t f ^ . (Hence i n view of 1 .3 .2 , [i i s an outer measure. .4 i f g i s c o u n t a b l y s u b a d d i t i v e on U, then.fa(A) = g(A) f o r e v e r y AeV. 1.4 Remark. Let p. be a Caratheodory measure (or a measure on a a - f i e l d c7) and l e t v be the outer measure generated by |a and 77^  (or d). Then by theorem 1.3, v agrees w i t h -l_t on ^ (or d) and. ( c 7 c ^ ) . For t h i s reason we s h a l l c o n c e n t r a t e on outer measures throughout. The f o l l o w i n g lemmas w i l l be u s e f u l f o r extendi n g set f u n c t i o n s t o r i n g s . 1.5 Lemma. Let c7 be a c l a s s of s e t s such t h a t G = 6Q(C7) and whenever a, 3ec7 there e x i s t s a f i n i t e d i s j o i n t f a m i l y .y 3 c-a w i t h a ~ f3 = U 7. I f h i s a non-negative and 24 f i n i t e l y a d d i t i v e s e t f u n c t i o n on CI, then h can be u n i q u e l y extended t o a non-negative and f i n i t e l y a d d i t i v e set f u n c t i o n h* on the r i n g 13 generated by (2. ' 1 . 6 C o r o l l a r y . Let ( X , l ) be a system of spaces and l e t %C be a system of f a m i l i e s of s e t s w . r . t . ( X , l ) such t h a t f o r e v e r y i e l , i s a a - f i e l d . I f g i s a non-negative f i n i t e l y a d d i t i v e s e t f u n c t i o n on R e c t ( y ) , then g can be u n i q u e l y extended t o a non-negative f i n i t e l y a d d i t i v e set f u n c t i o n g* on a Q ( R e c t ( ^ ) ) . 2. Approximation and G e n e r a t i o n of Measures. In t h i s s e c t i o n we. are concerned w i t h s e t f u n c t i o n s which can be approximated from below by elements of ^-compact f a m i l i e s . We a p p l y theorem 1 . 3 » t o such s e t f u n c t i o n s t o o b t a i n an e x t e n s i o n theorem s i m i l a r t o t h a t of Marczewski [ 9 ] , i n which" we i n c l u d e ' t h e n o n - a - f i n i t e case. We then t u r n t o the c o n s t r u c t i o n "of Radon-like measures, g e n e r a l i z i n g the d e f i n i t i o n of Rad6n measures t o the non-t o p o l o g i c a l case by u s i n g the f a m i l i e s V and as c l o s e d , and open f a m i l i e s r e s p e c t i v e l y , i n analogy w i t h the c o n s t r u c -t i o n of k-spaces (see e.g. K e l l e y [7]). The p a r t i c u l a r forms of our theorems on c o n s t r u c t i o n of Radon and Radon-like measures are motivated by the f a c t t h a t we w i l l a p p l y them to product spaces. There the s t r o n g "7 c o n d i t i o n s on the outer measures are q u i t e n a t u r a l l y s a t i s f i e d and we can thus dispense w i t h the u s u a l t o p o l o g i c a l requirement ( r e g u l a r i t y ) ' . 2 5 2 . 1 D e f i n i t i o n s . Let g be a non-negative s e t f u n c t i o n on a f a m i l y V of subsets of a space X. • . 1 C- i s an i n n e r f a m i l y f o r g on W i f f ccU and when Rett . 2 C i s an i n n e r f a m i l y f o r u i f f M- i s an outer measure on X- and <3 i s an i n n e r f a m i l y f o r u on 7h . .3 ^ i s an outer f a m i l y f o r g on V i f f 1 C V and f o r e v e r y ReK, g(H) = i n f { g ( G ) : Ge£, H<=G}. 4 . 4 i i s an outer f a m i l y f o r \i i f f u i s an outer measure on X and & i s an outer f a m i l y f o r u on 7f[ (and hence a l s o on the f a m i l y of a l l subsets of X ) . „ 5 [a i s Radon-like w.r.t. ( X, «£) i f f [a i s an outer measure on X, N i s an i n f i n i t e c a r d i n a l , and the f o l l o w i n g c o n d i t i o n s are s a t i s f i e d . , " .5.1 e = CTq(&) = 6^(3), . 5 . 2 C- i s an ^-compact f a m i l y of subsets of X, ' • . 5 . 4 J< = S Q U ) = 0 ^ ^ ) . . 5 . 5 (3 i s an i n n e r f a m i l y f o r |i on i , . 5 . 6 J' i s an outer f a m i l y f o r |j, . 5 . 7 ia(c) < co f o r e v e r y Cec3. I f <3 Is ^-compact f o r every c a r d i n a l we s h a l l say |j i s Radon-like w.r.t. (<3, JO • I f i n f a c t i i s a t o p o l o g y and (3 i s the f a m i l y of c l o s e d compact s e t s -i n the t o p o l o g y & then the above d e f i n i t i o n reduces t o g(H) = sup{g(c) : C<=H, Cec3}. 26 t h a t of a Radon outer measure and we s h a l l say [i i s Radon w.r.t.J', or simply t h a t u i s Radon i f the to p o l o g y i s c l e a r from the context. We f i r s t note c o n d i t i o n s under which the f a m i l i e s 3 , c o n s i s t of measurable s e t s . G-be an i n n e r f a m i l y f o r a. Then J^cT^ (hence ^^W^) • The next two theorems show the r o l e which ^-compact i n n e r f a m i l i e s p l a y i n the e x t e n s i o n of s e t f u n c t i o n s . The f i r s t theorem shows t h a t w i t h such a f a m i l y , f i n i t e s u b a d d i t i -v i t y guarantees countable s u b a d d i t i v i t y ; the second theorem uses t h i s and the f a c t t h a t f i n i t e a d d i t i v i t y y i e l d s measur-a b i l i t y t o show t h a t the Caratheodory e x t e n s i o n i s a measure w i t h s u i t a b l e approximation p r o p e r t i e s . T h i s w i l l a l l o w us t o e s t a b l i s h the e x i s t e n c e and some approximation p r o p e r t i e s -of i n v e r s e l i m i t measures. . 2.3 Theorem. Let X be a space and a r i n g of subsets of X/ and l e t g be a non-negative f i n i t e l y s u b a d d i t i v e s et f u n c t i o n on ft". I f there e x i s t s an ^-compact f a m i l y & which i s an i n n e r f a m i l y f o r g on U, then g i s co u n t a b l y s u b a d d i t i v e on U. ' 2.4 Theorem. Let g be a non-negative f i n i t e l y a d d i t i v e s et f u n c t i o n on a r i n g %C of subsets of a space X-. I f there e x i s t s an ^-compact s u b f a m i l y & c y which i s an i n n e r f a m i l y f o r g on .then the outer measure u. on X '.. 2.2 Lemma. Let u he an outer measure on a space X and l e t 27 generated by g and & has the f o l l o w i n g p r o p e r t i e s : . 1 ja i s an e x t e n s i o n of g, i . e . f o r a l l EeU, u.(H.)=g(H), n (A)=sup{u(c) : C e ^ G ) , C « = A } . • , In our work on product spaces, the c l a s s <3, which we use to prove the e x i s t e n c e of a l i m i t measure, i s not i n g e n e r a l a c o l l e c t i o n of c l o s e d compact se t s i n the product topology. The q u e s t i o n then a r i s e s as t o whether we can i n some way a d j u s t our measure to form a Radon measure w.r.t. the product topology. To show t h a t under c e r t a i n c o n d i t i o n s we can do t h i s , we prove the f o l l o w i n g theorems. They do not f o l l o w the u s u a l e x i s t e n c e p r o o f f o r Radon measures, which uses r e g u l a r i t y , but omit any t o p o l o g i c a l c o n d i t i o n s by u s i n g a p r o p e r t y which .arises n a t u r a l l y i n our product space measures..-2.5 Theorem. Let Y be a t o p o l o g i c a l space w i t h t o p o l o g y 3. Let X = ( A : A i s closed-and compact}, and l e t 13 be a base f o r J>' such t h a t 13 = aQ(B) .= 6 0 (/ ? ) . Let v be an outer measure on Y such t h a t v i s f i n i t e l y a d d i t i v e on X, f i n i t e , on X ^ 13, and such t h a t •'< f o r e very Be/? v(B)=sup{v(F) : F e B , F c l o s e d , Fefl? } I f we l e t h(G)=supC v(K) : YLeX, KcG} 28 f o r every GeJ-, and n(A ) = i n f {h(C-) : GeJ, A c G} f o r e very A e y , then: .1 [i i s a Radon outer measure, .2 \i(G) = h(G) f o r every GeA • 3 H ( K ) ^ v(K) f o r ev e r y KeK. The f o l l o w i n g theorem i s a g e n e r a l i z a t i o n of the above one t o the n o n - t o p o l o g i c a l case. The f a m i l y w i l l a c t as a base. 2 . 6 Theorem. Let X be a space and C- an K^-compact f a m i l y of subsets of X such t h a t 0e& and C = b^a) = a Q(c3). Let v be an outer measure on X which i s f i n i t e l y a d d i t i v e on'C-and f i n i t e on fluJ-, where J& = a Q U ) = 6 0 U ) c ^ , c aa + 1 U ) and f o r every Dei>, v(D)=sup{ v(F) : F c D , Fe ( ^ n«%'.)} . I f we now l e t h(G)=sup{v ( c ) : Cec3, C CG] ' ~ f o r every G s a ^ ^ i ) , and ji(A)=inf Ch(G-) : G e a ^ U ) , A C Q } , ;" f o r e very A c X, then 29 . 1 u i s an outer measure on X which i s Radon-like w.r.t. ( ^ C ; a a + 1 j U ) ) , .2 |a(G)=h(G) f o r e v e r y G e a ^ ^ ) , .3 v(c) ^ ji(c) f o r every CeC-. Since many approximation theorems r e q u i r e t h a t the measure be s e m i f i n i t e , we conclude t h i s s e c t i o n by I n d i c a t i n g a way of amending a measure t o make i t s e m i f i n i t e . 2.7 Theorem. Let v be an outer measure on a space X. I f f o r e very A ^ X we l e t v ' ( A ) = s u p { v ( B ) : -B c A and V ( B ) < «>} then: • .1 v' i s a s e m i f i n i t e outer measure, •2 v = v . \ ' 3. P r o o f s . Proof of 1.3 See S i o n [15] f o r .1, .2, .3. Since we always have \I(E) ^ g(H) f o r HeV; 1.3.4 f o l l o w s immediately. Proof of 1.5 By lemma. 1.6 Ch.I, every Be/3 can be w r i t t e n as U 7 where 3 Is a f i n i t e d i s j o i n t s u b f a m i l y Of G. Let h*(B) = 2 h ( F ) . F e 7 I f a l s o B= \JJ* where ^ i s another f i n i t e d i s j o i n t . s u b f a m i l y of 67, then 2 h.(F) = 2 ( 2 h ( F ^ G ) ) = 2 h(G), FeJ FeJ GeJ- Ge>£ -30 so t h a t h* i s w e l l d e f i n e d and c l e a r l y unique. I f A, B&I3, As\B=fi and A= UV, B= U 3 where M, 3 are f i n i t e d i s j o i n t s u b f a m i l i e s of then i s f i n i t e and d i s j o i n t , so t h a t h * ( A w B ) = 2 h(P) = 2 h(F) + 2 h(H) = h*(B) + h*(A) Hence h* i s c l e a r l y f i n i t e l y a d d i t i v e on 13. Proof of 1.6 By lemma 2.5 Ch.I, a ( R e c t ( ^ ) ) s a t i s f i e s the hypotheses of lemma 1.5. Proof of 2.2 To show t h a t a set i s ^ m e a s u r a b l e • i t i s s u f f i c i e n t t o show t h a t f o r e v e r y . T c X w i t h |a(T) < °°, ^ T ) = U ( T A P ) + | j ( T ~ F ) . Let B e ^ , T c B and U ( T ) = U ( B ) . Given £ > 0 , s i n c e c i s an i n n e r f a m i l y f o r \x, there e x i s t s CeC- such t h a t C «= B and u(B) < ^(c) + £ . Then F ^ C e C - c ^ - s o t h a t n ( c ) = ^ C / » F ) -1- | i ( C ~ F ) ' ' •• and t h e r e f o r e H ( T ) * u ( T A P) + n ( T ~ P) * [ i ( B r t F ) + U ( B ~ P) * ^(c/! P) + i a ( c ~ P) + 2 e . < M ( C ) + 2 £ ' £ |j(B) + 2 £ • / = I J ( T ) + 2 e. L e t t i n g £ -» 0 we get U ( T ) = U ( T „ P) •+ ',a(T ~ F) hence F i s measurable. 31 Proof of 2.3 Let H e % f o r ne UJ and A = U H . To see t h a t • ^ n n new g(A) =s 2 g(H ), l e t t < g(A), € > 0, and choose by r e c u r s i o n ne UJ N f o r every ne UJ, C e (3 and B e U so t h a t : n n C 0 c A and t < g ( C 0 ) , C n + 1 c B n = C n ~ H n ' «(c n + 1) > ' t i f g ( B n ) = « . Since fi C = j2f and C- i s ^Q-compact, there e x i s t s meuj such ne UJ that.C = 0 f o r n^m. I f k=max{n:g(C )>t}, then, f o r n ^ k , S ( B n ) * g ( 0 I H . 1 ) . + ^ T and s i n c e g ( C n ) * g ( H n ) + g ( B n ) , we havei t < g ( c k ) * g ( H k ) + g ( B k ) ^ g ( H k ) .+ g ( C k + 1 ) . + ^ T . Hence by i n d u c t i o n • • ' m .t < 2 g(H.) + g(C ) + £ * 2 g(H,) + £ . i=k 1 m ieuj 1 Thus g(A) ^ 2 g(H-). ieuj • Proof of 2 . 4 By lemma 2 . 2 , g i s c o u n t a b l y s u b a d d i t i v e on U, so t h a t g(H) = U ( H ) f o r every Hey. Since g i s a d d i t i v e on U, ^ c 5 ^ (theorem 1.3 ( 3 ) ) . To show 2 . 4 . 3 l e t keTh w i t h ^ ( A ) . < «. Then f o r C> 0 " 32 t h e r e e x i s t s a sequence H^H.... i n M such t h a t A c U H and new n 2 g ( H ) = 2 n ( H ) < u ( A ) + 8 . neuo neuu k There e x i s t s keuo, such t h a t i f H = U H , then u(A) < JJ(H) + £. ne cu C l e a r l y a l s o JJ(H ~ A) < £ , so t h a t there e x i s t s a sequence G - Q J G ^ . . . i n U such t h a t and H ~ A c: u G. ne uu n 2 g ( G ) = 2 n (G ) < £ neuu netu Let CQ£ C- be such t h a t C Q c H ~ G Q and u(C 0) > U ( H ~ G 0 ) - | . Then choose by r e c u r s i o n - f o r new C c C , ~ G n n - 1 n such t h a t n ( c n ) > n ( c n . r c ^ ) + 2 n + 1 Thus > u(H) - 2 n(Qm) - 2 - J E j - > n ( H ) - 2 £ . m=0 m m=0 2 m + 1 Then we have and n( n c ) ^(H) - 2 e ^ ^ A ) - 3 & , ne UJ n C c H ~ U G A, ne uu ne UJ from which we see u (A) = sup {ji(c) : C c A , C e e ^ e ) } . 33 Proof of 2 . 5 , 2 . 6 Since 2 . 5 i s c l e a r l y a s p e c i a l case of 2 . 6 , we proceed t o prove o n l y 2 . 6 . n • Let CeC-and C c u D where D N,D ,...D e £. . Then f o r m=0 m u 1 n £ > 0 there e x i s t s F e ^ y F e ^ such t h a t F c D Q and V ( D Q ~ F ) < £ . Hence C ^  FeC- and v ( c n D Q ) * v ( c n F ) + £ so t h a t v(c) =£ 'v(C ~ D Q) + v(C A F) + t * v(C ~ D Q) + h ( D Q ) + £ . m R e p l a c i n g C by C ~ U B. (which i s In C-), we get f o r any m^n k=0 K m m+1 v(0 ~ UD.) ^ v(c ~ U D ) + h ( D ,,) + £ , k=o k k=o n . m + 1 and by i n d u c t i o n , n v(c) * 2 h(D ) + n £ , m=0 m • • ' which shows t h a t n v(c) ^ 2 h ( D ) . m=0 • . Now we show t h a t h i s countably s u b a d d i t i v e on cr ..(s). Ctr I Let G- <=• U where G,G n,G„... are elements of cr id). For any new n .  crf"1 t < h(G), there e x i s t s CeG w i t h CCG and v(c) > t . Let for. each new, G = \JJL where JL c 3 and Card 3 ^ K . Then n n n n a C c U U ^ new n and Card U £ ^ K , so t h a t by lemma 3 . 6 . 2 of Ch.I, there new a e x i s t s a f i n i t e f a m i l y J-' c U,3 such t h a t C <= \J 3r. Let f o r ne w n each new, 34 D n = U (D : DeJ', D c G n ' D ^ G m f o r m < n - * and l e t ketu be such t h a t D = 0 f o r m>k. Then D eJ- f o r a l l k m n new and C <=• U D , so t h a t m=0 m k co t < v(c) ^ 2 h(D ) ^ 2 h(G_) m=0 m n=0 m s i n c e D c G m and h i s monotone. Thus h i s coun t a h l y sub-a d d i t i v e , hence u i s a Caratheodory measure and u(G) - h(G) f o r a l l Gea (.£) (theorem 1.3). • We next check t h a t c r ^ ^ ^ c ^ . Let G e a ^ ^ ) and T c X a+1 w i t h u(T) < <*>, and l e t £ > 0. Then there e x i s t s Uea .„(.#) 1 a+1 w i t h T<=U and u(u)< U(T) + £ . Then choose C^e (3, C ^ C T J A G such t h a t v C C j ) + £ > h ( U ^ G ) = p ( U ^ G ) , and C 2e <3, C g C t f ^ C ^ e CTa+1^^ s u c i l t h a - f c v ( C 2 ) + £ > h(U ~ C J = u(U ~ C j . Then v ( C 1 u C2.) * h(u) = u(u) -^ |i(T) + £ ^ ^ ( T A G ) + |i(T- ~ G) + £ ^ n ( U N G) + ^L(U ~ C^) + £ ^ v(C 1) + v(Cg) + 3 £ = v ( c ^ c 2) + 3 e .. Since £ > 0 i s a r b i t r a r y , u ( T ) •= u(Tn G) + n(T ~ G). Hence oa+^(£) <= and so C-c^.> (and i n f a c t J ^ c ^ ) . We next show \i(c) < °° f o r every Ce<3. Since 0 e (3, we have X e a ^ ^ (.#), hence there e x i s t s an.K - c o v e r i n g of the-35 space X by elements of A Hence e v e r y CeC* can be so c o v e r e d , and t h u s e v e r y Ce<3 has a f i n i t e c o v e r i n g b y elements of £. S i n c e elements of £• have f i n i t e |_r measure (h(G) ^  V ( G ) f o r e v e r y G e f f ^ ^ ) ) , i a ( c ) < « > . . C l e a r l y u ( c ) ^ v ( c ) f o r e v e r y Ce<3, hence u(G) = s u p { , u ( c ) : C c G , Ce&} f o r e v e r y Geo Thus \i i s R a d o n - l i k e w. r . t . ( Ka,<3, P r o o f o f 2.7 We f i r s t n o t e t h a t f o r any A c X , i f v ( A ) < oo t h e n v i ' ( A ) = v ( A ) . To see t h a t v ' i s a C a r a t h e o d o r y measure, l e t T <= U B Then f o r - a n y t < V'(T) t h e r e e x i s t s A<=T w i t h ne UJ t < v ( A ) < c o , and hence t < v ( A ) ^ 2 V ( A A B .) * 2 V ' ( B ) . ne uu ne UJ Thus v/(T) * 2 v:'(B ). new To see t h a t ^ / c r ^ , l e t kelf^t T<=X and V(T) < oo. . Then v(T) = v'(T) = V ; / (TA A ) + v ' ( T - A ) = v(To A ) + V ( T ~ A ) . To see t h a t %jz T/i^i, l e t Aetf? , T<=X and V ' ( T ) < ». • Then t h e r e e x i s t s B«= T w i t h V ; ' (T) = V ( B ) and, s i n c e v i s an o u t e r measure, C e % v, w i t h B<=C and V(B) = v ( c ) so t h a t V!'(T) V ( T A C ) ^ v(B) = v / ( T ) . I f we l e t D = T A C, t h e n D= T, v:'(T) = v(D) = v'(D) . . . and vi'(T ~ D) = 0 . Thus 36 V'(T) = v(D) = v ( D n A ) +. v(D ~ A ) = v'(DrtA) + v'(D ~ A) = V'(TA-A) + v ' ( T ~ A ) . hence A e ^ v / . Prom t h e above c o n s t r u c t i o n we see a l s o t h a t v ' i s an o u t e r measure, f o r T c C o ( T ~ D)., . • C c (T ~ D)- e 7H^t and vl'(T) - v'(c) = v'(C u (T ~ D ) ) . 37 CHAPTER I I I INVERSE SYSTEMS OP MEASURES I n t h i s c h a p t e r we d e v e l o p our n o t i o n s o f i n v e r s e systems of measure spaces ( t h e s e a re a l s o f r e q u e n t l y c a l l e d p r o j e c t i v e systems o f measure spaces) and g i v e s u f f i c i e n t c o n d i t i o n s f o r the e x i s t e n c e o f our b a s i c l i m i t measure. Some of the d e f i n i t i o n s we s h a l l use d i f f e r f r om those, used b y p r e v i o u s workers i n t h i s a r e a . F o r t h i s r e a s o n we b e g i n b y comparing our d e f i n i t i o n s w i t h o t h e r s and d i s c u s s i n g th e r e a s o n s f o r the changes we make. 1. Concepts o f I n v e r s e Systems. We b e g i n w i t h t h e s t a n d a r d d e f i n i t i o n (see e.g. B o u r b a k i [ 3 ]) o f an i n v e r s e ( p r o j e c t i v e ) system o f s p a c e s , i n w h i c h , however, we i n c l u d e the r e q u i r e m e n t t h a t the f u n c t i o n s be onto. 1.1 D e f i n i t i o n . (x,p,l) i s an i n v e r s e system of sp a c e s i f f .1 I i s a d i r e c t e d s e t (by_<), .2 ( X , l ) i s a system of spaces ( i . e . f o r e v e r y i e l , X^ i s a s p a c e ) , .3 f o r e v e r y i , j e l w i t h i < j , p. • i s a f u n c t i o n on X. onto X i J p i i i s ^'tle ^eir^^y f u n c t i o n , and p i k = P i j 0 P j k whenever i<j-<k. G i v e n an i n v e r s e system o f s p a c e s . ( x , p , l ) , a major problem i s t o determine whether one can i n some 3 8 way c l o s e t h e system, i . e . f i n d a " l i m i t " s e t X , and " l i m i t " f u n c t i o n s p^, f o r e v e r y i e l , such t h a t maps X onto X ^ and has t h e p r o p e r t y (a) P ± = P ± 3 o P j . whenever i < j . I t i s n o t h a r d t o see t h a t i f any such s e t X and f u n c t i o n s p^ e x i s t , t h e n X can be mapped i n t o a s u b s e t o f t h e -set d e f i n e d i n 1.2 below. 1.2 D e f i n i t i o n . The i n v e r s e l i m i t s e t of an i n v e r s e system of spaces (x,p,l) i s [xe n X . : TT. (x) = p. . ( T T . ( X ) ) whenever i < j } . • i e l 1 1 i J J (The n o t a t i o n Lim (x,p,l) i s o f t e n used t o denote the i n v e r s e l i m i t s e t . ) Fo r t h i s r e a s o n , t h e I n v e r s e l i m i t s e t L and the p r o j e c t i o n s :n\ r e s t r i c t e d t o L a r e t h e o n l y c a n d i d a t e s " u s u a l l y c o n s i d e r e d f o r X and p^. The problem i s t h e n reduced t o d e t e r m i n i n g whether L i s i n d e e d l a r g e enough so t h a t T T ^ [ L ] = X ^ . T h i s c o n d i t i o n i s u s u a l l y r e f e r r e d t o as' s i m p l e m a x i m a l i t y and i s not s a t i s f i e d by a l l I n v e r s e systems of spaces (see e.g. B o u r b a k i [ 3 ] ) . I n many i m p o r t a n t s i t u a t i o n s ( s t o c h a s t i c p r o c e s s e s , p r o d u c t s p a c e s , e t c . ) the spaces X ^ a l s o carry'measures' u. w h i c h a r e c o m p a t i b l e w i t h t h e f u n c t i o n s p. .. T h i s l e a d s t o the concept o f an i n v e r s e system of measures. 3 9 I t i s customary (see e.g. , C h o c k s i [*!•], M e t e v i e r do ] ) t o d e f i n e such a system b y a d d i n g t o the above d e f i n i t i o n o f an i n v e r s e system o f sp a c e s , \ f o r e v e r y i e l , ^ i s a measure on a a - f i e l d ( o r a - r i n g ) <7. o f s u b s e t s o f X^, and t o e x t e n d 1 . 1 . 3 t o i n c l u d e : whenever i-<j, p. • i s a measurable f u n c t i o n , a,nd H / P ^ M = n ± ( A ) f o r e v e r y Aec7^. The problem c o n s i d e r e d t h e n i s t h a t o f f i n d i n g , i n a d d i t i o n t o a " l i m i t " s e t X and " l i m i t " f u n c t i o n s p^, a " l i m i t " measure jj on the a - f i e l d g e n e r a t e d by the r i n g { p 7 1 [ A ] : i e l and Ae(7±) so t h a t H ( p T 1 [ A ] ) = n ± ( A ) f o r e v e r y i e l and Ae<7 One of t h e most u s e f u l p r o p e r t i e s o f such a measure jj i s the f o l l o w i n g . Suppose t h a t f o r each i e l , f ^  i s an a. -measurable f u n c t i o n on X. t o the r e a l s and I I f • = f - o p. . f o r i< j . Then f o r e v e r y Aec7., and j such ej U t h a t i-<j, 3 A" f , d n = -1 f i d ' u i > ~ ~ ~ and i f we l e t f = f i 0 Pj_ , t h e n f i s independent of i and ko A' f i d ^ i = r dfr P : 1 [A ] f o r e v e r y i e l and AecT^. I n p r a c t i c e , s i n c e , as we p o i n t e d out above, we can r e p l a c e X by L, a l l p r e v i o u s w o r kers i n t h i s f i e l d have c o n c e n t r a t e d on h a v i n g such a measure ja c a r r i e d by L. As a r e s u l t , t h e known e x i s t e n c e theorems r e q u i r e s t r o n g c o n d i t i o n s on t h e f u n c t i o n s p. . and on t h e r e l a t i o n s h i p between L and c o u n t a b l e subsystems of ( x , p , l ) . I n t h i s paper, we t r y t o a v o i d making the e x i s t e n c e o f a jj. dependent on a " l i m i t " s e t X w i t h f u n c t i o n s p^ s a t i s f y i n g c o n d i t i o n (a) above. We c o n s i d e r i n s t e a d t h e C a r t e s i a n p r o d u c t II X. and p r o j e c t i o n s , TT. , and concen-i e l 1 1 t r a t e a t f i r s t on f i n d i n g a measure ja h a v i n g the p r o p e r t y s u g g e s t e d by (b) above. Suppose a g a i n t h a t f o r each i e l , f ^ i s an ^ - m e a s u r -a b l e f u n c t i o n on X^ t o the r e a l s , and f o r i - \ j , ..' f . = f . o p. ., so t h a t f o r a l l Aec7 i. I f we now d e f i n e • f ^ = f ± o u±J i s n ot independent of i . However, i f we can f i n d a measure jj c a r r i e d b y II X. so t h a t f o r e v e r y j e l , Aec7. -• i e l 1 J / and k such t h a t j<k , the symmetric d i f f e r e n c e has j j measure z e r o , t h e n we have ; 1*1 whenever j e l and kec7.} independently of i , as i n (b) 3 above. We concentrate t h e r e f o r e on f i n d i n g such a jj. We note f u r t h e r t h a t i t i s s u f f i c i e n t t h a t the symmetric d i f f e r n e c e p - ^ C A ] A ( p . . o P . k ) " 1 [ A ] have u k measure zero f o r every Aec7 whenever i<j<k, i n order t h a t P i>] whenever i < j . Hence i n our d e f i n i t i o n of i n v e r s e system of measures we s h a l l r e p l a c e the requirement t h a t P i k = P i j o p by such a c o n d i t i o n . Thus an i n v e r s e system of measures i n our sense need not be an i n v e r s e system of spaces. D e f i n i t i o n . (x,p,u, I) i s an i n v e r s e system of outer measures (i.s.o.m.) ' i f f 1 I i s a d i r e c t e d set (by <), 2 f o r i e l , X^ i s a space and i s an outer measure on X. . 3 f o r i-<j, p ^ j i s ' a measurable f u n c t i o n on X-. onto X^, i . e . p .. : X. -> X ± and p - ] [ A ] e ^ _ . . f o r e very Ae57^; P i i i s . the i d e n t i t y f u n c t i o n ; ^ ( p - ^ C A ] A (p. o p . k ) " 1 [ A ] ) = 0 .3 ° and 4 2 H.(A) = ^ . ( p ~ ] [ A ] ) f o r e v e r y AeT/i whenever K j < k . To s i m p l i f y out n o t a t i o n we l e t We i n t r o d u c e n e x t our concept o f a " l i m i t " measure f o r (X,p,[j, i ) . . 1 4 D e f i n i t i o n . v i s a T T - l i m i t o u t e r measure f o r (x,p, p, i ) on D i f f ( x , p , j a , l ) i s an i.s.o.m., D c n X., v i s an i e l 1 • o u t e r measure on II X. w h i c h i s c a r r i e d by D, and t h e i e l f o l l o w i n g c o n d i t i o n s a re s a t i s f i e d : . 1 T T ~ 1 [ A ] e 7hv f o r e v e r y i e l and A e ^ , . 2 V ( T T " 1 [ A ] ) = |j^(A), ' • . : • and V ( T T - 1 [ A ] A ( P i j o T K ) " " 1 [ A ] ) = 0 whenever i e l , AeT/j^ and i < j . 2 . G e n e r a t i o n o f a T T - l i m i t Outer Measure. • I n t h i s s e c t i o n we f i r s t d e f i n e f o r an i.s.o.m. ( X , p , j j , I ) a s e t f u n c t i o n g on the f a m i l y R e c t ( ^ ) I n a manner su g g e s t e d b y c o n d i t i o n 1 . 4 . 2 o f t h e d e f i n i t i o n o f a. T T - l i m i t o u t e r measure. Any T T - l i m i t o u t e r measure w i l l have t o agree w i t h g on R e c t ( ^ ) . We t h e n g e n e r a t e • an o u t e r measure u on II X. by the s t a n d a r d C a r a t h e o d o r y i e l 1 . p r o c e s s and check under what c o n d i t i o n s jj i s i n d e e d an e x t e n s i o n o f g and a i r - l i m i t o u t e r measure. We b e g i n by e x h i b i t i n g i n the f o l l o w i n g lemma a c o n d i t i o n e q u i v a l e n t 4 3 t o 1 . 4 . 2 . 2 . 1 Lemma. L e t ( X , p j j a , l ) "be an i.s.o.m. and l e t v be an o u t e r measure on IT X. . Then v s a t i s f i e s c o n d i t i o n i e l 1 1 . 4 . 2 i f f f o r e v e r y cte R e c t ( ^ ) and- j e l w i t h i < j f o r e v e r y i e J , we have: J a (*) v(a) = n ( n p j l [TT [ a ] ] ) . ^ i e j a ^ ^ . We now use c o n d i t i o n ( * ) . t o d e f i n e a s e t f u n c t i o n g on R e c t ( ^ ) w i t h w h i c h we gen e r a t e a c a n d i d a t e f o r a r r - l i m i t o u t e r measure. 2 . 2 D e f i n i t i o n s . L e t (X,p,u.,l) be an i.s.o.m. . 1 For a e R e c t ( ^ ) , g(a) = u f n p-J [TT [ A ] ] ) J i e J 1 J ± a where j i s any element o f .1 such t h a t j > i f o r e v e r y i e J . (Note t h a t J i s f i n i t e and t h a t i n v i e w of a a c o n d i t i o n 1 . 3 - 3 o f the d e f i n i t i o n o f i.s.o.m., g i s independent of t h e c h o i c e of j . ) u i s the o u t e r measure on H X. g e n e r a t e d by g and i e l 1 R e c t ( ^ ) . W i t h no f u r t h e r c o n d i t i o n s we have t h e f o l l o w i n g lemma. 2 . 3 Lemma. For any i . s.o.m. ( x , p , a., I ) : . 1 g ( T T ? 1 [ A ] ) = u ^ A ) f o r e v e r y i e l and A e ^ , 2 . g ( TT. [ A ] ^ ( p . o T T . ) [ X . ~ A ] ) = 0 whenever i < j and A e ^ , kh • 3 I f J i s a f i n i t e d i r e c t e d subset" of I, then g( a) = 0 whenever c* e R^ct(%) a A { x e I X , : p , , ( x . ) = x, f o r i , o'ej, Kj) = 0. I e l .4 g i s f i n i t e l y a d d i t i v e on R e c t ( ^ ) . . 5 u i s an outer measure on X . . 6 Rect(^) c . . 4 Remarks. From the d e f i n i t i o n s i n v o l v e d and the above lemmas we have the f o l l o w i n g f a c t s : . 1 I f jj agrees w i t h g on Rect(ft|), then ij Is a i r - l i m i t o uter measure (note t h a t f o r i e l and Ae5^, rr~ 1 [ A ] A ( P i J o T T d ) " 1 [ A > ( T T - 1 [ A ] A ( P i j . o T T j J - 1 [ X i ^ A ] ) u ( T T - 1 C X i ~ A ] n ( p i j o T T j ) " 1 [A ]) . .2 I f jl does not agree w i t h g on Rect(#0, then no i r - l i m i t outer measure can e x i s t , s i n c e g c o u l d not be c o u n t a b l y s u b a d d i t i v e on Rect(^) and hence no outer measure c o u l d agree w i t h g. . 3 I f v i s a T T - l i m i t outer measure then f o r any A c n X., v(A) ^ jJ(A), f o r otherwise there.would e x i s t i e l a countable s u b f a m i l y 7 of Rect(^) which covers A and -2 V ( F ) < v(A), which i s i m p o s s i b l e . Fe7 We next s t a t e a c o n d i t i o n which i s s u f f i c i e n t t o ensure t h a t jj does i n f a c t agree w i t h g on Rect(#0. 4 5 . 2 . 5 D e f i n i t i o n . An i.s.o.m. (x,p,u-, i ) i s i n n e r r e g u l a r w . r . t . C i f f i s a system of f a m i l i e s o f s e t s w . r . t . • ( X , l ) such t h a t : .1 f o r e v e r y i e l t h e r e i s an KQ-compact f a m i l y of s e t s w h i c h i s an i n n e r f a m i l y f o r .2 f o r e v e r y j e l and CeC-., and e v e r y 1<3, u- i s a - f i n i t e o n P i . [ C ] . When we use the above c o n d i t i o n we s h a l l assume, w i t h o u t l o s s of g e n e r a l i t y , t h a t ja . (c) < oo f o r e v e r y j e l and Ce<3., f o r i f we l e t 3 C'. = {CeC : u . (c) < o o ) , 3 3 3 t h e n s a t i s f i e s 2 . 4 . 1 , s i n c e 2 . 4 . 2 shows t h a t n . 3 ^3 i s a - f i n i t e on e v e r y Cee-., and thus t h e r e e x i s t s a sequence Ae^. such t h a t A A and jj.(A„ ) < oo f o r j n n i i 1 j n e v e r y ne uu, and C = (J A. , hence ne uu n-^ . ( C ) .= s u p { ^ ( A n ) = sup{|a.(c^) * supCujVC^) = SUpCu.(C!') *Hj(G), ne uu] C 'e<3., C ; ' c A n f o r some neuu] C 7 e e C ' c C, n,(C>') < ~] •C 'ec' C '«= C] and t h u s f o r e v e r y Ae%., n.(A) = sup { n . ( c ) : C e a ' C c A } . J J J We can now s t a t e t h e main,theorem o f t h i s s e c t i o n . 2 . 6 Theorem. I f (x,p,ja,l) i s . a n i.s.o.m. w h i c h i s i n n e r r e g u l a r w.r.t.(3 f o r some <3, t h e n (I i s a T T - l i m i t o u t e r measure f o r (X,p,|_i,I) on II X. . i e l 1 46 Example 1 i n the Appendix shows t h a t without t h i s c o n d i t i o n s a r r - l i m i t outer measure may f a i l t o e x i s t . p. P r o o f s . Proof of 2 . 1 . Let A± = n^Ca] , J = J a s o t h a t a = n TT7 1[A. ] . i e J 1 1 Choose j e l so t h a t j > i f o r a l l I e J and l e t B = n P^.CA. ]. i e J . 1 J 1 Then V ( T T . 1 C B ] ) = n, (B), n: 1 [B] c a o U ((p. . o TT . ) 1 [A. ] ~ TT7 1 [A. ]), v i e J J J . acrr"- 1 [B ] u U ( TT7 1 [A, ] ~ (p, , o T T , ) " 1 [A. ]) 3 I e J 1 1 1 J 3 Since f o r each i e J , v ( n " 1 [ A ± ] A ( P j L J o T T ^ ) " 1 [A.]) = 0, we conclude v(a) = V (TT~. 1 CB]) = ia,(B). J u C o n v e r s e l y suppose f o r each aeRect (%?), v(a) = |a,(B) J where j and B are as ahove. Then f o r any i e l and A e ^ , — i l e t t i n g a = TT [A] and j = i , we have B=A and t h e r e f o r e - 1 v ( r r " ' [ A ] ) = n ± ( A ) Moreover, f o r any j">I, l e t t i n g A^= A, B. . = A .~ c 1 3 3 and A X p. [A], a = r r i 1 [ A ± ] n TT. 1 [A.. ] = TT. 1 [A] ~ ( p i ; j o T T . . ) " 1 [ A ] , we get 4 7 B = A . A P - I ^ ] = 0 and t h e r e f o r e V ( T T. [A] ~ ( P i . o T T ^ . ) " 1 [A]) = 0 . S i m i l a r l y , l e t t i n g A. = X. ~ A and A. = p. . [A ] we get V((p, , o TT ) " 1 [A] ~ TT [A]) = 0. X J J Thus . V(TT" 1 [A] A ( p ± k o T T K ) _ 1 [A]) = 0 whenever k>i and A e ^ . Proof of 2.3-1, 2.3.2, 2.3-3. S i m i l a r t o proof of 2 . 1 . To show t h a t g i s f i n i t e l y a d d i t i v e we w i l l use the f o l l o w i n g lemma. Lemma A. Let ( x , p , | j , l ) he an i.s.o.m., and l e t a, (3 be d i s j o i n t elements of Rect (%>?). Then f o r any k such t h a t k>i f o r a l l i e J a t > J g > A = n p'jJCrr.La]] ^  0 p" f e [TT. [0 ] ] = 0 \ , i e J i e J Q a |3 • Proof. By lemma 2 . 3 > Ch.I, there e x i s t s j e J a t / J g such t h a t T T . [ a ] N T T . [ @ ] = 0. I f , however, there e x i s t s xeA, then P A - K U ) e TT [ A ] N TTJ [0] = 0. hence no such x e x i s t s . Proof of 2.3-4 Let aeRect (ty)• and l e t £&Rect(#0 be. a f i n i t e d i s j o i n t f a m i l y such t h a t a = Let K = U J f f l and l e t j e l be such t h a t j > i f o r every ieK. Then J a «=• K so t h a t j > i f o r every i £ J a - A l s o l e t A = fl p ~ 1 [ % [ a ] ] i ? J a j 48 and f o r every pe#, -1 Then (Ag : ie/3} i s a f i n i t e d i s j o i n t s u b f a m i l y of ^.(from lemma A). Since f o r every i e J a and fie/3, [3 ] <^TT\'[a] and J <=• J 0, then A « c A j hence U A ^ A . a 3' P g e / 9 I On the other hand, f o r every xeA, c i o o s i n g ye II X. so i e l 1 t h a t y. = p. .(x) f o r every ieK, we see t h a t y e a and thus • ye 3 f o r some $e/3. Then {x} •« n p 7 j [ n , ( y ) ] - A g ) i € j g ^ . 3 hence xeA^ and A <= U A , . $e/3 p Thus g(a) = (i, (A) = 2 u.(A @) = 2 g(3). Proof of 2.3.5, 2.3.6. We f i r s t note t h a t by lemma 1.6, Ch.II, we can extend g t o a f i n i t e l y a d d i t i v e f u n c t i o n g* on the r i n g ( R e c t ( ^ ) ) . Now,-the Caratheodory measure \x generated by g and Rect(#?) i s the same as t h a t generated by g* and a (Rect ( , hence by theorem 1.3* Ch.II, fa Is an outer measure and a 0(Rect(590) .. ; Proof of 2.6. We f i r s t check some approximation p r o p e r t i e s of g when (x,.p,|j , l ) Is i n n e r r e g u l a r w.r.t. some system of f a m i l i e s of se t s 49 Lemma _B. I f ( X , p , j j , l ) I S an i.s.o.m. w h i c h i s i n n e r r e g u l a r w . r . t . <3, t h e n f o r ,any aeRect(/?0, t < g( a) and i e l , t h e r e e x i s t s CeC-^ such t h a t C c TT^ [ a] and g ( a A T T~ 1 [C]) > t . P r o o f . L e t k e l w i t h k>j f o r a l l j e o [ i ] , and l e t A = n p~.JCTT.Ca]].' j e J ^ ^ ° a Then u k ( A ) = g( a) and t h e r e e x i s t s C 'e e^., C '<= A w i t h 1-^(0') >.t. Now choose B e ^ , B <= T^'Ca] such t h a t p i k C c ' ] c B and [a^ i s o - f i n i t e on B. Then t h e r e e x i s t CQ,C^, . . . ec- such t h a t f o r such new, c <= C , „, C <= B and n n+1 n H k(B ~ U C ) = 0 ne uu hence M C ' ~ U P l k [ C n ] ) - 0 • ne uu so t h a t f o r some meuu ^ C ^ P i k [ C m ] ) >*' • . L e t t i n g C = C m, we see t h a t C c TN [a] and t h a t g ( a nTT - 1 C c ] ) = n k ( _ n p - J C n j C a ] ] A p ~ ] J [ C ] ) ^" 6^a * u k ( C i A P i k [ C ] ) > t . • Lemma C. I f ( x , p , | j , i ) I s an i.s.o.m. w h i c h i s i n n e r r e g u l a r - ' w . r . t . <3, t h e n Rect(<3) I s an i n n e r f a m i l y f o r g on R e c t ( ^ ) . P r o o f . L e t a e R e c t ( ^ ) and J a = { i Q , i i } . G i v e n t<g( a). Choose C ne e , C n c TT. [ a ] such t h a t u i 0 u x Q g( a /\ TT7 1 [ C n ] ) > t , 1 Q U 50 and by r e c u r s i o n on m, C e (3. , C <= TT^ [a] such t h a t m m g ( t t n n TT, 1 [c. ]) > t. 1-0 x l x I f n c = n T T T 1 [C, ], 1=0 1 i x then CeRect(a), C « a and g(c) > t . JLemma D. Let g* be the extension:.of 'g t o a f i n i t e l y a d d i t i v e f u n c t i o n on CTQ(Rect(/??)) • I f (X,p,|_i, i ) i s i n n e r r e g u l a r w.r.t. <3, then a Q(Rect(C-)) i s an Inner f a m i l y f o r g* on a Q ( R e c t ( ^ ) ) . Proof. Let aea (Rect ( ) . Then there e x i s t s a f i n i t e d i s j o i n t f a m i l y ^ c R e c t ( ^ ) such t h a t a = Us. I f t<g*(a), choose f o r each Be/?, C Re Rect(e) so t h a t C R c B and t< 2 g(CL.). Then Theorem 2 . 6 now f o l l o w s from theorem 2 . 4 , Ch.II, s i n c e o n(Rect(<3) i s K -compact (by lemmas 3 - 5 - 1 and 3 - 4 . 3 , C h . l ) . Be/? g' : ) = 2 gMCLJ = 2 g C ) . , B Be/? B Be/? B^ > 5 1 CHAPTER IV APPROXIMATION PROPERTIES OF LIMIT MEASURES ON PRODUCT SPACES I n t h i s c h a p t e r we d i s c u s s problems r e l a t e d t o the f o l l o w i n g : g i v e n an i.s.o.m. ( x , p , u , i ) i n w h i c h f o r each i e l , X^ i s a t o p o l o g i c a l space and i s Radon, under what c i r c u m s t a n c e s can we get a T T - l i m i t o u t e r measure w h i c h i s Radon f o r the p r o d u c t t o p o l o g y ? T h i s l e a d s us t o seek g e n e r a l a p p r o x i m a t i o n p r o p e r t i e s - of j j and t o amend jj when ja i s not s a t i s f a c t o r y . F o r t he sake o f m o t i v a t i o n t h e t o p o l o g i c a l case i s t r e a t e d f i r s t . We show t h a t a system o f bounded Radon measures always has a T T - l i m i t o u t e r measure w h i c h i s Radon w . r . t an a p p r o p r i a t e t o p o l o g y . T h i s t o p o l o g y i s o b t a i n e d from a f a m i l y of s e t s C,whose elements a r e c l o s e d b u t n o t i n g e n e r a l compact i n the p r o d u c t t o p o l o g y . C- i s , however, ft-compact f o r e v e r y c a r d i n a l We t h e n t u r n t o t h e problem of f i n d i n g -T T - l i m i t measures w h i c h are Radon w . r . t the p r o d u c t t o p o l g o y , and g i v e some c o n d i t i o n s under w h i c h such measures e x i s t . I n t h e second s e c t i o n we g e n e r a l i z e t h e r e s u l t s t o the n o n - t o p o l o g i c a l case (the r e s u l t s of the, f i r s t s e c t i o n a r e i n f a c t c o r o l l a r i e s of r e s u l t s i n t h e second s e c t i o n ) i n w h i c h the i n n e r f a m i l i e s a re ft-compact i n s t e a d of b e i n g the c l o s e d compact s e t s of some t o p o l o g y . 52 In "both the t o p o l o g i c a l and n o n - t o p o l o g i c a l cases we o b t a i n measures which are Radon or come v e r y c l o s e t o b e i n g Radon under c o n d i t i o n s considerably.weaker than those needed by p r e v i o u s workers who r e s t r i c t e d t h e i r a t t e n t i o n t o the Inverse l i m i t s e t . 1. The T o p o l o g i c a l S i t u a t i o n . 1.1 General Assumptions and N o t a t i o n . Throughout t h i s s e c t i o n we suppose .1 (X,p.,]a, i ) i s an i.s.o.m., .2 f o r every i e l , .2.1 X i i s a t o p o l o g i c a l space. .2.2 i s the f a m i l y of open subsets of X^, .2.3 7^ i s the f a m i l y of c l o s e d subsets of X^, .2.4 i s the f a m i l y of c l o s e d compact subsets of X i, .2.5 i s an i n n e r f a m i l y f o r j j ^ , .2.6 HJJCK)-- < 0 ° f o r every ¥LeK±, .2.7 p. i s a - f i n i t e on p. . [ K ] f o r every whenever l < j . .3 X = II X. . i e l 1 .4 J- i s the product, t o p o l o g y on X . •5 ^ i s the f a m i l y of c l o s e d s e t s i n the product t o p o l o g y on X . .6 X i s the f a m i l y of c l o s e d compact s e t s i n the product t o p o l o g y on X. 5 3 . 7 6 1 ( o 0 ( R e c t ( ^ ) ) ) . . 8 jj i s the outer measure on X i n t r o d u c e d i n d e f i n i t i o n 2 . 2 . 2 C h . I I I . . 9 g i s the set f u n c t i o n on Rect(^) i n t r o d u c e d i n d e f . 2 . 2 . Ch.III. 1 . 2 Remarks. The f o l l o w i n g are immediate consequences of our assumptions. . 1 (X,p,|jjl) i s i n n e r r e g u l a r w.r.t. K (see d e f . 2 . 5 C h . I I l ) hence u i s a n - l i m i t outer measure (theorem 2 . 6 C h . I l ) . . 2 The f a m i l y <3 c o n s i s t s of c l o s e d s e t s i n the'product t o p o l o g y {C-c-T) which are not i n g e n e r a l compact, hut C- i s an ^-compact f a m i l y f o r every c a r d i n a l K (lemmas 3 . 3 , 3 - 4 , 3 - 5 C h . l ) . . 3 The outer measures are not n e c e s s a r i l y Radon, but i f t hey are bounded, then they are Radon. We note a l s o t h a t our assumption 1 . 1 . 2 . 5 r e q u i r e s t h a t f o r every Aer/^, u ±(A) = sup{ ki i(K) : Ke^ ±, K=A}. Th i s f o r c e s ^ t o be s e m i - f i n i t e , s i n c e we r e q u i r e c l o s e d compact se t s t o have f i n i t e measure and f o r an -outer measure v i t i s o n l y n e c e s s a r y t o check t h a t v(A) = sup{v(B) : B^A, V(B) < <*} f o r a l l A e^ . . v We now check approximation p r o p e r t i e s of jj under our g e n e r a l assumptions ( 1 . 1 ) . ' " 5 4 1 . 3 Theorem. i f Ae7ft~ and jj(A) < then ja ( A ) = sup{[1(C) : CeC, C c A } . C l e a r l y the c o n d i t i o n t h a t j7(A) be f i n i t e can be r e p l a c e d by: _ (a) J T( A) = sup{iT(B) : B c A ; JT(B) < oo}. Thus the theorem f a i l s t o h o l d e s s e n t i a l l y o n l y i n the p a t h o l o g i c a l case i n which a l l subsets of A have e i t h e r i n f i n i t e measure or measure zero. The f o l l o w i n g propo-s i t i o n s show t h a t under our assumptions such cases are l i m i t e d and t h a t they do not occur i n the s e t s i n which we are p r i m a r i l y i n t e r e s t e d . 1 . 4 Pro-position. For any Ae7T(~, at l e a s t one of A,X ~ A s a t i s f i e s c o n d i t i o n (a) above. 1 . 5 P r o p o s i t i o n . Every Aeo^(Rect(#0) s a t i s f i e s c o n d i t i o n (a) above. In view of p r o p o s i t i o n 1 . 5 we can a d j u s t our measure so as t o e l i m i n a t e the p a t h o l o g i c a l cases. 1 . 6 Theorem. Let, f o r every A X , i r ' ( A ) = sup£iI(B) : B^A and £(B) < <»}; then: . 1 . ja' i s a s e m i f i n i t e i r - l i m i t outer measure, .2 fr^t = n~ , . 3 f o r every Ae^~/ , j I ' ( A ) = sup{JT'(c) : Ce2^, C c A } . 55 Now we c o n s i d e r c o n d i t i o n s "under which a Radon i r - l i m i t outer measure e x i s t s . The f o l l o w i n g theorem shows t h a t under the s o l e a d d i t i o n a l c o n d i t i o n , t h a t the measures are bounded (hence Radon) j l can always be r e g u l a r i z e d t o y i e l d a T T - l i m i t outer measure which i s Radon w.r.t an a p p r o p r i a t e t o p o l o g y on X.'( (Example 6 seems t o i n d i c a t e t h a t t h i s t o p o l o g y r a t h e r than the product t o p o l o g y i s a n a t u r a l one t o c o n s i d e r i n c o n n e c t i o n •with i n v e r s e l i m i t s ) . 1.7 Theorem. I f f o r each i e l , ^ i s bounded, and i f we l e t £ be the t o p o l o g y h a v i n g as a base then there e x i s t s c. a i r - l i m i t outer measure which i s Rad6n w.r.t the t o p o l o g y 3*. 1.8 Remarks. The t o p o l o g y JJ, generated by i s not i n g e n e r a l the product t o p o l o g y . (J") although i t c o n t a i n s the product t o p o l o g y i f the spaces are compact. The u s u a l base f o r the product t o p o l o g y - i s , however, co n t a i n e d i n Rect(^) whenever the measures are Radon,, hence the elements of the base are measurable f o r any T T - l i m i t outer measure. We note a l s o t h a t i f ^u(X^) = <»_, then f o r any KeX ~ u^X^ ~ K) = o o , so t h a t i s not Radon w.r.t.c{%^). We next seek c o n d i t i o n s under which we can f i n d a . T T-limit outer measure which i s Radon w.r.t. the product t o p o l o g y J>. 56 F i r s t we d e f i n e a candidate f o r such a measure by r e g u l a r i z i n g jj w i t h r e s p e c t t o A 1.9 D e f i n i t i o n . (T* i s the set f u n c t i o n on the subsets of X d e f i n e d by p>(A) = lnf£h(G) j A c G, GeJ} f o r every A X, where • . • h(G) = sup{[I(K) ; K c Gj Ke£} f o r every open set G. 1.10 Theorem. I f f o r each i e l , i s Radon, then: .1 jj* i s Radon w.r.t. the t o p o l o g y & , .2 i f i s a T T - l i m i t outer measure then f o r ev e r y E e a Q ( R e c t ( J ' ) ) w i t h |T(E) < <», £ * ( E ) = [T(E) = sup{£(K) : K e £ K c E } . .3 i f . Y i s any r r - l i m i t outer measure which i s Radon w.r.t. 1 , then Y = £*. The problem now i s to f i n d c o n d i t i o n s on our system which w i l l guarantee t h a t jo* i s a T T - l i m i t outer measure. In order t o o b t a i n the r e s u l t i n theorem 1.10.2 we f i n d t h a t we have t o impose r e s t r i c t i o n s on the index set I and/or the f u n c t i o n s p. .. The a - f i n i t e n e s s of the measures u. ensures t h a t f o r every aeRect(#0,' g(a) = i n f C g ( B ) : BeRect(.£),; ..acB}, so t h a t jo*(a) ^ g( a). Our main theorem i s then the f o l l o w i n g . 57 1.11 Theorem. I f f o r each i e l , i s a a - f i n i t e Radon o u t e r measure, t h e n j j ^ I s a T T - l i m i t o u t e r measure Radon w.r.t.& whenever any one o f t h e f o l l o w i n g c o n d i t i o n s h o l d s : . 1 Card I ^ NQ, .2 t h e r e e x i s t s a c o f i n a l s e t I n c I w i t h Card I n £ and p. .[A.] e 9(. f o r e v e r y Aefif. whenever i ^ j . 1 J X J .3 p. .[A] e ?r. f o r e v e r y Ae?f. and p " l C B ] e ?C f o r e v e r y Be?^ whenever i-<o". 1.12 Remark. F o r any AeRect(TTi), i f l>(A) = sup{jo^(K) : KeX , K = A ) t h e n j>(A) = s u p C J X'K) : Ketf', K < = A } where X' = ( n K. : K.e f o r e v e r y i e l } , i e l 1 1 1 hence the i m p o r t a n t f a m i l y o f compact s e t s f o r our i n v e r s e systems i s 2. The N o n - T o p o l o g i c a l Case. 2. 1 C-eneral Assumptions and N o t a t i o n . Throughout t h i s s e c t i o n we assume: .1 (x,p,u,l) i s an i.s.o.m. -.2 a i s an o r d i n a l and f o r each i e l , C% i s an K a~compact f a m i l y of s u b s e t s o f X^ such t h a t : .2. 1 &± = oQ{c±) = 6 ^ ^ ) , 5 8 .2.2 ^ i s an i n n e r f a m i l y f o r , .2.3 ^ ( c ) < 0 0 f o r e v e r y Cec^,-.2.4 i s a - f i n i t e on P^-j^C] f o r e v e r y CeC-^ . whenever i < j . .3 X = ' n X. . • i e l 1 .4 <3 = 6 1 ( a 0 ( R e c t ( < 3 ) ) ) .5 & n = 6.(a ({ n C, : C,eC-, f o r e v e r y i e l } ) ) . v i e l 1 1 .6 (T i s t h e measure i n t r o d u c e d i n d e f i n i t i o n 2.2 C h . I I I . 2.2 Remarks. ' .1 (x,p,u,l) i s i n n e r r e g u l a r w.r.t..<3 (def . 2 . 5 C h . I I I ) , hence j j i s a T T - l i m i t o u t e r measure. .2 The d e f i n i t i o n of &• i s e n t i r e l y analogous t o the d e f i n i t i o n o f the same symbol i n s e c t i o n 1. S i n c e , however, our space X i s n o t now endowed w i t h a p r o d u c t t o p o l o g y , we s h a l l n o t attempt t o o b t a i n r e s u l t s i n terms o f an analogue o f t h e f a m i l y K i n s e c t i o n 1 ( a l t h o u g h such an analogue c o u l d be c o n s t r u c t e d ) . I n s t e a d we s h a l l s t a t e our. r e s u l t s i n terms o f S Q , w h i c h i s analogous t o t h e f a m i l y KQ d i s c u s s e d i n remark 1.12. .3 The o u t e r measures a r e not n e c e s s a r i l y R a d o n - l i k e w . r . t . (^,(3^,A) f o r some J ^ , though, s i m i l a r l y t o s e c t i o n 1, t h e y a r e ( w i t h = cjc - ^ ) ) whenever the are bounded. We n o t e a l s o t h a t i n any case we r e t a i n ' f r o m our g e n e r a l a s sumptions (2.1) t h e . f a c t t h a t f o r e v e r y A e ^ u ± ( A ) = supCia i(c) : Ce&±, C c A } . 59 We now show t h a t u has s i g n i f i c a n t a p p r o x i m a t i o n p r o p e r t i e s under our g e n e r a l a s s u m p t i o n s . 2.J5 Theorem. I f A e 5 ^ ~ and JI(A) < °° t h e n (b) 11(A) = sup{fl(c) : C e % C < = A } . As i n s e c t i o n 1 we can e x t e n d t h e above theorem t o a l l elements of 7J\~ e x c e p t t h o s e whose s u b s e t s a l l have i n f i n i t e measure or measure z e r o , and we can a g a i n show t h a t such cases a r e l i m i t e d . 2.4 P r o p o s i t i o n . F o r any Ae#?~ , a t l e a s t one o f A,X ~ A s a t i s f i e s c o n d i t i o n (b) above. 2 . 5 P r o p o s i t i o n . E v e r y A e ( R e c t (7h) ) s a t i s f i e s c o n d i t i o n (b) above. I n v i e w of p r o p o s i t i o n 2 . 5 we can a d j u s t our measure so as t o e l i m i n a t e the p a t h o l o g i c a l cases and s t i l l have a T T - l i m i t o u t e r measure w i t h i n n e r a p p r o x i m a t i o n by <3 . 2 . 6 Theorem. L e t f o r e v e r y A X , jT'(A) = s u p ( i i ( B ) : B e A and [1(B) < <»}, t h e n : . 1 {j>' i s a s e m i f i n i t e T T - l i m i t o u t e r measure, .3 f o r e v e r y A e ^ / ^'(A) = sup{£ ' (c) : C e e , CcA}. 60 We now t u r n t o the problem of f i n d i n g when t h e r e e x i s t T T - l i m i t o u t e r measures w h i c h a re R a d o n - l i k e w . r . t . ( ^ ^ 1 ) f o r a p p r o p r i a t e f a m i l i e s ft", J>. We s h a l l seek such r e s u l t s when & i s e i t h e r S or . As p r e v i o u s l y s t a t e d (remark 2 . 2 ) t h e measures a r e R a d o n - l i k e w.r.t.(K i f t h e y a r e bounded. We t h e r e -f o r e c o n s i d e r f i r s t systems w i t h o n l y t h i s c o n d i t i o n added. 2 . 7 Theorem. I f f o r each i e l , ^ i s bounded, t h e n ja i s R a d o n - l i k e w. r . t . ( N , e, c_(S)). I t i s not as a r u l e p o s s i b l e t o e x t e n d theorem 2 . 7 t o the case i n w h i c h the measures are i n f i n i t e - see remark 1 . 8 . We n e x t t r y t o o b t a i n r e s u l t s s i m i l a r t o theorem 2 . 7 f o r the f a m i l y SQ. We ar e ".able t o do t h i s f o r some i n f i n i t e measures b u t we need o t h e r c o n d i t i o n s on the system. The f a c t o r s w h i c h govern our c h o i c e of c o n d i t i o n s a r e e s s e n t i a l l y t h e same as i n the t o p o l o g i c a l case (see d i s c u s s i o n p r e c e d i n g theorem 1 . 1 1 ) b u t i n a d d i t i o n we have t o r e s t r i c t t h e c a r d i n -a l i t y o f I t o a l l o w f o r t h e f a c t t h a t i s ' o n l y - N -compact.-2 . 8 Theorem. F o r each i e l , l e t be a a - f i n i t e o u t e r measure w h i c h i s R a d o n - l i k e w . r . t . ( N ,C>., A ) f o r some CC IL. ~L w i t h c_(c\) «= A c j z ^ . Then, whenever any one o f t h e .,. f o l l o w i n g c o n d i t i o n s h o l d s , t h e r e e x i s t s a T T - l i m i t o u t e r measure w h i c h i s R a d o n - l i k e w . r . t . ( & A,CQ , -a a + 1 ( R e c t U ) ) ) . 61 . 1 Card I ^ KQ. .2 Card I ^ N , and t h e r e e x i s t s a c o f i n a l s e t I N I w i t h Card I Q <: &Q and, p^ . [A ] e f o r e v e r y Ae<3 whenever i - \ j . . 3 Card I £ Na, and p. .[A] e f o r e v e r y Aec~ and p7 . [B ] e f o r e v e r y Be<3. whenever K j . 3 . P r o o f s . S i n c e theorems 1.3* 1.4, 1.5, 1-6 a r e s p e c i a l c a ses of 2 . 3 , 2 . 4 , 2 . 5 , 2 . 6 , r e s p e c t i v e l y we s h a l l prove o n l y t h e l a t t e r . P r o o f o f 2 . 3 ( 1 . 3 ) . L e t g* be t h e f i n i t e l y a d d i t i v e e x t e n s i o n of g t o CTQ(Rect(#0). Then a n(Rect(cO) i s an i n n e r f a m i l y f o r g* on a (Rect(#0) (from lemma D i n the p r o o f of theorem 2 . 6 - C h . I I l ) , and i s ^ - c o m p a c t by lemmas 3 - 5 - 1 and 3 . 4 . 3 i n Ch.I. The r e s u l t t h e n f o l l o w s f r o m theorem 2 . 4 I n C h . I I , b y c h o o s i n g CT (Rect(<3)) f o r the f a m i l y »<3« i n theorem 2 . 4 C h . I I . P r o o f o f 2 . 4 ( 1 . 4 ) . I f £(x) < t h e r e s u l t f o l l o w s f rom theorem 2 . 3 . O t h e r w i s e , f o r any i e l , t h e r e e x i s t s a sequence CQ,C.J,..'. e C-^ such t h a t , u i ^ X i ^ = s u P ^ j [ _ ( c n ) '• n £ U J} = 0 0 • a - f i n i t e on A A T T - 1 [ U C N ] or on (X ~ A ) ^ rr7 ' [ U C ], and .hence th e r e s u l t f o l l o w s e a s i l y f r om theorem 2 . 3 . Proof of 2 . 5 ( 1 . 5 ) . By lemma C i n the pr o o f of theorem 2 . 6 Ch.TII, f o r every AeRect(^) g(A) = sup{g (c) : C c A , ceRect ( c - ) } , and s i n c e Rect(C-) <= Rect(ftj) and ja(A) = g(A) f o r every A e R e c t ( ^ ) , then iI(A) = sup{[1(C) :• CeRect(e>), C<=A} which i m p l i e s c o n d i t i o n (b) s i n c e Rect(<3) <3 . Proof of 2 . 6 ( 1 . 6 ) . By theorem 2 . 7 Ch.II, 2 . 6 . 1 and 2 . 6 . 2 h o l d . To see 2 . 6 . 3 l e t keTn~', then jT'(A) = sup{£(B) : B«= A,' Be%~ , JT(B) < 0 0 } = sup{sup{{I(c) : CciB, C e S ] : Be^~ . |T(B) < ~ } = sup{£'(c) : C^A, CecT} s i n c e f o r every CeC , u ( 0 ) = i / ( c ) < oo. We next prove theorem 2 . 7 , which we w i l l use t o prove theorem 1 .7. ' 3.% 3.5 Proof of 2 . 7 . By lemma 3.3s, Ch.I, <3 i s N -compact. By theorem 2 . 3 cT i s an i n n e r f a m i l y f o r Z. so t h a t J ~ <c (lemma 2 , 2 , C h . I I ) , hence iT(G) = sup{[1(C) : C<^G, CecT} f o r e very G e J ~ . . C l e a r l y a l s o Jg* i s an outer f a m i l y f o r j l s i n c e f o r ev e r y A«=-X, there e x i s t s Be^~ , A c B w i t h u (T(A) = JI(B) and, s i n c e ja(x) < °°, f o r every £ > 0 there a e x i s t s Ce&, C c X ~ B w i t h ptX ~ B ~ C) < £. 63 so t h a t B c X ~CeJ<~and. £T(X ^ C) < jI(B") + £. P r o o f o f 1.7. From theorem 2.7 we see t h a t jo s a t i s f i e s the r e q u i r e m e n t s imposed on v i n the hypotheses o f theorem 2.5, Ch.II' ("by u s i n g ^ f o r the f a m i l y o f base elements 13, and n o t i n g t h a t <3 c o n s i s t s of s e t s w h i c h are c l o s e d and compact i n t h e t o p o l o g y 3, and & 7/i~). I f we l e t , as i n theorem 2.5, C h . I I , X = [A A i s c l o s e d and compact i n 3"} and h(G-) = sup{^T(K) : KeX, K ^ G ] , f o r Ge3_, t h e n h ( u ) = ia(u)v' f o r e v e r y U e J ^ (from theorem 1.3 s i n c e ^ c ^ ). I f we now l e t Y(A) = i n f { h ( G ) : Ge3, A c G}, t h e n , f r om theorem 2.5, C h . I I , ¥ i s Radon w . r . t . 3 and f o r e v e r y GeJg , Y(G) = h(G) = ij(G). C l e a r l y a l s o f o r e v e r y Cec^, Y(c) * (1(c). F o r any £ >0 and A e R e c t ( ^ ) , t h e r e e x i s t s by theorem 1.3 and t h e f a c t t h a t i s an o u t e r f a m i l y , CeC and G e J ^ such t h a t C ^ A ^ G and |T(C) + S * ~(A) '2: £(&) - £ , so t h a t ~T(G) = Y(G) 2: Y(A) 2: Y(C) 2: £(Q) - 2 £ . Thus Y(A) = j j ( A ) , and s i n c e c ^ , S <= $ , we have Aeflty and Y i s a T T - l i m i t o u t e r measure. To prove theorem 1.10 we w i l l need the f o l l o w i n g lemmas. 64 Lemma A. L e t f o r each I e l , be Radon w . r . t . A , and l e t K ^ K g e X w i t h K ^ K g = 0. Then ,u(K1 u K 2 ) = j!(K 1) 4- jKKg). P r o o f . ' L e t X ~ K = U £ where £ e R e c t (.£<). Then K p c U 13 1 n hence t h e r e e x i s t B n,B.,...B e 13 such t h a t K_ <=• U B and „ n ° 1 n n 2 m = 0 m . K 1 C X ~ A V S i n c e _ U A e V m = 0 1U m = 0 (K 1 u K 2) = pK.j) + ja(Kg) Lemma B. L e t v be a Radon measure on a t o p o l o g i c a l space Y w i t h t o p o l o g y %C. Then v i s c o m p l e t e l y d e t e r m i n e d by i t s v a l u e s on any base 13 w i t h a {B) = 13. P r o o f . F o r e v e r y c l o s e d compact, s e t C, v(c). = infCv(G)- : G open, C = C-}, = i n f {v(B) : Be/3, c <= B} s i n c e i f C <= Ge^, t h e r e e x i s t s B e ^ w i t h C«=BcQ. Thus the va l u e , on e v e r y c l o s e d compact s e t i s d e t e r m i n e d b y t h e v a l u e s on 13. But t h e v a l u e on e v e r y open set. i s d e t e r m i n e d b y the v a l u e s on t h e c l o s e d compact s e t s , and s i n c e the v a l u e of v on e v e r y s e t A«=Y i s . d e t e r m i n e d by t h e v a l u e on the open s e t s , v i s c o m p l e t e l y d e t e r m i n e d . P r o o f o f 1 . 1 0 . L e t <$ = [A : A e a 0 ( R e c t O ) and jT(A) < °o}. Then i n v i e w o f theorem 1 . 3 , we see t h a t t h e h y p o t h e s e s on v,/3,X In theorem 2.5 o f C h . I I , a r e s a t i s f i e d i f we r e p l a c e v (j., 13 b y o and X b y X. (Note t h a t s i n g l e t o n s i n X i have f i n i t e [i^ measure, hence <$ i s a base f o r J<). Thus ^ i s Radon w.r. t . % . 65 To see 1.10.2., 1.10.5 l e t ¥ be any T T - l i m i t outer measure which i s Radon w.r.t. A Then for. any GeJ' , Y(G) = sup{Y(K) : Ke^ and K<= G} p>(G) = sup {jj (K) ; Ke?T and K=G] £ £(£) and, s i n c e Y ^ jj (remark 2.4.3, C h . I I I ) , we get ¥(JB) * p* (J6) * pT(i&). But f o r every A e a ^ R e c t (7li)) we must have Y ( A ) = j j ( A ) , hence f o r e very Ee#, ¥(E) = £*(E) = 11(E) = sup { pM : KeK, K c E ) . 1.10.3 f o l l o w s immediately by lemma B. We w i l l use theorem 2.8 i n the proof of theorem 1.11, hence we prove theorem 2.8 f i r s t . The f o l l o w i n g lemmas w i l l be used i n the proof of theorem 2.8. Lemma C. I f , f o r each i e l , ^ i s a - f i n i t e and Radon-like w.r.t. (& ,<3., ) f o r some such t h a t c ( f i . ) c i , c l , , a i l 1 - — 1 ' 1 ~ o . . 1 then f o r every £ > 0 and . aeRect (7I\) there e x i s t s GeRect(j<) such t h a t a < G and ja(G ~ a) < £. . Proof. Let £ > 0 and aeRect (77i). Let n be the number of elements of J • , and f o r each j e J choose G .e Jr. such t h a t a . a 3 j-TT^ . [a] «• G . and n°3 ~ V a ] ) < 5 • : Then G = n_ T T ~ . 1 [ G . ] e R e c t ( j ) , j e j J J . a -66 and etc G. Then G ~ a «= U' TT"1 [G. ~ rr. [ a ] ] , hence ia'(G ~ a) £ 2 ,U(TT". 1[G, ~ TT, [ A ] ] ) < n~ = C . Lemma D. Let i<j<k , A<= X^ . and a = .. [p [ A ] ] e , then | i k ( A ~ p ^ [ a ] ) = 0 Proof- A ~ P T ^ [ a ] c ( p 0 p i k ) ~ 1 [a] ~ p"^ Ca], and, by d e f i n i t i o n 1.4, C h . I I I • u k ( ( p i j ° p j k r 1 [ a ] - P i k [ a 3 ) = 0-Proof of 2 . 8 . Suppose f i r s t t h a t Card I £ &a and l e t £ = {A : A e a Q ( R e c t ( ^ ) ) and j!(A) < «>}. We s h a l l check t h a t the hypotheses of theorem 2 . 6 , Ch.II are s a t i s f i e d w i t h "v" r e p l a c e d by "ja", "&0" and as above. (a) From lemmas 3 . 5 . 2 , 3 . 4 . 3 and 3 . 3 Ch.I, C-Q i s K -compact and c l e a r l y 3 Q = 6 ^ 5 ^ ) = oQ(&0). (b) To see t h a t £^J^, l e t G 0 5" = { II C. : C.e<3. f o r every i e l } . i e l 1 1 1 • Then f o r a n y . i e l and GeJ^, by lemma 3 - 5 . 3 , Ch.I TTT1 [ X ± ~ G] e ^ , hence C y l J ^ J , . By lemmas 1 . 9 - 3 * 1 .9 -4 of Ch.I, C y l J ' e J ^ . Then by lemmas 1 . 8 . 3 and 1 . 7 . 3 of Ch.I, a 0 ( R e c t U ) ) <= J% , hence 3 . C l e a r l y a l s o & = c r Q ( ^ ) = 6 Q(.fr). (c) To see t h a t c ( & 0 ) e o ^ ^ ) j l e t f o r each i e l , C^eC-and l e t C = n C.. Then f o r each i e l , i e : -1 e l x X ~ TT" 1 [ C 1 ] = TT" 1 CX i~ C ± ] e O^JB), s i n c e u- i s a - f i n i t e and X . ~ C . e i . . S i n c e Card I ^ N , ^ i i x i a J C = n TTT 1 [C . ] and i e l 1 1 we have Thus s i n c e a , ,(.fr) i s c l o s e d under f i n i t e i n t e r s e c t i o n s a+1 ( f r o m lemma 1.8.3, Ch.I, and the f a c t t h a t -fr = 6Q(.#)), we have £• (50> C aa + 1 ( ^ ' (d) To check t h a t f o r e v e r y De.fr, ,u(D) = sup{£(F) : F e ( j ~ * 7R~ ), F<=D}, o 0 p. _ -i n o t e t h a t , f o r e v e r y i e l and Cec-., TT. [C ] e Jp? , and t h a t 1 ' 1 °0 by lemmas 1.7, 1.8 of Ch.I, Hence £ = 6 1 ( a 0 ( R e c t ( e ) ) ) o c ^ , so t h a t by theorem 2.3, f o r e v e r y De.fr, |T(D) - sup{£(F) : F e J ~ , FeW~ , FcD}'. °0 ^ 68 (e) To check t h a t jj i s f i n i t e l y a d d i t i v e on <3Q, l e t C^CgeC^ be such t h a t C , ^ C 0 = 0. Then X ~ C,e -a .(-£), so t h a t t h e r e 1 c. 1 C t r l e x i s t s A w i t h Card £ .N and .X ~ C; = U £'. Then c*, t C g c U i ' , and s i n c e <=• &Q , t h e r e e x i s t D Q , D ^ , . . .D^e J>{ n 0 n n. such t h a t C„ cr U D . S i n c e U D e %~ and C, r\ U D = 0 , 2 m=0 m m=0 m . » 1 m=0 m we have rr(clW c 2 ) . = .pr(c1). + ;r(c2). Thus i n v i e w of ( a ) , ( b ) , ( c ) , ( d ) , (e) above,cand theorem 2 . 6 , C h . I I , i f we l e t . h ( G ) = sup{ £ ( c ) : CeCQ, C C G ) f o r e v e r y Gea ,.{£), and ^ a+1 Y(A) = i n f C h ( G ) : A c G , Gea L. (.#)}, a + 1 f o r e v e r y A X, t h e n Y i s an o u t e r measure on X, and i s Radon-, l i k e w . r . t . &Q, a a + 1 U ) ) / Y(G) = h(G) ^ jT(G) f o r e v e r y Gea and Y(c) ^ JT(c) f o r e v e r y Ce&Q. We show now t h a t Y i s a r r - l i m i t o u t e r measure' i f the f o l l o w i n g c o n d i t i o n i s s a t i s f i e d f o r e v e r y A e R e c t ( ^ ) , (*) j!(A) _= supine) : C e e 0 , C c A ) . C l e a r l y i f (*) h o l d s f o r a l l A e R e c t ( ^ ) , i t ' a l s o h o l d s f o r a l l A e a 1 (Rect ( % ) ) . Hence i f A e a Q ( R e c t ( A c G where G e a a + 1 ( ^ ) , and (*) h o l d s , t h e n h(G-) ^ jI(A) and t h e r e f o r e T(A) jJ(A). S i n c e A c a Q ( R e c t (in) ), we have h(D) = 11(D) f o r a l l DeA Thus' f o r any A e R e c t ( ^ ) , by lemma C, Y(A) ^ i n f Ch(D) : DeA, A c D} ^ £f(A), so t h a t Y(A) = u(A) =. sup{ £ ( c ) : C e g C c ' A } 69 £ (supY(c) : C e e 0 , C c A } = Y ( A ) , and s i n c e Y i s a - f i n i t e , t h e r e e x i s t s Cea^S^) w i t h C c A and Y(A ~ C) = 0 , so t h a t Ae??iy. Thus f i s a T T - l i m i t o u t e r measure i f (*) h o l d s . ( l ) Suppose 2 . 8 . 1 h o l d s . L e t coeRect(%>?) and l e t I = [ 1 Q , i , . . .}. Then, f o r t < j I ( a ) , choose b y r e c u r s i o n C ne such t h a t C c- rr^ [ a ] and n n n |j'(a* n. TT" 1 [C ]) > t m=0 -(by lemma A, C h . I I l ) and l e t C = n TT" 1 [C ]. new n Then CeC^, C <= a and £(c) > t . Hence (*) h o l d s , (b) Suppose t h a t 2 . 8 . 2 h o l d s . L e t cteRect(#0" and t<jj( a). Choose a c o f i n a l s u b s e t { 1 ^ , 1 ^ , . . . } o f I w i t h J ^ I Q f o r e v e r y j e J ^ and ± n < i n + 1 f o r e v e r y new. F o r each new, l e t A = n ' p"..1 [ T T . [ a ] ] , 1 1 ^ e J a ^ ° so t h a t H (A ) = £(a) > t , n and n+1 n n+1 By recursion, choose C e so that C A c A A , n • l . 0 0 ' n n c , „ <= A , „ a r i p . . [ c ] n + 1 n + 1 m=0 P V n + 1 m . 7 0 ana i M- > t . F o r any j e l , l e t n be the s m a l l e s t i n t e g e r "n w i t h j < i - and s e t K . = p .. [G ], and K = n K. . Then n 3 ^ n N J . i e l 1 K .e CV. and T T j [a] so t h a t Kec^ and K <= a. To check t h a t ja(K) ^ t , we f i r s t note t h a t f o r any f i n i t e J c l we have jT( n T T ~ 1 OK.]) > t . Indeed, f o r each j e J , l e t n . be t h e s m a l l e s t i n t e g e r w i t h J j < i and k=max n.. 'Then, f o r any j e J , C ~ p - J ' [K . ] = C^~ p".1 [p.. [C ] ] c V ^ i & 1 i 0 P i i j ° k ^ ^ k ^ n . ^ V A 3 3 so t h a t by lemma D, and t h e r e f o r e n T T " 1 [ K ]) = n ( n P ; ! OK.]) * n k (c k )> t . ' j e J J J • * j e J J 1 k J K . Let ft" = { H Q , ^ , . . . ) R e c t ( ^ ) be a c o v e r of K. By lemma C, g i v e n £ > 0 , f o r each new l e t G ne Rect(J') be such t h a t H n c G n and jT(Gn) £ (TCH^) + - | . S i n c e G Q ,G 1 , . . . eJ<~ , t h e r e m 2 ' m . ^ e x i s t s menu such t h a t K <=• U G 1 . L e t J = U J . Then J i s 1 = 0 1 1 = 0 u l * f i n i t e , and f o r any xe fl TT~J [K . ] t h e r e e x i s t s yeK w i t h y-= x . f o r a l l j e J . From lemma 2.4, Ch.I, we have J J xe n TT"1 [ { T f . ( y ) ) ] c G v • j e J J J x f o r some 1 = 0 , 1 , . . . m , 'so t h a t . m n TT". 1 [K ] <r u G. . j e J ^ ^ 1 = 0 • . 7 1 Thus m 2 jI(G, ) ^ u( U S, ). > t , lew ± 1-0 J" hence, 2 £(H n) * t - £ . ne UJ S i n c e £ i s a r b i t r a r y and f o r e v e r y HeRect (/??), g(H) = ja(H), we have f7(K) > t so t h a t (*) h o l d s . (c) F i n a l l y , suppose t h a t 2 . 8 . 3 h o l d s . L e t aeRect (Til) and k>j f o r e v e r y J^J . F o r t<jx( cc) choose Cec^. w i t h J a and such t h a t M ^ C ) > t . F o r any i e l l e t c_ = P i k C c ] i f i < k " 1 P i j fc^j CC]] f o r some j > i , j>k o t h e r w i s e . As i n 2 . 8 . 2 we see t h a t i f K= II C. t h e n KeCV , K c a and i e l 1 u ja(K) 2 : t , so t h a t a g a i n (*) h o l d s . P r o o f of 1 . 1 1 . From the d e f i n i t i o n of• 8 Q i n s e c t i o n 2 , i t f o l l o w s t h a t i n t h e t o p o l o g i c a l c a s e , i n w h i c h th e < 2 ~ are c l o s e d compact s e t s , S Q - c o n s i s t s o f c l o s e d compact s e t s . F u r t h e r m o r e , i f i n the h y p otheses of theorem 2 . 8 , the f a m i l i e s a r e t o p o l o g i e s , t h e n by t a k i n g a t o be a ' s u f f i c i e n t l y l a r g e o r d i n a l , a a + 1 ( R e c t U ) ) = J , where 2 i s the p r o d u c t t o p o l o g y . Thus theorem 2 . 8 shows t h a t under the h y potheses of.. 72 theorem 1 . 1 1 t h e r e e x i s t s a T T - l i m i t o u t e r measure v such t h a t % i s an o u t e r f a m i l y f o r v on X, and f o r e v e r y Ge2? , v(G)-= sup{v(K) : Ke%, K t f G } . F u r thermore '{G : G e R e c t O ) and V ( G ) < <*>} i s a base f o r 2, so t h a t s i n c e e v e r y Ke?^ i s c o n t a i n e d i n the u n i o n of a f i n i t e number of elements of t h i s b a s e, V ( K ) < 0 0 f o r e v e r y KeX . Hence v. i s Radon w . r . t . 2 . Then by theorem 1 . 1 0 . 3 , v=~*.. 7 3 CHAPTER V LIMIT MEASURES ON THE INVERSE LIMIT SET I n t h i s c h a p t e r we t r y t o answer the f o l l o w i n g q u e s t i o n s . When does a •TT-limit o u t e r measure e x i s t on t h e i n v e r s e l i m i t s e t , L j and what ' r e g u l a r i t y 1 c o n d i t i o n s can i t p o s s e s s (e.g. when i s i t Radon)? Our approach i s from the p o i n t o f v i e w o f r e s t r i c t i n g ja t o L. 1. D e f i n i t i o n s and N o t a t i o n . I n t h i s s e c t i o n we c o l l e c t d e f i n i t i o n s and n o t a t i o n used i n t h e s e q u e l . 1 . 1 B a s i c A s s u m p t i o n s . Throughout t h i s s e c t i o n we assume (X, p , j a , l ) i s an i.s.o.m. and P i k ' = P i o ° P j k whenever i < j <k , so t h a t ( x , p , l ) i s a c t u a l l y an i n v e r s e system of spaces. We a l s o assume t h a t the i n v e r s e l i m i t ' s e t , L-j-, i s such t h a t f o r e v e r y i e l , TT^CL-J-] = X^ ( s i m p l e m a x i m a l i t y ) . 1 . 2 D e f i n i t i o n s (Subsystems). F o r any d i r e c t e d s u b s e t J of I , • 1 (X,p,p.,j) w i l l denote the subsystem o b t a i n e d by r e s t r i c t i n g X and [i t o J and p t o { ( i , j ) : i<3 and i , j e j } . C l e a r l y (X,p, p., J) i s a l s o an I.s.o.m. 7-.2 X T = I X . . d i e J 1 ( i n case J=I we may w r i t e X f o r X^-.). . ~> L T = ( x e X T : TT. ( X ) = p. . ( T T . ( X ) ) whenever i - < j a n d J J x x j j i , j e J } . Thus L j i s . t he i n v e r s e l i m i t s e t of (X,p, [ x , J ) . . + r j i s the f u n c t i o n on X t o X j such t h a t f o r e v e r y xeX, r j ( x ) = x | J . .5 v i s an i n v e r s e l i m i t o u t e r measure f o r (X,p,u, J) i f f v i s an o u t e r measure on X j such t h a t : . 5.1 v i s c a r r i e d by L j , i . e . v ( X j ~ L j ) = 0, .5.2 f o r . e v e r y i e J and AeT^, T T" 1 [A] e D\v and V ( T T " 1 [ A ] ) = H ( A ) . {This i s e q u i v a l e n t t o the s t a n d a r d d e f i n i t i o n of i n v e r s e l i m i t measure.). I n the n e x t two d e f i n i t i o n s we i n t r o d u c e p r o p e r t i e s of the system and of measures wh i c h we w i l l use i n the theorems t o f o l l o w . 1.3 D e f i n i t i o n . (X,p , i x , i ) s a t i s f i e s s e q u e n t i a l m a x i m a l i t y i f f , f o r e v e r y c o u n t a b l e d i r e c t e d s u b s e t J of I , t h e range o f r j | i s ' a l l . o f L j , . i . e . : f o r e v e r y sequence i n J i , j . . . ' i n I w i t h i < i , „ and sequence y w i t h y eX. Cr 1 n n+1 ^ 0 •'n 1 n and p. . (y ) = y. f o r e v e r y neuu, t h e r e e x i s t s V n + I A l 1 n 75 x e L T such t h a t x^ . = y n f o r e v e r y new. ~n 1.4 D e f i n i t i o n . An o u t e r measure cp on a space S i s al m o s t s e p a r a b l e i f f t h e r e e x i s t s a c o u n t a b l e f a m i l y Be fn } and a s e t T<=s such t h a t cp(T) = 0 and f o r e v e r y x,yeS ~ T w i t h x/y t h e r e e x i s t s BeB w i t h xeB and y^B. I n many cases we w i l l w i s h t o work w i t h s e m i f i n i t e measures a l t h o u g h the measure produced by our e x t e n s i o n p r o c e s s may not be s e m i f i n i t e . The f o l l o w i n g d e f i n i t i o n i n d i c a t e s how we o b t a i n such a measure from t h e g i v e n one. 1.5 D e f i n i t i o n . I f v i s an outer.measure on a space S, v ' i s t h e s e m i f i n i t e o u t e r measure on S d e r i v e d f r o m v b y t a k i n g v'(A) = sup{v(B) : B c i and V(B) <«>},_ f o r e v e r y A c S. 1.6 Remark. I t f o l l o w s from theorem 2.7 C h . I I t h a t v ' I s i n d e e d a s e m i f i n i t e o u t e r measure. I f v i s i t s e l f a s e m i f i n i t e o u t e r measure t h e n v - v'. 2. E x i s t e n c e of an I n v e r s e L i m i t ^ M e a s u r e . I n t h i s s e c t i o n we c o n s i d e r the problem' of the e x i s t e n c e of an I n v e r s e l i m i t o u t e r measure. We b e g i n b y i n d i c a t i n g t h e r e l a t i o n between such a measure and the measure jj i n t r o d u c e d . i n 2\2. 1 C h . I I I . 76 2 . 1 lemma. An i n v e r s e l i m i t outer measure e x i s t s i f f ja | L j i s an i n v e r s e l i m i t outer measure. 2 . 2 Lemma. I f f o r each i e l , U- i s a s e m i f i n i t e outer measure, then ja | L^ i s an Inverse l i m i t outer measure ( i . e . such a measure e x i s t s ) i f f ja i s a T T-limit outer measure such t h a t the s e m i f i n i t e outer measure ja' d e r i v e d from ja i s p s e u d o - c a r r i e d by L j . For an example of the p a t h o l o g i c a l s e t s we a v o i d by c o n s i d e r i n g \x' see example 3 i n the appendix. In view of the above lemmas we devote the r e s t of t h i s s e c t i o n t o deter m i n i n g c o n d i t i o n s under which ja or ja 7 i s c a r r i e d or p s e u d o - c a r r i e d by L^. We have two types of c o n d i t i o n s under which t h i s occurs and we d i s c u s s them sep-a r a t e l y . F i r s t we c o n s i d e r " s e p a r a b i l i t y " c o n d i t i o n s . 2-3 Lemma. Suppose t h a t f o r every i e l , • ^ i s almost se p a r a b l e . .1 I f I i s countable, then ja i s c a r r i e d by L^. . 2 ' I f s e q u e n t i a l m a x i m a l i t y i s s a t i s f i e d and f o r each i e l , ^ i s s e m i f i n i t e , then j l ' i s p s e u d o - c a r r i e d by L-j-. 2A Remark. 2.3. 1 remains t r u e even i f (X,p,ja, I) i s an I.s.o.m. f o r which (X,p,l) i s not n e c e s s a r i l y an i n v e r s e system of spaces. In e f f e c t the r e s t of the hypotheses f o r c e Lj to be l a r g e enough t o c a r r y ja . • 77 By combining lemma 2 . 3 w i t h the fundamental e x i s t e n c e theorem 2 . 6 of C h . l l l we o b t a i n the f o l l o w i n g theorem. 2 . 5 Theorem. I f (X,p,|i, I) i s i n n e r r e g u l a r w.r.t.C f o r some C3 ( d e f i n i t i o n 2 . 5 Ch. I l l ) and i f f o r each i e l , i s almost separable, then an i n v e r s e l i m i t outer measure e x i s t s whenever one of the f o l l o w i n g c o n d i t i o n s h o l d s . .1 I i s countable, .2 (X,p,u,l) s a t i s f i e s s e q u e n t i a l maxirnality. 2 . 6 Remarks. P r e v i o u s l y known e x i s t e n c e theorems r e q u i r e f u r t h e r c o n d i t i o n s on the images and i n v e r s e images of the f u n c t i o n s p. . than are used i n theorem 2 . 5 (see e.g. Choksi [ 4 ] , Metevier [10]). In view of lemma 2 . 3 . 1 we can conclude t h a t i n 2 . 5 . 1 j l i s a TT-limit outer measure which i s c a r r i e d by L T and not j u s t p s e u d o - c a r r i e d . The f o l l o w i n g theorem shows t h a t t h i s i s not the case f o r any non-t r i v i a l system when Card I > KQ. _ _ V h 2 . 7 Theorem. i f 1 i s .countable, and X. c o n t a i n s at l e a s t two p o i n t s f o r uncountably many i e l , then f o r every A e R e c t ( ^ ) , Ia(A) = p(A ~ L I ) , hence jj Is not c a r r i e d by L T whenever ja r 0 . 78 From the above theorem we see t h a t i n many s i g n i f i c a n t c a s es where an i n v e r s e l i m i t o u t e r measure does e x i s t , L j i s no t ^-measurable. T h i s may e x p l a i n many of the d i f f i c u l t i e s e n c o u n t e r e d by i n v e r s e l i m i t measures. For.example, even when \X i s Radon, i t s ' r e s t r i c t i o n t o L j may n o t be. We now examine a n o t h e r t y p e o f c o n d i t i o n s under w h i c h an i n v e r s e l i m i t o u t e r measure e x i s t s . Here.we e s t a b l i s h a " t o p o l o g i c a l " r e l a t i o n s h i p between L j and i n n e r f a m i l i e s C j f o r the measures ( X j . C o n d i t i o n s s i m i l a r t o ours have been used b y p r e v i o u s w o r k ers (e.g. Bochner [2], C h o k s i [4], M e t e v i e r [10]) who worked o n l y w i t h L j (not c o n s i d e r i n g i t s r e l a t i o n w i t h X ) . The f o l l o w i n g theorem i s a b a s i c e x i s t e n c e theorem from t h i s p o i n t of view . 2.8 Theorem. Suppose t h a t t h e assumptions of C h . I I s e c t i o n 2 a r e s a t i s f i e d (see 2.1 C h . I V ) . I f f o r e v e r y sequence i e l i n I w i t h I < I. , , f o r a l l neuu, t h e f a m i l y n n~r i ' { TT~ 1 [C ] ^  Lj. : CeC. f o r some .new}: n n i s ^ - c o m p a c t , t h e n j j / ' i s p s e u d o - c a r r i e d b y L j , hence la'7 | L j i s an i n v e r s e l i m i t o u t e r measure. 2.9 Remarks. The h y p o t h e s e s of theorem 2.8 a r e o b v i o u s l y ' s a t i s f i e d i f the spaces C j are compact H a u s d o r f f , the measures U j Radon, and the f u n c t i o n s p^ . c o n t i n u o u s . I n t h i s case L j i s compact so t h a t { T T T 1 [ C ] A L X % CeC. f o r some i e l } 79 c o n s i s t s o f s e t s w h i c h a r e compact i n the p r o d u c t t o p o l o g y , and thus a r e c e r t a i n l y ^ - c o m p a c t . S i n c e i n o t h e r c ases i t may be d i f f i c u l t t o check the hypotheses o f theorem 2 . 8 d i r e c t l y , we g i v e i n t h e f o l l o w i n g theorem a c o n d i t i o n on c o u n t a b l e subsystems w h i c h , i f sequen-t i a l m a x i m a l i t y h o l d s , w i l l ensure t h e e x i s t e n c e o f an i n v e r s e l i m i t o u t e r measure. We s h o u l d n o te t h a t the- f o l l o w i n g s theorem i s e s s e n t i a l l y t h a t of M e t e v i e r [ 1 0 ] , though we i n c l u d e the s e m i f i n i t e case. 2 . 1 0 Theorem. Suppose t h a t t h e assumptions o f Ch.XV s e c t i o n 2 a r e s a t i s f i e d (see 2 .1 C h . I V ) , and t h a t s e q u e n t i a l m a x i m a l i t y i s s a t i s f i e d . Then \x' i s p s e u d o - c a r r i e d b y L j , hence fa1'! L^ . i s an i n v e r s e l i m i t o u t e r measure, whenever the f o l l o w i n g c o n d i t i o n s h o l d : i f 1 i s a sequence i n I w i t h i < i , . f o r e v e r y neuu, and n n~r i ^m= ^ p i I C e C i f o r some neuu w i t h rn'^n}, m n n . 1 t h e n % m i s K -compact f o r e v e r y meuu_, and .2 (p. . [{x} ] /\ K : KeX } i s K -compact f o r e v e r y 1 l I L m m u l,meuu w i t h K m , and xeX. . x l We c o n c l u d e t h i s s e c t i o n by i n d i c a t i n g how one can t r a n s f e r an i n v e r s e l i m i t o u t e r measure f o r a system t o one f o r a subsystem, and v i c e - v e r s a . 80 2.11 Theorem. Suppose v i s an i n v e r s e l i m i t outer measure. Then f o r any d i r e c t e d subset J of I, the set f u n c t i o n Y generated by the f a m i l y a = [ A c X j : f ~ 1 [A] e \ \ and the s e t f u n c t i o n h on CI 3 d e f i n e d by h(A) = v ( r ~ 1 [ A ] ) f o r a l l keC7, i s an i n v e r s e l i m i t outer measure f o r (X,p,u,J). 2.12 Theorem. Let J be a c o f i n a l subset of I. Then: -.1 r j | L-j- i s one-to-one and onto L j . .2 i f v i s an i n v e r s e l i m i t outer measure f o r (x,p,|j, j ) , the s e t f u n c t i o n Y d e f i n e d by ' Y(A) = v ( r j C A n L I ] ) f o r e very A^X-j-, i s an i n v e r s e l i m i t outer measure f o r (x,p,u,l). From the above theorems we see t h a t an i.s.o.m. has an i n v e r s e l i m i t outer measure i f i t can be imbedded"in a system which does have one, and t h a t theorems 2.5, 2.8 and 2.10 can be somewhat extended by r e q u i r i n g t h a t t h e i r hypotheses be s a t i s f i e d o n l y f o r a c o f i n a l subsystem.-3. Approximation P r o p e r t i e s of Inverse L i m i t Measures. We now t u r n t o the problem of f i n d i n g c o n d i t i o n s under,, which we can f i n d approximating f a m i l i e s f o r i n v e r s e l i m i t outer measures. The f o l l o w i n g lemmas show t h a t i t may be s u f f i c i e n t t o f i n d such f a m i l i e s f o r c o f i n a l subsystems. 8f 3 . 1 Lemma. L e t J be a c o f i n a l s u b s e t of I . . 1 I f <3 i s an ^-compact f a m i l y o f subsets- o f L j t h e n C L j ^ r j 1 [C] : Ce£} i s an ^-compact f a m i l y o f s u b s e t s o f L^. . 2 I f • v i s an i n v e r s e l i m i t o u t e r measure f o r (X,p, \i, J ) and i f C, & a r e r e s p e c t i v e l y i n n e r and o u t e r f a m i l i e s f o r v t h e n { L j A r j 1 [C] : CeC% C c L j ) and C L-j - r i r " . 1 [G-] : Ge^} u (X ~ L-j-} " a r e r e s p e c t i v e l y i n n e r and o u t e r f a m i l i e s f o r the i n v e r s e l i m i t o u t e r measure Y d e f i n e d b y Y(A ) = v ( r J [ A A L I ] ) f o r e v e r y A c X j . -3 . 2 Lemma. L e t J be a c o f i n a l s u b s e t of I . I f v i s an i n v e r s e l i m i t o u t e r measure f o r (X^p, ja, j ) w h i c h i s . R a d o n - l i k e w. r . t . ( N,<3, &) t h e n the o u t e r measure Y d e f i n e d by i Y(A ) = v C r j C A o L j ] ) f o r a l l A c i s an i n v e r s e l i m i t o u t e r measure f o r ;.(X,p, \x, i ) w h i c h i s R a d o n - l i k e w . r . t . ( ^ C - ^ l ' ) where o' = ( r ~ 1 [C] : CeC-}3 G-' = { r ~ 1 [G] : G e £ . 82 3-3 Remark. Note t h a t even i f a l l t h e X^ a r e t o p o l o g i c a l spaces and. v i s Radon w . r . t . the p r o d u c t t o p o l o g y o f X j , we cannot c o n c l u d e t h a t ¥ i s Radon w.r.t the p r o d u c t t o p o l o g y on X-j- s i n c e t h e images o f compact s e t s under -1 r j may n o t be compact. I n - t h i s case we c o u l d use some o t h e r t o p o l o g y as i n d i c a t e d i n 3.8 below and i n example 6 i n t h e appendix. We now t u r n our a t t e n t i o n t o d e t e r m i n i n g when an i n v e r s e l i m i t o u t e r measure i s Radon or R a d o n - l i k e . We b e g i n w i t h t h e case i n w h i c h we have t o p o l o g i c a l i n f o r m a t i o n about L j . 3.4 Theorem. L e t the ass u m p t i o n s 1.1 o f Ch.IV h o l d , and suppose t h a t f o r each i e l , f_i^ i s Radon. Suppose a l s o • t h a t L-j- i s a c l o s e d s e t i n the p r o d u c t t o p o l o g y . Then t h e s e t f u n c t i o n . p>'(see d e f i n i t i o n 1.9 Ch.rv) i s a Radon o u t e r measure s u p p o r t e d by L^. I t i s c l e a r t h a t . L j i s c l o s e d whenever t h e spaces X^ are H a u s d o r f f and the f u n c t i o n s p. . c o n t i n u o u s . ^ Thus, we can combine 3-4 w i t h 1.11 Ch.IV t o o b t a i n t h e f o l l o w i n g theorem. 3.5 Theorem. Suppose t h a t f o r each i e l , X^ i s a H a u s d o r f f space, U . i s a a - f i n i t e Radon o u t e r measure,, and p. . •'' i s c o n t i n u o u s whenever i-<j . Then j i * i s an i n v e r s e ^ l i m i t o u t e r measure w h i c h i s . Radon w . r . t t h e p r o d u c t t o p o l o g y whenever any one o f t h e f o l l o w i n g c o n d i t i o n s _ h o l d : 83 .1 There e x i s t s a c o u n t a b l e c o f i n a l s e t I Q C I . — 1 .2 p. .[A] i s compact whenever i < j and A i s a compact s u b s e t o f X^. 3.6 Theorem. Suppose t h a t I i s c o u n t a b l e and t h a t f o r each i e l , .1 X^ i s a t o p o l o g i c a l space, ' .2 [ i . i s a a - f i n i t e , a l m o s t s e p a r a b l e Radon o u t e r measure. Then fx* i s a Radon i n v e r s e l i m i t o u t e r measure w . r . t . the, p r o d u c t t o p o l o g y . N o t i c e t h a t the above theorem r e q u i r e s no c o n d i t i o n s on the f u n c t i o n s p^ . beyond t h o s e n e c e s s a r y f o r an i.s.o.m. (We c o u l d i n f a c t d i s p e n s e w i t h the f a c t t h a t (x,p,l) forms an i n v e r s e system of s p a c e s ) . We may now use lemma 3-2 t o e x t e n d theorem 3-6 t o .systems w i t h a c o u n t a b l e c o f i n a l s u b s e t , p o s s i b l y l o s i n g , however, t h e f a c t t h a t t h e measure i s a c t u a l l y Radon. 3-7 Theorem. Suppose t h e r e e x i s t s a c o u n t a b l e c o f i n a l s u b s e t J o f I such t h a t f o r each i e J , .1 X^ i s a t o p o l o g i c a l space, .2 i s a a - f i n i t e , a l m o s t s e p a r a b l e Radon o u t e r measure. Then t h e o u t e r measure v d e f i n e d b y v(A) = j > ( r j [ A * L j J ) " i s an i n v e r s e l i m i t o u t e r measure w h i c h i s R a d o n - l i k e w.r. t . C3 & where 84 C- = ( r [C ] : C c l o s e d and compact i n L j } . J>~ 1[G] : G ° P e n i n L j ^ -3 . 8 Remark. I n theorem 3 - 7 , we c o u l d use a t o p o l o g y w . r . t . w h i c h v i s Radon. One way o f d o i n g t h i s i s t o use the t o p o l o g y &' . F o r a n o t h e r approach w h i c h may he s u c c e s s -es f u l see example 6 i n t h e appendix. There t h e p r o d u c t t o p o l o g y i s t o o f i n e t o a l l o w any b u t t r i v i a l i n v e r s e l i m i t Radon measures. The f o l l o w i n g theorems a r e n o n - t o p o l o g i c a l a n a l o g u e s o f theorems 3-6" and 3 . 7 . 3 . 9 Theorem. Suppose t h a t I i s c o u n t a b l e and t h a t f o r each i e l , i s a c r - f i n i t e a l m o s t s e p a r a b l e o u t e r measure on X i w h i c h i s R a d o n - l i k e w . r . t . ( N ^ C ^ , A ) f o r some f a m i l i e s C j , (see d e f i n i t i o n 2 . 1 . 5 C h . I I ) . Then t h e r e e x i s t s an i n v e r s e l i m i t o u t e r measure which, i s R a d o n - l i k e w . r . t . ( K a , 6 1 ( a 0 ( R e c t ( e ) ) ) , a a + 1 ( R e c t U ) ) ) . 3 . 1 0 Theorem. Suppose t h a t t h e r e e x i s t s a c o u n t a b l e c o f i n a l s u b s e t J of I such t h a t f o r each i e J , i s a a - f i n i t e a l m o s t s e p a r a b l e o u t e r measure on w h i c h i s Radon- -l i k e w.r. t . ( ^ a>&^>\) • Then t h e r e e x i s t s an i n v e r s e l i m i t "'Outer measure w h i c h i s R a d o n - l i k e w . r . t . ( K a , w h e r e S = { r j H C ' l ^ L - j . : C<=Lj, Ce6., ( a Q ( R e c t (evl J ) ) ) y= { r ~ 1 [ G l n L j : G X j G e a ^ (Rect ( J < | j ) ) } u 8 5 I n the d e s c r i p t i o n o f C-' and above, (3 | J means the r e s t r i c t i o n of t h e system of f a m i l i e s o f s e t s C- t o th e s e t J , and | J i s s i m i l a r . The o p e r a t i o n s ***** i n v o l v i n g t h e s e f a m i l i e s t a k e p l a c e e n t i r e l y i n X j . k. P r o o f s . P r o o f of 2 . 1 C l e a r l y we need" o n l y t o show t h a t i f j ! | L j i s not an i n v e r s e l i m i t o u t e r measure t h e n no such-measure e x i s t s . To do t h i s we f i r s t e s t a b l i s h t h e f o l l o w i n g lemma. Lemma A. L e t aeRect(/?0, and k e l be such t h a t k>j f o r a l l j e J . I f we l e t ° a B = fl p " 1 [TT.CCO]], j e J ^ J t h e n , a ' . . 1 Be^., .2 • TT"1 CB ] nLx = a ^ L j , • 3 g(a) = f a k ( B ) . P r o o f . Immediate from t h e d e f i n i t i o n s . Suppose t h a t ja f L-j- i s n o t an i n v e r s e l i m i t o u t e r measure. We know f r o m lemma 2 . 3 . 6 C h . I I I t h a t Rect (57;) e , hence f o r ' e v e r y i e l and A e ^ , Thus i t must be t h a t f o r some j e l and Be771. ;r | L J . ( T T - 1 C B ] V g(TT~ 1 c B ] ) = ^ . ( B ) . S i n c e C B ] ) £ g(Tr-'! [ B ] ) 86 we must have L j U " 1 [B])= i ^ ( r r " 1 LB] A L X ) < ^ ( B ) . 0 Then b y t h e d e f i n i t i o n of. j j , t h e r e e x i s t s a c o u n t a b l e f a m i l y AeRect{n) such t h a t TT" 1 [B ] n L T c U £• <J *-* and 2 g(D) < g ( T T - 1 L B ] ) . De.fi- • J L e t 3-= {DQJD^J...} and f o r each neuu l e t i N E I and B ne#^ be such t h a t V L ] . = T T . 1 [ B n ] n L I n and g ( L n ) = [i± ( B n ) ( t h i s i s p o s s i b l e b y lemma A ) , n Then TT" 1 fe ] A L T c u TT? 1 [B„ ] A L and d new 2 JJ. (B ) < n [B] - ' " neuu n Hence t h e r e cannot e x i s t an o u t e r measure v c a r r i e d by L^ f o r w h i c h V ( T T . 1 [B]) = ^ ( B ) and V ( T T . " 1 [B ] ) = M . (B ) i n K i ^ n n n f o r e v e r y neuu, i . e . t h e r e cannot e x i s t an i n v e r s e l i m i t o u t e r measure. 8 7 P r o o f of 2.2 Suppose f i r s t t h a t u i s a i r - l i m i t o u t e r measure and t h a t j j ' i s p s e u d o - c a r r i e d b y L j . L e t AeT/i^. Then ^ T r " 1 [A]) = ^ ( A ) = sup(M i(B) : B<=A, BeTT^, ^ ( B ) < 0 0 } = s u p { j l ( r r j 1 [ B ] ) : , B C A , B e ^ , ^ ( B ) < 0 0 } Thus ^ ( A ) = 5 ' ( r r - 1 [ A ] ) = J ' d r ^ A ] , ^ ) = ja( rr7 1 [A ] ^  L-j.) = ja | L I ( n7 1 [A ] ) , and t h u s £T | L-j. i s an i n v e r s e l i m i t o u t e r measure. Now suppose t h a t pT J L-j- i s an i n v e r s e l i m i t ' o u t e r measure. Then s i n c e an:'.inverse l i m i t o u t e r measure i s a l s o a T T - l i m i t o u t e r measure i t f o l l o w s f r o m remark 2.4.2 C h . I I t h a t jj i s a T T - l i m i t o u t e r measure. Suppose a l s o t h a t t h e r e e x i s t s Ae#^,/ (=77h) such t h a t A c X ~ L-j- and 0 < JI,-(A). Then from the d e f i n i t i o n o f j ] } ' t h e r e e x i s t s a s e t B e A, B e ^ ~ ( s i n c e A e ^ j ) w i t h 0 < !I(B) < 0 0 . Then "by d e f i n i t i o n of jj t h e r e e x i s t s f o r 0 < £ < a f i n i t e f a m i l y A R e c t ( ^ ) w i t h jj( UJ9- ~ B) < £. and t< J T ( B ) - c < £( u^). Furthermore b y lemma 2.5.3 Ch.I we-can choose 3 t o be a d i s j o i n t f a m i l y . Then . ' 2 £(D* L T ) * Z jT(D ~ B) < £ . • DCS- De^ 88 B u t , s i n c e j j I L j i s an i n v e r s e l i m i t o u t e r measure, lemma A shows t h a t jT(D/* L T ) = jx(D) f o r e v e r y De„fr, hence 2 [ T ( D n L T ) = 2 pT(D) = £ ( U J > ) > £ . Dej^ x De.fr Hence no such s e t s A,B e x i s t , so jx' i s p s e u d o - c a r r i e d b y L-j.. P r o o f of 2.3 We f i r s t e s t a b l i s h t h e f o l l o w i n g lemma. Lemma B. L e t f o r each i e l , (j. be alm o s t s e p a r a b l e , and l e t I Q be a c o u n t a b l e d i r e c t e d s u b s e t o f I . Then {T(X ~ r ~ 1 [ L T ]) = 0. P r o o f . F o r each i e I Q l e t T ^ c X^ and 13^ «= ^  be such t h a t Hj(T^) = 0, 13^ i s c o u n t a b l e , and f o r e v e r y x,yeX^ ~ T^ w i t h x^y, t h e r e e x i s t s Be/3^ such t h a t xeB, y^B. F o r each i , j e I Q w i t h Kj and Be/9^ l e t B i 3 .- T , - 1 Q 3 ] . T T - ^ p - J ^ - B ] ] . Then, PT(B1;J) = g ( p " t [ B ] n p : l [ X . ~ B ] ) = g ( ^ ) = 0 f o r e v e r y such i , j . S i n c e X ~ r ~ 1 [L ] c U(B • i, j e l i < j and Be£ }. v U- TT"1 [T. ], j - 0 x Q I J u i i e I i 1 we have iltX ~ r ~ 1 [ L T ]) = 0. -o 0 Lemma 2 . 3 . 1 f o l l o w s i m m e d i a t e l y f r o m lemma B. ; To prove 2 . 3 . 2 we w i l l use the f o l l o w i n g lemma. Lemma C. L e t ^ be a l m o s t s e p a r a b l e f o r e v e r y I e l and l e t s e q u e n t i a l m a x i m a l i t y h o l d . Then f o r e v e r y a e R e c t ( ^ ) , ja( a) = jl( a r\ Lj). P r o o f . F o r £ > 0 l e t ft'cRect(^) be a c o u n t a b l e c o v e r o f a /> L j such t h a t 2 g(H) * jj( a A L T ) + £ , He,V ± and l e t T = U J„ v J and l e t K c l be a c o u n t a b l e d i r e c t e d He^ H a . s e t w i t h T c K . " By s e q u e n t i a l m a x i m a l i t y , f o r each x e a ^ T j t h e r e e x i s t s xs'e L-j- such t h a t f o r e v e r y 'k.eK^x^ = x^. Then rr .(x';/) e TT, [ a ] f o r e v e r y j e J , hence J J . cc sp b y lemma 2.4 Ch.I, n TT"1 [{X,7} ] c y y , keK hence xe and a n r ~ 1 [Lg] c U # . Thus, u s i n g lemma B, • H ( a ) * iT(U^) + [T(X - r " 1 ! ! ^ ] ) £ ja( a n L-j.) + £ + 0. S i n c e £. i s a r b i t r a r y , '{1(a) = jKa^L-j.). T u r n i n g now t o t h e p r o o f o f 2.3.2, suppose t h a t AeT/U, (=?/iJ) i s such t h a t A c x ~ L T and Q J ' C A ) > 0. Then as i n t h e p r o o f of lemma 2.2 t h e r e e x i s t s B e A, B e % ^ w i t h 0 < J J ( B ) < oo, a n d , , f o r 0 < £ < ^|^- a . f i n i t e d i s j o i n t ^ c R e c t ( ^ ) w i t h j!( ~ B) < £ and •pT(B) - e * (T(u^). Thus we have a g a i n 2 jT(D* L T ) * 2 pT(D ~ B) < £ , De.fr De.fr a,nd from lemma C, 2 • jj(D A L T ) = 2 pT(D) > ~T(B) - £ > E , De.fr " De.fr w h i c h i s a c o n t r a d i c t i o n . ' Hence no such A e x i s t s and j a ' i s p s e u d o - c a r r i e d by L j . P r o o f o f 2.5. By theorem 2.6 C h . I I I , j l i s a T T - l i m i t o u t e r measure, and by lemma 2.3, j ? 7 i s p s e u d o - c a r r i e d by L-j- under c o n d i t i o n s 2.5.1 or 2.5.2. Then lemma 2.2 shows t h a t ja | L j i s an i n v e r s e l i m i t o u t e r measure. P r o o f of 2.7 L e t A e R e c t ( ^ ) , AeRect(#0 be a c o u n t a b l e c o v e r of A ~ L T , and l e t T = U J„uJ, and xeA A L t . Then, s i n c e He^ -I I s u n c o u n t a b l e l e t i e l ~ T a n d y e X ^ be such t h a t y^x^. I f we d e f i n e x'e X by l e t t i n g x '. = x . f o r j / i and xi/=y, t h e n J J i x i ' e A ~ L j . Hence x 'e U U. Thus ' xe n r r ~ 1 [ { x , } ] = 0 TT". 1 [{x '} ] c U -jeT J J . jeT J J (from lemma 2.4 C h . l ) . .Hence A / > L - j - c U ^ s o t h a t A c and t h e r e f o r e from t h e d e f i n i t i o n of ja , ja(A - L j ) = p t A ) l 9 1 P r o o f o f 2 . 8 By theorem 1 . 6 Ch.IV, i s a r r - l i m i t o u t e r measure, and f o r e v e r y A e ^ / , pi'(A) = sup {£(c) . : C « A , CeC,}, where &• = 6 1 ( a Q(Rect ( c - ) ) . . Suppose B e ^ / , $ ' ( B ) > 0 and B c X ~ L j . Then t h e r e e x i s t s f o r t < J J " ( B ) a'sequence CQ , C ^ , . . . i n . o ^ R e c t ( < 2 - ) ) such t h a t c n + - | ^ c n f o r each new ( s i n c e Rect(c-) = 6 0 ( R e c t ( & ) ) ) , H C c B, and jj( fl C ) > t . F o r each new t h e r e e x i s t s b y l e new ' new lemma 2 . 5 . 3 Ch.I, a f i n i t e d i s j o i n t f a m i l y 13 R e c t ( ^ ) such t h a t C = 13 . Furthermore we can choose t h e f a m i l i e s 13, so n n t h a t i f m<n, e v e r y Be/? n i s a subset o f some element of 13 . L e t i n be such t h a t ln>3 e v e r y j e U J R and choose b y u u Be£ • r e c u r s i o n i , ,.e I so t h a t 1 , > i and i" , ,>• 3 f o r a l l n + 1 n + 1 • n n + 1 0 j e u J R . F o r each new l e t B e / ? n + 1 D.. = U 0 p " 1 [ r r . CB]]. n Be/3^ j e J B J 1 n J Then f o r each m,neuu w i t h m<n D <? p 7 1 , [D ] n 1 m m n and H ( D J = ^ Cm> > t • ' " m . ( t h i s f o l l o w s from lemma A above and lemma A i n . t h e p r o o f s o f C h . I I I , s i n c e we have ftcj = 2 ia(B) - 2 g(B) = ^  ( D ) ) / m Be/3 ' Be/3 V m m m • 9 2 L e t 0 < £ < |- and f o r each neuu choose K «: D n, K^e c, such t h a t (D ~ K ) < n 2 F o r each neuu l e t n _ 1 E = fl p. [K ]. n r, - ^ I x L m m=0 m n Then N - 1 E a» D' .  ~ U p. ' . CD ~ K • ] n n m=0 V n m m hence, ^i ( E n } a * " 2 > | . •^n n m=0 2 N + L ^ Thus, f r o m s i m p l e m a x i m a l ! t y n i 1 [ E n U L I ^ A l s o f rom lemma A i t i s c l e a r t h a t n T T " 1 C E „ ] A L t = n TTT1 CK ] A L n n x m=0 m 1 so t h a t n -1 -0 TT.'CK ] A L m=0 V m 1 f o r any ne uu, hence n TT - 1 CKr ] rtL ^ 0. meuu m B u t , f rom lemma A, f o r e v e r y n e w TT-^ C K ^ . L j C C ^ L j hence 0 j£ n C A L T « B A L T neuu n c o n t r a d i c t i n g the f a c t t h a t B « X ' ~ L j . . Hence no such B e x i s t s and j j ' i s - p s e u d o - c a r r i e d by L^. 9 3 P r o o f o f 2. 10 We s h a l l check t h a t t h e h y p o t h e s i s o f theorem 2.8 i s s a t i s f i e d . L e t i be a sequence i n I w i t h i < i ,, ^ n n+1 f o r e v e r y ne w, 7 = £ ^ 1 | C Q : Ce& , new}, n n and F be a sequence i n 3 w i t h H F ^ 0 f o r e v e r y new. We m=0 m have t o show t h a t fl F m ^ 0. me CD F o r each mew, l e t j(m) be t h e s m a l l e s t i n t e g e r k w i t h F m = r r - 1 [ C ] ^ L I f o r some Ce<3. , and l e t C e c\. be such t h a t X k • m ^ ( m ) F = TT-T 1 [C ] A L T . m i f c m J ' 1 I Le t K'• = f l ' f P i i CC.] : new and j ( n ) > m}. m j ( n ) S i n c e , f o r each mew, t h e f a m i l y *C = Cpi ± [C] : Cec3. and new, n^m} m n n i s ^ - c o m p a c t and, f o r each new n p [c ] = n •[ n r r " 1 re ] A L -\^0 , . .1=0 1 0 1 j ( l ) L x0 1=0 1 j ( l ) 1 1_ we see t h a t ^ 0. S i m i l a r c o n s i d e r a t i o n s , show t h a t f o r any new, K n / 0 . L e t x Q e K Q. Then T?l\ [ { x Q } ] A K / 0 ^ l 1 • o t h e r w i s e , b y c o n d i t i o n 2. 10-. 1, t h e r e would e x i s t m w i t h 1 m ' V i U 1=0 l 1 l j ( l ) 1 94 hence, m X C ^ 0 P i i M [ C 1 ] ' u 1=0 ^o^-jd) x so t h a t x A4 K n, c o n t r a d i c t i n g the c h o i c e o f x n . Thus t h e r e e x i s t s x ^ K 1 w i t h ( x ) = x Q , a n d b y s i m i l a r arguments we can choose, by r e c u r s i o n , f o r each neuu, x n e such t h a t whenever m^n.' L e t (by s e q u e n t i a l m a x i m a l i t y ) yeL-j. be' such t h a t y. = x f o r e v e r y neuu. C l e a r l y f o r -every neuu n y e n - n c ^ L j t h u s y e n ( T T , " 1 [ c J / i L T ) n e . \ L n X' so t h a t C T T " 1 [ C ] n L T : CeC- , new} n i s ^ - c o m p a c t . Thus t h e c o n d i t i o n s o f theorem 2.8 a r e s a t i s f i e d , hence j ! | L-j. i s an i n v e r s e l i m i t o u t e r measure. P r o o f of 2.11' I t i s c l e a r t h a t h i s c o u n t a b l y a d d i t i v e on C? and t h a t G i s a r i n g , hence Y i s an o u t e r measure on X j , and C?c7l\^ . Y i s s u p p o r t e d b y L j s i n c e ' r j 1 <XJ ~V c V Li- :; and s i n c e ( u s i n g T T f o r p r o j e c t i o n i n X T ) Y ( T T". 1[B ] ) = h ( T r " 1 [ B ] ) = V ( T T " 1 [ B ] ) = | X , ( B ) i f j e j and Be#?. , Y i s an i n v e r s e . l i m i t o u t e r measure f o r 95 ( X , p , u , j ) . P r o o f o f 2. 12. 1 Lmnediate f r o m the d e f i n i t i o n s . P r o o f o f 2. 12.2 L e t TT. denote p r o j e c t i o n onto t h e j c o o r d i n a t e from Then f o r e v e r y j e J , i < j and Beft^, * ( T T ~ 1 C B ] ) = Y ( T T ^ 1 t p ^ . [ B ] ] ) = V ( T T ~ 1 [ P ~ 1 [ B ] ] ) , hence Y ( n - 1 [ B ] ) = U j ( P i J CB ]) = ^ ( B ) . S i n c e r j i s 1:1 on L-j- and Y(x ~ L-j.) = 0 , Y i s an o u t e r measure and w i t h i , j , B as above, C B ] e flfy s i n c e Y I ^ C B ] ~ L-J.) = 0 and r J C n ^ 1 [ B ] r i L I ] = T T j 1 C p ^ . C B ] 3 n L j w h i c h i s i n . Hence Y i s an i n v e r s e l i m i t o u t e r measure. P r o o f o f 3.1 Immediate f r o m the d e f i n i t i o n s and theorem 2 . 1 2 . 1 . P r o o f of 3.2 Immediate-from t h e d e f i n i t i o n s and lemma 3 » 1 . P r o o f of 5.4 Rect(j') forms a base f o r the p r o d u c t t o p o l o g y on X. Hence t h e r e e x i s t s ^c.Rect(J') such t h a t X ~ Lx = U3. S i n c e f o r e v e r y i e l , c 7l\^} we have Rect ( £ ) c Rect {°fh) . . I t t h e n f o l l o w s from lemma 2 . 3 - 3 C h . I I I and s i m p l e m a x i m a l i t y t h a t f o r each BeB, g(B) '= 0 . L e t C c X ~ L^ he c l o s e d and compact. ' Then t h e r e e x i s t s a f i n i t e s u b f a m i l y B/CB such 96 t h a t C «• U/?'. Thus pr(c) * 2 g(B) = o, BeB and so h(X ~ L-j.) = sup{jj(c) : C = X ~ L j , C c l o s e d and compact}= 0. Then j>(X ~ Lj) = i n f (h(C-) : X ~ Lx G, G open] = 0, so t h a t jo* i s s u p p o r t e d by L^. By lemma- 1.10 Ch.IV, jo* i s Radon w . r . t . t h e p r o d u c t t o p o l o g y . P r o o f o f 3.5 By theorem 1.11 Ch.IV, j j * i s a Radon r r - l i m i t o u t e r measure ( w . r . t . t h e p r o d u c t t o p o l o g y ) . By theorem 3-4 i s s u p p o r t e d by L-j-. Hence j j * i s an i n v e r s e l i m i t ' o u t e r measure w h i c h i s Radon w . r . t . t h e p r o d u c t t o p o l o g y . P r o o f of 3.6 By theorem 1.11.1 Ch.IV, |? i s a T T - l i m i t o u t e r measure which, i s Radon w . r . t . t h e p r o d u c t t o p o l o g y . By lemma 2.3 above jj(X ~ L-j.) = 0. Hence from remark- 2.4.3 C h . I I I , ja*(X ~ L j ) = 0. Thus- £fr i s a l s o an i n v e r s e l i m i t o u t e r measure. P r o o f of 5.7 Immediate from theorem 3-6 and-lemma 3.2. P r o o f of 3.9 By theorem 2.8.1 Ch.IV, t h e r e e x i s t s a T T - l i m i t o u t e r measure w h i c h i s R a d o n - l i k e w . r . t . ( K , 6 , (a n(Rect(c<))), a , (Rect ( j ) ) ) . L e t v be such a measure. Then f o r every A X, v(A) < jj(A) (remark 2 . 4 . 3 C h . I I I ) . By lemma 2 . 3 . 1, jj(x ~ L-j-) = 0 , thus v(}T ~ L-j.) = 0 and v i s an i n v e r s e l i m i t outer measure. Proof of 3 - 1 0 Immediate from theorem 3 - 9 and lemma 3 - 2 . 98 APPENDIX (EXAMPLES) T h i s a p p e n d i x c o n s i s t s o f examples o f i n v e r s e systems of o u t e r measures w h i c h i l l u s t r a t e v a r i o u s p o i n t s such as the n o n - e x i s t e n c e o f r r - l i m i t o u t e r measures or i n v e r s e l i m i t o u t e r measures, or t h e i r r e l a t i o n s h i p t o the t o p o l o g i e s i n v o l v e d . R e f e r e n c e i s made t o t h e s e examples a t a p p r o p r i a t e p l a c e s i n t h e t e x t and a , b r i e f summary of the r o l e s o f each example f o l l o w s . 1. ' F o r t h i s system, no r r - l i m i t o u t e r measure e x i s t s . 2. A r r - l i m i t o u t e r measure e x i s t s b u t no i n v e r s e l i m i t o u t e r measure can e x i s t . 3- I n t h i s system the T T - l i m i t o u t e r measure |a i s n o t s e m i f i n i t e even though each i s s e m i f i n i t e . 4. Here the T T - l i m i t o u t e r measure jj i s n o t Rad6n w . r . t . the p r o d u c t t o p o l o g y , b u t t h e r e g u l a r i z - e d o u t e r measure {i* (see d e f i n i t i o n 1.9 Ch.IV) i s a T T - l i m i t o u t e r measure and i s Radon w . r . t . t h e p r o d u c t t o p o l o g y . 5. A T T - l i m i t o u t e r measure e x i s t s b u t no such measure can be •Radon w . r . t . the p r o d u c t t o p o l o g y , a l t h o u g h t h e f u n c t i o n s p a r e c o n t i n u o u s , i j " 6 . I n t h i s example b o t h a T T - l i m i t o u t e r measure and an i n v e r s e l i m i t o u t e r measure e x i s t , b u t n e i t h e r can be Radon w . r . t . t h e p r o d u c t t o p o l o g y , even though the system has an "upper bound" ( i . e . t h e r e e x i s t s j e l ' s u c h t h a t f o r e v e r y i e l ) and t h e f u n c t i o n s p. . a r e c o n t i n u o u s -e x c e p t a t one p o i n t i n each space X.. 99 Here, however, i f t h e t o p o l o g y o f complements o f c l o s e d compacts i s used, t h e n jj and ja | L a r e Radon and t h i s t o p o l o g y r e l a t i v i z e d t o the .inverse l i m i t s e t L i s e s s e n t i a l l y the one w h i c h would n a t u r a l l y be e x p e c t e d . The i.s.o.m. i n example 1 has no r r - l i m i t o u t e r measure, hence no i n v e r s e l i m i t measure. E s s e n t i a l l y t h e same example appears I n Halmos [ s ] , where i t i s used t o show t h a t i n d i r e c t p r o d u c t measures may n o t e x i s t . 1. Example. L e t n be Lebesque o u t e r measure, | j * Lebesque i n n e r measure, and l e t A Q , A ,... be a sequence o f p a i r w i s e d i s j o i n t s u b s e t s of [0,1] such t h a t f o r each neuu, u(A )= 1, [ i * ( A N ) = 0. Then l e t B m= U A N f o r each meuu (hence f o r m^1, n ( B m ) = 1, | i * ( B m ) =0). D e f i n e t h e i.s.o.m. (X,p,|j, i ) as f o l l o w s : l e t I = w(with t h e u s u a l o r d e r i n g ) , and f o r e v e r y j e l , l e t • X. = [0, 1 ], ^ = V I B. , .. ' p. . be the i d e n t i t y mapping whenever i ^ j . Then f o r e v e r y i e l , 7)\^ c o n t a i n s t h e B o r e l s e t s i n [0, 1 ], hence ^ i s s e p a r a b l e f o r e v e r y i e l . By lemma 2.3, Ch.V, the measure j l i s c a r r i e d by t h e i n v e r s e l i m i t s e t L, i . e . £( n X, ~ L) = 0. i e l 1 100 F u r t h e r m o r e . g ( T T ^ 1 ' C x I ~ B I ] ) = f i i ( X I ~ B I ) = 0, and, L={xe n X. : x , = x . f o r e v e r y I , j e l } = u r r " 1 [X. ~ B, ], i e l 1 1 d i e l 1 x x s i n c e U •( [0, 1 ] ~ B.) = [0, 1 ]. i e l x Thus, i!(L) £ 2 £ ( T T 7 1 C X , ~ B , ]) * 2 g ( r r 7 1 EX, ~ B, ]) = 0 i e l . 1 x- 1 i e l 1 1 1 so t h a t pT = 0. I t t h e n f o l l o w s (see remark 2.4.2 C h . I I l ) t h a t no ' r r - l i m i t o u t e r measure c o u l d e x i s t f o r (x,p,[j,l) hence a l s o no i n v e r s e l i m i t o u t e r measure. The n e x t example shows t h a t an i.s.o.m. may have a r r - l i m i t o u t e r measure w i t h o u t h a v i n g an i n v e r s e l i m i t o u t e r measure a l t h o u g h the system i s an i n v e r s e system of spaces 'and i t s i n v e r s e l i m i t s e t s a t i s f i e s s e q u e n t i a l m a x i m a l i t y . 2.' Example. L e t S he an u n c o u n t a b l e space and l e t BQ,B,J,... be a sequence of s u b s e t s o f S such t h a t f o r each ne uu, B . . e B . B^ i s u n c o u n t a b l e and D B = -0 . n + 1 n n n e u u n ;/ D e f i n e t h e i.s.o.m. ( X , p , j j , I ) as f o l l o w s : l e t I = uu ( w i t h t h e u s u a l o r d e r i n g ) and f o r e v e r y j e l , l e t 101 \i . be the o u t e r measure on X . d e f i n e d b y 3 3 u . ( A ) =• 1 i f Card A o B . > Kfl • I J - ( A ) = 0 o t h e r w i s e , J p. . be t h e i d e n t i t y mapping whenever i * \ J . 3- 3 F o r each j e l , {A<= X . : C a r d ( B . - A ) ^ *L} forms an ^ - c o m p a c t c l a s s w h i c h i s an i n n e r f a m i l y f o r Hence (theorem 2.6, C h . I I I ) ja i s a T T - l i m i t o u t e r measure. However f o r each i e l , g(-nr~1 CXj- B ± ] ) = 0 and s i n c e S = U (S ~ B.) i e l 1 . L = {ye 17 X . : y.= y. f o r a l l i , J e l } c u TT"1 LX, ~ B. ], i e l 1 ± J i e l 1 1 . 1 • hence i l ( L ) ^ 2 [TUT 1 CX,~B, ]) = 2 g U ? 1 [ X . ~ B , ]) = 0. i e l i e l 1 1 1 Thus jj | L i s n o t an i n v e r s e l i m i t o u t e r measure and so by lemma 2.1, Ch.V, no i n v e r s e l i m i t o u t e r measure e x i s t s , We now g i v e an example of an i.s.o.m. i n w h i c h a l l t he measures a r e s e m i f i n i t e (though n ot a - f i n i t e ) , and f o r w h i c h t h e r e e x i s t s a " p a t h o l o g i c a l " s e t A <= n X. such t h a t A has ~ i e l 1 I n f i n i t e ja-measure and a l l o f i t s s u b s e t s have i n f i n i t e ^-measure z e r o , so t h a t ja I s not s e m i f i n i t e . 102 Example. D e f i n e t h e i.s.o.m. (x,p,|j, i ) as f o l l o w s : l e t 1 = [0*11 ( w i t h t h e u s u a l o r d e r i n g ) and f o r each j e l , l e t X . = [0, 1 ], J ja - be c o u n t i n g measure on X . p. . be the i d e n t i t y f u n c t i o n whenever i ^ j . L e t A={xe .n X. : f o r .some ae(0, 1), x = 0 and x."= a f o r e v e r y i e l w i t h " ! ^ a}. To see t h a t jJ(A) = «>, l e t /?<=Rect(#?) be a c o u n t a b l e c o v e r of A, T = U " J r , and Be/? B D = (xeA : x = 0 f o r some a .4 T} a Then TT (D) = (0, 1) ~ T and, f o r e v e r y aerr.-^D), t h e r e e x i s t s xeD w i t h x^= a f o r e v e r y i e T , and Be/S1 w i t h xeB, so t h a t aerr.(B) f o r e v e r y J e J g , hence ae n p~.J [ T T . C B ] ] ) £ 2 g ( B ) , j e J B J'' J Be/? From t h i s we see t h a t TT (D)c u n p",J [TT, C B ] ] . 1 Be/3 j e J B Then s i n c e ^ ( D ) i s u n c o u n t a b l e , PO = n (TT C D ] ) ^ p ( u n p~!Crr,CB]]) ^ 2 g ( B ) , 4 4 Be/? jeCL J- 1 J Be£ and- so j J(A) = O O . We n e x t w i s h t o show t h a t e v e r y s u b s e t of A w h i c h has f i n i t e ja measure has i n f a c t ja-measure z e r o . F i r s t we note t h a t i f xeA and 103 C(x) = T T " 1 [ { 0 } ] A T r - 1 [ { x . } ] where i e l i s such t h a t x^= 0, t h e n C(x) e Rect(#?) and g ( c ( x ) ) = 0, so t h a t ja((x}) = 0. Now suppose t h a t E c A and [1(E) < a>. Then t h e r e e x i s t s a c o u n t a b l e f a m i l y W Rect(Tn) such t h a t E c U ^ a n d 2 g(H) < oo. F u r t h e r m o r e EeK we may assume t h a t TT^(H) c (0 , l ) s i n c e TT^  (E) c , ( 0 , ' 1 ) . L e t EeU and D W = ' 0 P " 1 [ T T . [ H ] ] . Then g ( H ) = M ^ C D J J ) < hence c o n t a i n s o n l y a f i n i t e number o f p o i n t s . Now, f o r each j e ( 0 , 1), t h e r e i s a unique xeA w i t h x .= 0, so i f J E ; / = {xeE' : x .= 0 f o r some j e U J„} 3 EeU t h e n E ' i s c o u n t a b l e and, f o r e v e r y xeE ~ Ei', HeV, and j e J H , we have x .= x^ . Thus i f x e ( E ~ El-';) n H, t h e n x i € DH' S i n c e D H ' i s f i n i t e and f o r e v e r y t e D H t h e r e i s a unique xeA w i t h x^= t , we c o n c l u d e t h a t (E ~ E . ' ) n H i s f i n i t e , hence E = E:'y. U ( ( E ~ E ') n H) , ' EeU i s c o u n t a b l e and t h e r e f o r e jj(E) = 0. From t h e e x i s t e n c e o f such a s e t ,A, i t f o l l o w s t h a t Jl i s n o t s e m i f i n i t e . 104 We next e x h i b i t a case i n w h i c h \± i s a T T - l i m i t o u t e r measure b u t i s not Radon w . r . t . the p r o d u c t t o p o l o g y . The measure jo* ( d e f i n i t i o n 1.9* C h . I V ) i s however, a T T - l i m i t o u t e r measure . - Note a l s o t h a t no i n v e r s e l i m i t o u t e r measure e x i s t s . 4. Example . F o r each j e ( 0 , 1) l e t B . = ( j , l ) . L e t (x , p , | j , l ) J be the i . s . o . m . d e f i n e d as f o l l o w s : l e t I = (0, . l ) , o r d e r e d b y , and f o r each j e l l e t X = (0 ,1) w i t h t o p o l o g y 3 = (X , ( 0 , j ] , ( j , 1), 0}, u . be the o u t e r measure on X . d e f i n e d b y , J J u . ( A ) = 1 i f A n ' B , / 0 p..(A) = 0 o t h e r w i s e , J p . - . be the i d e n t i t y f u n c t i o n whenever i ^ j . <J Then f o r each j e l 7n. = { A c r x i : A c ( 0 , j ] or ( j , 1 ] ^ A } . From t h i s i t ' i s e a s i l y checked t h a t . | j . i s Radon, hence b y theorem 2 . 6 , C h . I I l , jj i s a r r - l i m i t o u t e r measure . C o n s i d e r the se t B = I B . . B i s a p r o d u c t of c l o s e d i e l 1 compact s e t s hence c l o s e d and- compact . Suppose t h a t tfeRect(^) i s a c o u n t a b l e f a m i l y of s e t s such t h a t B <= U d. Then f o r any p o i n t xeB t h e r e e x i s t s ked w i t h x e A , hence f o r e v e r y J e J A * - - ' T T . ( X ) e rr. [ A ] , J j and an e x a m i n a t i o n of'5ft. shows t h a t B . C T T . [ A ] , «J d d so t h a t i n f a c t B<=-A. L e t : k = max C j : j e J f l } . 105 Then B(A) = u k( n E^EwjM]) n u. ]) " A " A = u k C B k ] = 1 . Thus ja(B) = 1. Now c o n s i d e r B l ' = nx. ~ B . ' i e l 1 L e t -frcRect(tf?) be a c o u n t a b l e f a m i l y of s e t s w i t h B ' ' c U ^ and l e t T = U J*. Then T i s " c o u n t a b l e , hence De.fr u t h e r e e x i s t s i e l w i t h i ^ T. L e t xeB 1' be such t h a t TT. (x) e X. ~ B . , and, f o r e v e r y j / i, T T . ( X ) e B .. Then X X X d o f o r some De.fr, xeD, so t h a t f o r e v e r y j e J ^ j T T . ( X ) e rr.[D]. By a p r e v i o u s argument i t f o l l o w s t h a t B . c TT. [D] f o r e v e r y j e J D , hence B<=- D so t h a t jx(D) = g(D) = 1, and hence jx(B/) = 1. Prom t h i s and the f a c t t h a t jI(B) = 1, i t f o l l o w s t h a t B 4" ^ ~ > hence j l i s n o t Radon w . r . t . t h e p r o d u c t t o p o l o g y . I t i s e a s i l y seen t h a t f o r e v e r y a e R e c t { % ) , |I( a) = 0 o r 1, and jl(a ) = 1 i f f B a. From t h i s i t f o l l o w s t h a t jx( a) = supCjx(K) :-.K c l o s e d and compact, K«=a), f o r e v e r y aeRect(#0, and t h a t e v e r y c l o s e d compact s u b s e t o f B ' has jj-measure z e r o . S i n c e B^ i s open, the r e g u l a r i z e d measure j i * ( d e f i n i t i o n 1. 9., Ch. IV") must be such 1 0 6 t h a t J J M B ' ) = 0. F o r e v e r y |3eRect(/ft), we have e i t h e r 3 c B ' o r B «• 3, so t h a t i t i s e a s i l y checked t h a t j ] * i s a T T - l i m i t ^ o u t e r measure. By theorem 1. 10, Ch.IV, i i * i s Radon w.r.t. t h e p r o d u c t t o p o l o g y . We note a l s o t h a t t h e i n v e r s e l i m i t s e t , L , i s a s u b s e t o f -1 U T T X [X 1 ~ B 1 ] new n n . n so t h a t -1 (T(L) ^ 2 g ( T T 1 J L ~ B 1 ] ) = o ne u) n n - hence no i n v e r s e l i m i t o u t e r measure e x i s t s . The f o l l o w i n g s i m p l e example shows t h a t t h e r e may be no T T - l i m i t o u t e r measure or i n v e r s e l i m i t o u t e r measure w h i c h i s Radon w . r . t . t h e p r o d u c t t o p o l o g y even when th e f u n c t i o n s p^-. a r e c o n t i n u o u s and t h e system i s o t h e r w i s e u n e x c e p t i o n a l . 5- Example. L e t ia be a Radon o u t e r measure on the r e a l l i n e R, such t h a t (j.(R) = 1, ja i s Rad6n and ja does not .have compact s u p p o r t (e.g. a G a u s s i a n measure). A l s o l e t • J be an u n c o u n t a b l e s e t . D e f i n e the i.s.o.m. ( X , p , | j , l ) as f o l l o w s : l e t I = f i n i t e s u b s e t s of J } , o r d e r e d b y i n c l u s i o n , • and f o r each k e l , l e t X, = n R, w i t h t h e u s u a l t o p o l o g y j • K t e k ii, be t h e p r o d u c t measure on I , 107 p i k b e P r o O e c " k l o n o f X k o n"fco whenever i ^ k . Then by theorem 2.6, Ch.III, i s a T T - l i m i t outer measure. Let C be a compact subset of J J X.. Then f o r i e l 1 . each j e j , TT [C ] i s compact. Let S i> 0 be such t h a t J C a r d C j e J : ^ . ( T ^ C C ] ) < 1 - £ } ^ N Q , a n d ' l e t J Q , ^ , . . . be a sequence i n Cj e J : • u.j( TT. [C ]) '< 1 - £ } For each ne UJ, l e t i n = C'V : a * n } and l e t n K = n T T " 1 [ T T. [C]] . m=0 Jm Jm Then g(K ) = H (TT CK ]) ^ (1 - £ ) n . n n Hence f o r any 6>0 th e r e e x i s t s meuo such t h a t g(K ). < 6, and s i n c e C c K , ja(c) = 0. Thus f o r any compact set K and T T - l i m i t outer measure Vj V(K) = 0, and thus no T T - l i m i t outer measure can be Radon f o r the product topology, and a l s o no i n v e r s e l i m i t outer measure can be Rad6n f o r the product t o p o l o g y ( i n t h i s case the i n v e r s e l i m i t measure i s a product measure). The next example shows t h a t an i.s.o.m.'may be such t h a t no T T-limit outer measure or i n v e r s e l i m i t outer measure may •-• be Radon w.r.t. the product t o p o l o g y even though the spaces X^ are t o p o l o g i c a l (and l o c a l l y compact), the measures ^ 108 a r e Radon, t h e f u n c t i o n s p. . are c o n t i n u o u s e x c e p t a t one p o i n t , and the system has i n f a c t an upper bound. However, i f we use the r e l a t i v i z e d t o p o l o g y o f complements of compact s e t s , we can check t h a t the r e s t r i c t i o n o f jj t o t h e i n v e r s e l i m i t s e t i s an i n v e r s e l i m i t o u t e r measure w h i c h i s Radon w . r . t . t h i s t o p o l o g y . Moreover t h i s t o p o l o g y i s e s s e n t i a l l y t h e t o p o l o g y we n a t u r a l l y e x p e c t on t h e i n v e r s e l i m i t s e t , namely t h a t o f t h e upper bound space. A l s o , theorem 1.6, Ch.IV shows t h a t a T T - l i m i t o u t e r measure e x i s t s w h i c h i s Radon w . r . t . t h e t o p o l o g y o f complements o f compact s e t s on n X.. i e l 1 6. Example. L e t S = [0,1) and A. be Lebesque outer, measure on S. D e f i n e t h e i.s.o.m. (X,p, j j / l ) . as f o l l o w s : l e t I = (0., 1 ] w i t h t h e u s u a l o r d e r i n g and f o r each j e l l e t X j - . S , nf V be the u s u a l t o p o l o g y on X, 3 C-^  the c l o s e d compact s u b s e t s o f S, " " . p ^ . be t h e f u n c t i o n d e f i n e d by P-,(x) = x + j - i (modulo 1), whenever i ^ j . Note t h a t p. . i s c o n t i n u o u s e x c e p t a t . -1- 3 S i n c e t h e s y s t e m ' i s a l s o an i n v e r s e system o f spac e s , and has an upper bound, we e x p e c t (S, x) to be t h e " l i m i t " . I t i s c l e a r t h a t t h e i n v e r s e l i m i t s e t , L, can be i d e n t i f i e d w i t h X,j = S b y t h e 1:1 mapping cp : L -* X^ d e f i n e d b y cp(x) = TT (x) = x. f o r e v e r y xeL, s i n c e ) . -109 L = [xe UX. : x. = p. .(x.) f o r a l l i , j e l w i t h i < j } i e l 1 3 3 ='{xe E X , : x ± = P i 1 ( x 1 ) f o r a l l i e l } . i e l F u r t h e r m o r e , i t i s n o t d i f f i c u l t t o see t h a t the o u t e r measure v on' E X . , g i v e n by v(A) = X ( c p[AnL]) f o r e v e r y i e l 1 A ^ n x . , i s c a r r i e d b y L and i s an i n v e r s e l i m i t o u t e r i e l 1 measure, w h i c h again' i s as we would e x p e c t ( i n f a c t v = fa | L ) . However t h e t o p o l o g y i n d u c e d on L b y t h e p r o d u c t t o p o l o g y i s much f i n e r t h a n t h e o r d i n a r y t o p o l o g y on S. I t i s i n f a c t e q u i v a l e n t t o t h e " h a l f - o p e n " i n t e r v a l t o p o l o g y on S. To see t h i s l e t seS and l e t i = s . .Then p^^^s) = 0, and f o r 0 < h < s, [0,h) i s a ne i g h b o r h o o d o f 0 , so t h a t c p ^ E p ^ L L O , ! ! ) ] ] = cp" 1 EEs,s+h) ] i s a n e i g h b o r h o o d of cp E ( s } ] . T h i s t o p o l o g y has a v e r y r e s t r i c t e d c l a s s . o f compact s e t s ( i n f a c t no compact s e t -can be uncountable)' and no c o n t i n u o u s n o n - z e r o measure can be Radon w . r . t . t h i s t o p o l o g y . I f , on t h e o t h e r hand, we use t h e t o p o l o g y 3" w i t h -&~ as a base (cT=6,lq1 (Rect ( c 3 ) ) ) , we i n d u c e a t o p o l o g y w h i c h c o r r e s p o n d s t o a more n a t u r a l t o p o l o g y on S. F o r -each i e l , i n t h e t o p o l o g y of complements of compact s u b s e t s o f X±, s e t s o f t h e form E0,h)•u'(1-h,1) form a -base f o r t h e neig h b o r h o o d system a t {0}. Thus f o r seS, ' s / 0, i f we l e t i=S and choose h s u f f i c i e n t l y s m a l l , we 110 o b t a i n L A TT"1 CCo,h) y (1-h, 1) 3 = cp"1 [pT] [fo,h) y (1-h, r) ]] = cp"1 [ ( s - h , s+h) ] mm A as a n e i g h b o r h o o d o f cp ( s ) i n the r e l a t i v e t o p o l o g y o f 3" t o L. Thus, s t a n d a r d neighborhoods of s i n the o r d i n a r y t o p o l o g y on S a r e l i f t e d i n t o n eighborhoods of cp ( s ) i n — A t h e new t o p o l o g y on L. F o r cp" (o), we o b t a i n neighborhoods A o f the' form cp [ L " 0,h ) u ( l - h , 1) ] f o r h > 0 , w h i c h a re s u f f i c i e n t l y d i f f e r e n t f r om the u s u a l n e i g h b o r h o o d s t o make t h e t o p o l o g y compact (and s t i l l H a u s d o r f f ) . L e t <$ be t h e f a m i l y o f open neighborhoods on L d e s c r i b e d above. We n e x t check t h a t <$ a c t u a l l y forms a base f o r the t o p o l o g y i n d u c e d on L by 3. From lemma A w h i c h f o l l o w s t h i s example we see t h a t = 2, hence = c(S0 = a 1 ( R e c t ( c 3 ) ) , so t h a t 3 has f o r a subbase C y l J , where f o r each i e l , J>/ i s any base f o r t h e t o p o l o g y c j e v ) on X^. I n p a r t i c u l a r we s h a l l t a k e f o r Jpf- t h e f a m i l y o f s e t s of t h e form [0,a) v (b,c) u ( c , 1 ) , where 0 a ^ b =s c ^ 1. F o r any such s e t G, and l e i , i t — 1 ~ i s c l e a r t h a t p ^ [GO i s open i n the u s u a l t o p o l o g y on X i= S and t h a t i f O e p ^ f e ] , t h e n P j ^ [G]" c o n t a i n s a s e t o f the f o r m [0,h) v (1 .h, 1) f o r some h>0. C o n s e q u e n t l y 9 t p i 1 [ G ] ] i s c o n t a i n e d i n t h e - t o p o l o g y g e n e r a t e d by t h e f a m i l y S d e s c r i b e d above, hence S i s a base ( i t i s c l e a r l y c l o s e d under f i n i t e i n t e r s e c t i o n s ) f o r t h e t o p o l o g y 3/ -i n d u c e d on L by 3. 111 I f we now d e f i n e t h e t o p o l o g y 3" on X b y 3^' = 3"' v (x ~ L} and e x t e n d v t o a l l of X by l e t t i n g v(x ~ L) = 0, t h e n v i s an i n v e r s e l i m i t o u t e r measure w h i c h i s Radon w . r . t . Note a l s o t h a t s i n c e L i s a c l o s e d s e t i n t h e p r o d u c t t o p o l o g y , any T T - l i m i t o u t e r measure w h i c h i s Radon w.r.t the p r o d u c t t o p o l o g y must be s u p p o r t e d b y L, (theorem J>.k, Ch.IV). S i n c e t h i s i s i m p o s s i b l e no such n - l i m i t o u t e r measure e x i s t s . However, theorem 1.5, Ch.IV shows t h a t t h e r e e x i s t s a " r r - l i m i t o u t e r measure w h i c h i s Radon w . r . t . 3". The.next lemma shows t h a t I n many p r o d u c t space s i t u a t i o n s we do n o t i n c r e a s e t h e number of s e t s under c o n s i d e r a t i o n by u s i n g 3 i n s t e a d o f <3. Lemma A. L e t (x,l) be a system o f spaces w i t h I u n c o u n t a b l e and l e t C be a system o f f a m i l i e s of s e t s w . r . t . ( x , l ) such t h a t f o r e v e r y i e l , i s -compact, {x} e <2^  f o r e v e r y xeX^ and t h e r e e x i s t s a sequence C Q , C ^ , . . . e <3^  such t h a t X± = U C.. Then i f we l e t . " neuu (3 = 6 1 ( a 0 ( R e c t ( c 3 ) ) ) , f " . we have 7 g = & . P r o o f . L e t Fe,7~ , F / 0. Then s i n c e f o r i e l , X. = U C 0 1 new n f o r some sequence C n,C i n <3., i f we l e t f o r each neuu, • 112 t h e n K e (3 f o r each neuu and F = U K . Then f o r e v e r y K n new n n t h e r e e x i s t f i n i t e f a m i l i e s d^ m ^ c Rect(C-) f o r e v e r y . me uu such t h a t n mew ^ n ' m ) L e t T = m,.ne w Aed, = U .II J A , (n.m) i e l ~ T and x^^e X^. Then rr" [{x^} ] Fec3 , hence t h e r e e x i s t f i n i t e f a m i l i e s 13 c R e c t ( c ) f o r e v e r y meuu such t h a t TT^ CB 3 = {x i} f o r e v e r y meuu and Be#m., and . TTT 1 [ { X I } 3 A P - = n u/? . • mew . F o r e v e r y meuu and Be/? m l e t • • B 7 = n TT" 1 [rr. [B ] ] j e ( J B ~ i ) 3 3 and 13^ - (B ' : B e ^ } . S i n c e f o r e v e r y mew and Be£?m we have B!'e R e c t ( e ) j we w i l l have shown t h a t FeC- i f we show t h a t F = n \JB' m mew L e t yeF. Then f o r some ne yeK^,, hence f o r each mew t h e r e e x i s t s A e d( \ w i t h yeA . L e t zeX he such t h a t m [o.,m) ° m z .= y. f o r j / i and z.= x. . Then .since i ^ J f l zeA . f o r e v e r y J J 1 1 A m m _ *i mew, and t h u s zeP, hence zerr^ [{x^} 3 F. C o n s e q u e n t l y f o r e v e r y mew t h e r e e x i s t s B- e 13 w i t h zeB , hence zeB ° m m irr m But s i n c e i ^ J g ' f ° r any mew, we must have yeB ' f o r e v e r y m m me mi, hence " • ye n B m ' . me w 113 Now l e t ye fl U/ ? ' . Then t h e r e e x i s t s f o r e v e r y mew, me uu m Bm e m^ y e Bm* H e n c e ^ a g a i n w e l e t zeX be such t h a t z .? = J ^ 1J and z. = x., we must have zerr" C{x.}] Bm = Bm f o r e v e r y m e c u - Hence z e fl and zeF. Then a g a i n t h e r e e x i s t s K such t h a t zeK , and hence f o r e v e r y meuu, A e a, \ w i t h zeA . S i n c e i l J * f o r any me uu, m in,m; m T A ° yeA^. Hence Y £ K n ^ and t h e r e f o r e yeF. Thus F= n UB1, ~ ^ me UJ M hence FecV and hence 3Q> C,. S i n c e C, i s c l o s e d under i n t e r s e c t i o n , C-C 3Q> and so 3^? = &. ( 1 1 4 BIBLIOGRAPHY 1. E. S p a r r e A n d e r s e n and B. J e s s e n , On t h e i n t r o d u c t i o n o f measures i n i n f i n i t e p r o d u c t s e t s , D a n s k i V i d . S e l s k a b Mat. Pys. Medd. 25 (1948) No.4. 2. S. Bochner, Harmonic a n a l y s i s and p r o b a b i l i t y t h e o r y , U n i v . o f C a l . P r e s s , B e r k e l e y , 1955-3. N. B o u r b a k i , T h e o r i e des Ensembles L i v r e I C h . I l l , Hermann, P a r i s . 4. J.R. C h o k s i , I n v e r s e l i m i t s of measure s p a c e s , P r o c . London .Math. Soc. , 8 (1958 3 2 1 - 3 4 2 . 5. P.R. Halmos, Measure t h e o r y , Van N o s t r a n d , New York, (1950). 6. M. J i r i n a , C o n d i t i o n a l p r o b a b i l i t i e s on s t r i c t l y s e p a r a b l e a - a l g e b r a ( R u s s i a n ) Czech. Math. J . T ( 7 9 ) ( 1 9 5 4 ) 3 7 2 - 8 0 . 7. J.L. K e l l e y , G e n e r a l Topology, Van N o s t r a n d , New York .'(1955). . 8. A.N. K o l m o g o r o f f , Grundebe gr i f f d er W a h r s . c h e i n l i c h h e i t ( B e r l i n , 1933), ( E n g l i s h t r a n s l o a t i o n : C h e l s e a , New York 1956). 9. E. M a r c z e w s k i , On Compact Measures, Fund. Math. 4 0 ( 1 9 5 3 ) 1 1 3 - 2 4 . 10. M. M e t i v i e r , L i m i t es p r o , j e c t i v e s de me sure s, M a r t i n g a l e s . A p p l i c a t i o n s . Ann. d i Mathematica 63, CT963) 225-352. 1 1 . P.A. Meyer, P r o b a b i l i t i e s and p o t e n t i a l s , B l a i s d e l l (1966). 12. J.P. R a o u l t , Sur une g e n e r a l i z a t i o n s d'uh theorem d ' l o n e s e a T u l c e a , C R . Acad. Sc. P a r i s 259( 1964), 2769-2772. 13- C.L. S c h e f f e r , G e n e r a l i z a t i o n s of the t h e o r y of Lebesque -•spaces and of t h e d e f i n i t i o n of e n t r o p y i n e r g o d i c t h e o r y , t h e s i s , U n i v e r s i t y of U t r i c h t \9Uo. 14. C.L. S c h e f f e r , P r o j e c t i v e l i m i t s of d i r e c t e d p r o j e c t i v e -systems o f p r o b a b i l i t y spaces ( t o a p p e a r ) . 15. M. S i o n , L e c t u r e Notes on Measure Theory, B i e n n i a l Seminar of t h e Canadian M a t h e m a t i c a l Congress, (1965). 

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