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Cylinder measures over vector spaces 1971

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-CYLINDER MEASURES OVER VECTOR SPACES by HUGH GLADSTONE ROY MILLINGTON B.S c , U n i v e r s i t y of West Indies, Jamaica, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of MATHEMATICS We accept t h i s t h e s i s as conforming to the required standard. The U n i v e r s i t y of B r i t i s h Columbia March 1971 In present ing th i s thes i s in p a r t i a l f u l f i lmen t of the requirements for an advanced degree at the Un iver s i t y of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f r e e l y ava i l ab le for reference and study. I fu r ther agree that permission for extens ive copying of th i s thes i s fo r s cho la r l y purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i ca t i on of th i s thes i s fo r f i nanc i a l gain sha l l not be allowed without my wr i t ten permiss ion. Department of The Un ivers i ty o f B r i t i s h Columbia Vancouver 8, Canada i i . S u p e r v i s o r : P r o f e s s o r M. S i o n ABSTRACT I n t h i s p a p e r we p r e s e n t a t h e o r y o f c y l i n d e r measures f r o m t h e v i e w p o i n t o f i n v e r s e systems o f measure s p a c e s . S p e c i f i c a l l y , we c o n s i d e r t h e p r o b l e m o f f i n d i n g l i m i t s f o r t h e i n v e r s e s y s t e m o f measure s p a c e s d e t e r m i n e d by a c y l i n d e r measure y o v e r a v e c t o r s p a c e X * F o r any su b s p a c e Q o f t h e a l g e b r a i c d u a l X s u c h t h a t ( X , n ) i s a d u a l p a i r , we e s t a b l i s h c o n d i t i o n s on u w h i c h e n s u r e t h e e x i s t e n c e o f a l i m i t measure on Q . F o r any r e g u l a r t o p o l o g y G on £2 , f i n e r t h a n t h e t o p o l o g y o f p o i n t w i s e c o n v e r g e n c e , we g i v e a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n on y f o r i t t o have a l i m i t measure on £2 Radon w i t h r e s p e c t t o G We i n t r o d u c e the c o n c e p t o f a w e i g h t e d s y s t e m i n a l o c a l l y c onvex s p a c e . When X i s a H a u s d o r f f , l o c a l l y convex s p a c e , and . £2 i s t h e t o p o l o g i c a l d u a l o f X , we use t h i s c o n c e p t i n d e r i v i n g f u r t h e r c o n d i t i o n s under w h i c h y w i l l have a l i m i t measure on £2 Radon w i t h r e s p e c t t o G We a p p l y o u r t h e o r y t o t h e s t u d y o f c y l i n d e r measures o v e r H i l b e r t i a n s p a c e s and J l ^ - s p a c e s , o b t a i n i n g s i g n i f i c a n t e x t e n s i o n s and c l a r i f i c a t i o n s o f many p r e v i o u s l y known r e s u l t s . i i i . TABLE OF CONTENTS Pages INTRODUCTION 1 CHAPTER 0: PRELIMINARIES 3 1. S e t - t h e o r e t i c N o t a t i o n 3 2. Outer Measures and I n t e g r a l s 4 3. Radon Measures 6 4. Induced Radon Measures 6 CHAPTER I : CYLINDER MEASURES OVER VECTOR SPACES 9 1. I n v e r s e Systems o f Measure Spaces 9 2. C y l i n d e r Measures o v e r V e c t o r Spaces 16 3. N o n - t o p o l o g i c a l L i m i t Measures 20 4. Radon L i m i t Measures 33 5. F i n i t e C y l i n d e r Measures 45 CHAPTER I I : CYLINDER MEASURES OVER TOPOLOGICAL VECTOR SPACES . 52 1. N o t a t i o n 52 2. E - t i g h t C y l i n d e r Measures 54 3. L i m i t s o f C o n t i n u o u s C y l i n d e r M easures 64 4. Induced C y l i n d e r Measures 74 CHAPTER I I I : APPLICATIONS 77 1. P r e l i m i n a r i e s 77 2. H i l b e r t i a n Spaces 82 3. N u c l e a r Spaces 94 5. £P-spaces. 101 APPENDIX: 114 1. S p e c i a l Measures on F i n i t e - d i m e n s i o n a l s p a c e s 114 2. P o s i t i v e - d e f i n i t e F u n c t i o n s on V e c t o r Spaces 126 3. CM-spaces 133 4. Examples 144 BIBLIOGRAPHY: 153 ACKNOWLEDGEMENTS I w i s h t o thank Dr. S i o n f o r h i s i n v a l u a b l e g u i d a n c e t h r o u g h o u t t h e w r i t i n g o f t h i s t h e s i s . I w i s h t o thank a l s o Dr. Greenwood and Dr. S c h e f f e r f o r t h e i r h e l p f u l s u g g e s t i o n s . F i n a l l y , I w o u l d l i k e t o than k M i s s B a r b a r a K i l b r a y f o r h e r a c c u r a t e t y p i n g o f t h i s t h e s i s . INTRODUCTION C y l i n d e r measures were f i r s t i n t r o d u c e d i n d e p e n d e n t l y by I.M. G e l f a n d ( G e n e r a l i z e d random p r o c e s s e s , [ 1 0 ] ) and K. I t o ( S t a - t i o n a r y random d i s t r i b u t i o n s [ 1 5]) as a more g e n e r a l k i n d o f s t o c h a s t i c p r o c e s s ( J . Doob [ 7 ] ) , and a r i s e n a t u r a l l y i n p r o b a b i l i t y t h e o r y when one d e f i n e s a s t o c h a s t i c i n t e g r a l ( [ 4 ] p. 137, [7] p. 426, [15] p. 2 1 1 ) . On t h e o t h e r hand, t h e demands o f t h e o r e t i c a l p h y s i c s ( i n p a r t i c u l a r , quantum f i e l d t h e o r y and s t a t i s t i c a l m e c h a n i c s ) have l e d t o a c o n s i d e r a b l e i n t e r e s t i n t h e t h e o r y o f i n t e g r a t i o n o v e r f u n c t i o n s paces ( [ 1 3 ] , I.M. G e l f a n d and, A.M. Yaglom [ 1 2 ] , I . S e g a l [ 4 3 ] ) , where t h e i n t e g r a l s c o n s i d e r e d a r e d e f i n e d w i t h r e s p e c t t o some c y l i n d e r measure ( e . g . as i n L. G r o s s [13] p. 5 3 - 5 4 ) . I n t he s t u d y o f c y l i n d e r measures r e s e a r c h e r s have concen- t r a t e d on two main a p p r o a c h e s : i n one, a c y l i n d e r measure i s v i e w e d as a l i n e a r map on a v e c t o r s p a c e i n t o t h e s p a c e o f m e a s u r a b l e f u n c t i o n s on some p r o b a b i l i t y space ( G e l f a n d [ 1 0 ] , I t o [ 1 5 ] , G e l f a n d and V i l e n k i n [11] Ch. IV , F e r n i q u e [ 9 ] ) ; i n t h e o t h e r , i t i s v i e w e d as a s e t f u n c t i o n on a f a m i l y o f c y l i n d e r s e t s o f a v e c t o r space ft ( M i n l o s [ 2 5 ] , G e l f a n d and V i l e n k i n [11] Ch. I V , P r o h o r o v [ 3 3 ] , B a d r i k i a n [ 1 ] , L. S c h w a r t z [ 3 9 ] ) . I n h e r e n t i n b o t h a p p r o a c h e s i s t h e n o t i o n o f an i n v e r s e ( o r p r o j e c t i v e ) s y s t e m o f measure spaces ( [ 2 5 ] p. 293, [11] p. 309, [33] p. 409, [1] p. 2, [3.9] p. 832, [9] p. 3 4 ) . I n t h i s t h e s i s we v i e w a c y l i n d e r measure as an i n v e r s e s y s t e m o f measure spaces i n d e x e d by the f i n i t e d i m e n s i o n a l s u b s p a c e s o f a v e c t o r space X . M o r e o v e r , we do so w i t h o u t any a p r i o r i c h o i c e o f the " t a r g e t " space ft on w h i c h the l i m i t measure i s t o l i v e . The b a s i c p r o b l e m o f f i n d i n g a l i m i t o f the s y s t e m on a space ft o f l i n e a r f u n c t i o n a l s i s t h e n a n a l y z e d w i t h v a r i a b l e ft i n C h a p t e r I . The key i d e a t h e r e i s t o examine t h e measure t h e o r e t i c s i z e o f ft i n r e l a t i o n to t h e a l g e b r a i c d u a l X . To t h i s end the n o t i o n o f " a l m o s t " s e q u e n t i a l m a x i m a l i t y i s i n t r o d u c e d . N e x t , i n C h a p t e r I I , we c o n s i d e r the more s t a n d a r d p r o b l e m o f f i n d i n g a Radon l i m i t measure on ft when X i s a t o p o l o g i c a l v e c t o r s p a c e and ft i s i t s t o p o l o g i c a l d u a l . When X i s H a u s d o r f f and l o c a l l y convex, by i n t r o d u c i n g the c o n c e p t o f a w e i g h t e d s y s t e m i n X , we e s t a b l i s h a c o n d i t i o n f o r t h e e x i s t e n c e o f such a Radon l i m i t i n terms o f the n o t i o n o f c o n t i n u i t y w i t h r e s p e c t to a w e i g h t e d s y s t e m . I n C h a p t e r I I I we a p p l y the t h e o r y o f C h a p t e r I I t o t h e s t u d y o f c y l i n d e r measures o v e r H i l b e r t i a n , n u c l e a r , and £^-spaces, t h e r e b y e x t e n d i n g and c l a r i f y i n g s e v e r a l p r e v i o u s l y known r e s u l t s . I n t he a p p e n d i x we e s t a b l i s h m a i n l y t e c h n i c a l r e s u l t s used i n the p r o o f s of C h a p t e r I I I and p r e s e n t s e v e r a l c o u n t e r - e x a m p l e s . 3. CHAPTER 0 PRELIMINARIES 1. S e t - t h e o r e t i c N o t a t i o n . I n t h i s work we s h a l l use t h e f o l l o w i n g n o t a t i o n . (•1) 0 i s the empty s e t . F o r any s e t s A and B , A ~ B = {x £ A : x i B} . w i s the s e t o f f i n i t e o r d i n a l s . R i s t h e f i e l d o f r e a l numbers. R + = { t e R : t _> 0} . <£ i s the f i e l d o f complex numbers. I n p r o o f s we s h a l l a b b r e v i a t e " s u c h t h a t " t o " s . t . " (2) F o r any s e t X and f a m i l y H o f s u b s e t s o f X , - LJH = u H , ]~\H = n H , HeH HeH P(H) = {H'C. H : H' i s c o u n t a b l e , d i s j o i n t , and = [_|H} . F o r any A c X , H | A = {HA A : H E ff} . (if i s a compact f a m i l y i f f f o r any n" c H , i f A c h " i s f i n i t e => (I H , th e n [\H* ^ 0 . 4. For any topology G on X , K(G) = {K c X : K i s closed and compact i n G} (3) For any set X and A c x > 1, : x e X -> 1 e <E i f x e A , A O e S i f x e X ~ A , For any f : X -> Y , B c Y , f|A : x E A + f(x) e Y , f [A] = {f (x) : x e A} f _ 1 [ B ] = {x e X : f(x) e B} For any sets X and Y , I c X x Y , x e X , y e Y , I x = (Y e Y : (x,y) e 1} , I Y = {x £ X : (x,y) e l } . 2. Outer Measures and Integrals. Our measure-theoretic approach i s e s s e n t i a l l y that of Caratheodory, as given by M. Sion i n [44] and [45]. (1) For any set X and Caratheodory measure n on X , M i s the family of n~measurable sets. n i s an A-outer measure i f f A c. M , and for any A c X , n ^ ( A ) = inf{n(A') : A d A' e rt} n i s an outer measure i f f n i s an M -outer measure. n Throughout this work a l l measures considered w i l l be outer measures, n i s the C a r a t h e o d o r y measure on X g e n e r a t e d by T and A i f f A i s a f a m i l y o f s u b s e t s o f X w i t h 0 e A , + T : A -> R w i t h T(0) = 0 , and f o r any B C X , n(B) = i n f { E T(H) : f / c A i s c o u n t a b l e , B c r | j H } . HeH (X,n) i s a measure space i f f X i s a s e t and n i s an o u t e r measure on X (2) I n t e g r a t i o n . We o b s e r v e t h a t f o r any measure sp a c e (X,n) , P(M ) i s d i r e c t e d by r e f i n e m e n t . I n g e n e r a l , we s h a l l be c o n s i d e r i n g c o m p l e x - v a l u e d f u n c t i o n s on X and t h e r e f o r e a l s o c o m p l e x - v a l u e d i n t e g r a l s . However, we p o i n t o u t t h a t f o r any n-measurable f : X -> R + , ' / f d = l i m I ( i n f f [ B ] ) * n ( B ) n PeP(M ) BeP n l i m E' (sup f [ B ] ) - n ( B ) . PeP(M ) BeP F u r t h e r , f o r . any f : X ->• R + , the o u t e r i n t e g r a l / * f d = l i m Z (sup f [ B ] ) ' n ( B ) n PeP(M ) BeP n i s a w e l l - d e f i n e d p o i n t i n R U {°°} 6. (3) Radon Measures I n t h i s p a p e r , many o f the measures we c o n s i d e r w i l l i n f a c t be Radon measures. We g i v e t h e r e l e v a n t d e f i n i t i o n s b elow. F o r any s e t X and t o p o l o g y G on X , ri i s a G-Radon measure on X i f f ( i ) n i s a G-outer measure on X , ( i i ) K e K(G) => n(K) < ». , and f o r e v e r y G e G , ( i i i ) n(G) = sup'{n(K) : K C G , K e K(G)\ . F o r any G-Radon measure n on X , supp n = s u p p o r t o f n (4) I n d u c e d Radon Measures L e t Y be an a b s t r a c t s pace. F o r any f i n i t e measure space (X,n) and T : X -> Y , T[n] i s the C a r a t h e o d o r y measure on Y g e n e r a t e d by n 0 T - 1 and { A C Y : T _ 1 [ A ] e M^} We s h a l l use the f o l l o w i n g lemmas. Lemmas (1) F o r any A e M ^ - j > T 1 [ A ] e M and t h e r e f o r e T[n](A) = n(T 1 [ A ] ) 7. (2) For any space Z and U : Y -> Z , U[T[ n]] = (U 0 T)[n] (3) If X and Y are t o p o l o g i c a l spaces, T i s continuous, and r\ i s Radon, then T[n] i s Radon. Proof of Lemma 4.1. Let & = ( A c ! : T _ 1 [ A ] E M } . n F i r s t we note that for a l l A e A , T[nJ(A) = n ( T _ 1 [ A ] ) . Let A e Hpj- j • Since A i s a a - f i e l d and T[nJ i s an . A-outer measure, there e x i s t s A' e A s.t. A c A' and T[n](A) = T [ r | ] ( A ' ) . If T[n](A) = 0 then (1) 0 < n ( T _ 1 [ A ] ) <_ n ( T _ 1 [ A ' J ) = T [ n ] ( A ' ) = 0 . In general, since T[n](Y) < 0 0 , T[n](A' ~ A) = 0 and therefore by (1), n(T~ 1[A' ~ A]) = 0 and T _ 1 [ A ' ~ A] e M n Hence, since T ^[A'] e M , T _ 1 J A ] = T _ 1 [ A ' ] - T _ 1 [ A ' ~ A] e M n The second a s s e r t i o n now follows immediately from the f a c t that T[n]|A = n o T " 1 . 1 Proof of Lemma 4.2. We need only observe that, by 4.1 above, ' { B C Z : (U 0 T ) _ 1 [ B ] e M̂ } = {B C Z : U _ 1[B] e and for any B CZ Z s.t. (U 0 T ) _ 1 [ B ] e , n((U o T ) _ 1 [ B ] ) = T [ n ] ( U _ 1 [ B ] ) . I Proof of Lemma 4.3. Let A e M T r n ] a n c i £ > 0 . By Lemma 4.1, T _ 1 [ A ] e , and therefore, since n i s a f i n i t e Radon measure, there e x i s t s a compact K c T "*"[A] s .t. n(T_1[A.]) - n(K) < e . ' • Then T[K] c A i s compact and T[n](A) - T[ n ] ( K ) <.n(T _ 1[A]) - n(K) < e Hence, since e > 0 was a r b i t r a r y , (1) T[n](A) = sup{T[ri](C) : C C A i s compact} . Since T[n] i s f i n i t e i t then also follows that T[n](A) = i n f {T[n] (G) : G ^ A i s open} , and since T[n] i s an outer measure we therefore conclude that for a l l B C Y , (2) T[n](B) = i n f {T[n](G) : G O B i s open} . consequently T[n] i s Radon. ® CHAPTER I CYLINDER MEASURES OVER VECTOR SPACES As indicated i n the introduction, we s h a l l treat cylinder measures as being s p e c i a l inverse systems of measure spaces (Choksi [6]). In the following section we introduce the basic notions and r e s u l t s that we s h a l l require about such systems. 1. Inverse Systems of Measure Spaces. Throughout t h i s section, F i s an index set directed by a r e l a t i o n < For any E e F , (Xp,y ) i s a measure space, E For any E and F i n F with E < F r_, „ : X_, X„ i s s u r j e c t i v e , L, r r L with being the i d e n t i t y map. 10. D e f i n i t i o n s (X , p ) E i s an inverse sj'Stem of measure spaces r e l a t i v e r r c £ r to the maps r , E, t i f f , f o r any E,F , and G i n F w i t h E < F < G , rE,G = r E , F ° rF,G ' and, for a l l A e M , r ~ ^ [ A ] £ M F , y F ( r ^ F [ A ] ) = u E ( A ) L e t (XLpjPp)^,^^ be an i n v e r s e s y s t e m o f measure s p a c e s r e l a t i v e t o t h e maps r E, b I f ft i s a s e t , and f o r each F e F , p_ : ft -> X „ i s s u r j e c t i v e , t h e n , we c a l l (ft,0 a r r l i m i t r e l a t i v e t o t h e maps p o f t h e g i v e n i n v e r s e s y s t e m o f F measure s p a c e s , i f f f o r each E and F i n F w i t h E < F P E = r E , F ° P F ' and, F i s an o u t e r measure on ft such t h a t f o r a l l A e M , E p'V] e M , ap'V]) = U E ( A ) . F o r the r e s t o f t h i s s e c t i o n we assume t h a t (X , p ) r r r r £ r i s an i n v e r s e system o f measure s p a c e s r e l a t i v e t o t h e maps r . b,.b 1.2 D e f i n i t i o n s For any set ft , and s u r j e c t i v e maps p : ft ->• X such F F that f o r any E and F i n F with E < F , P E = rE,F ° P F ' '(1) Cyl(ft, P) = {P~V] : F e F, A e Mp} , (2) T(ft,p) : p ' 1 ^ ] e Cyi(n,p) -»• u p(A) e R + , (3) n„ i s the Caratheodory measure on Q generated by ft, p T _ and Cyl(ft,p) . ii, p When there can be no ambiguity we s h a l l omit the subscripts ft and p Remarks The following assertions are r e a d i l y established. (Choksi [6], Mallory and Sion [23]). (1) Cyl(ft,p) i s a f i e l d . (2) T i s well-defined and i s f i n i t e l y a d d i t i v e on Cyl(ft,p) (3) Cyl(ft, P) c M r . (4) (ft,n) i s a l i m i t r e l a t i v e to the maps p of the given r inverse system of measure spaces i f f n|Cyl(ft,p) = x . (5) There e x i s t s an outer measure E, on ft such that (ft,£) i s a l i m i t r e l a t i v e to the maps p of the given inverse system r of measure spaces <=> 12. T i s countably a d d i t i v e , i n which case n i s such a measure. We now suppose that (ft>n— —) i s a l i m i t r e l a t i v e to the maps p , ft,p F ft C ft , and f o r each F e F , Pp = Pp|ft i s s u r j e c t i v e . We s h a l l be interested i n determining when (ft,n n ) i t s e l f " 5 P i s a l i m i t r e l a t i v e to the maps p F 1.3 Lemmas. With the above notation and hypotheses, (1) ) i s a l i m i t r e l a t i v e to the maps p of the given ft, p F inverse system of measure spaces i f f rrr- ( A ) = n— —(A f\ ft) for a l l A e C v l (ft, p) . Si,p ft,P v ' (2) For any F'c F , l e t A(F') be the set of a l l f e ft such that there does not exist g e ft with Pp(g) = P F ( f ) for every F e F' . I f , f o r every {F } C F with F < F for each n e w n new n n+1 then n--(A({F } )) = 0 , ft,p n new n--(A) = rnr-(A O ft) f o r a l l A e Cyl(ft,p) Proofs 1. Let Cyl(fl.p) = Cyl , Cyl(n,p) = Cyl , T Tft,p ' T Tft,p ' ft, p ft,p 1.3.1 Since the maps p are s u r j e c t i v e , f o r each A i n C y l , — — — Q there e x i s t s a unique A e Cyl s.t. A = A A ft • Then, A e C y l -> A e Cyl i s b i j e c t i v e and T(A) = 7 ( A ) f o r a l l A e Cyl . Hence, for any B o fi , n ( B n f i ) = i n f { _ E _ 7(H) : He Cyl i s countable, H e H B n f t c j j H } = inf{ E T(H) : H e Cyl i s countable, B A ft c j j H } HeH = n ( B n n) . Consequently, i f (ft,n) i s a l i m i t , then, by Remark 1.2.4., f o r any A e Cyl , 7(Ac\ ft) = n(A) = T(A) = 7(A) = 7(A) . On the other hand, i f 7(A) = n(A f\ ft) for a l l A e C y l , then, again by Remark 1.2.4., f o r any A e Cyl , n (A) = "7(1 A 0) = 7(A) = 7(A) = x (A) , and therefore (ft,n) i s a l i m i t . 14. 2. F o r any s u b f a m i l y F' o f F ; l e t , — 1 . C y l ( F ' ) = { P p [B] : F e F' , B e M p} We s h a l l show t h a t f o r any A e C y l t h e r e e x i s t s A'c ft s . t . rf(A') = 0 and rf(A ~ A') = rf (A (\ ft) . I n w h i c h c a s e , rf(A) = rf(A ~ A') + rf(A a A') = Tf(A ~ A') = rf(A f\ ft) and t h e lemma f o l l o w s . F o r each n e w , l e t C y l be c o u n t a b l e w i t h A r\ ft C I lH and £ 7(H) < 7f(A f\ ft) + 1/n . n Heff F o r each n e w , choose c o u n t a b l e F c F w i t h n (1) {A} u H c Cyl(F ) . n n S i n c e (X^ ^ p ) p e p ^ s a n i n v e r s e s y s t e m o f measure s p a c e s r e l a t i v e t o t h e maps r , we may f u r t h e r assume t h a t (2) F i s a sequence {F .}. i n F w i t h F . < F . n n , j jew n , j n , j + l f o r each j e w L e t A = A(F ) n n and A' = U A n new S i n c e Tl(A ) = 0 f o r e v e r y n e w , then (3) n(A') = 0 .' L e t n e w . F o r each f e A ~ t h e r e e x i s t s g e ft s . t . 7 (g) = ? ( f ) f o r a l l F e F . 15, In p a r t i c u l a r , by (1),. g e A . Hence, for some H e H , with H = p [p-''"[H] ] n (J " G for some G e F , n g e H , and consequently, f e p ^ t p g C f ) ] = P G 1 [ p G ( g ) ] c p / t P g t H ] ] = H . It follows that A ~ A n c U H n , and therefore (4) 7(A ~ A ) < n ( A a ft) + 1/n . n — Since A c ft ~ ft for each n e w , n A' C ft ~ ft • Hence A <A ft c A ~ A' , and therefore, by (4), for each n e w , "n(A r\ ft) <_ n"(A ~ A') <_ n(A ~ A ) <_ n"(A r\ ft) + 1/n Consequently, ~r\(A n ft) = ri(A ~ A') . 16. 2. Cylinder Measures over Vector Spaces.- We s h a l l view a cylinder measure over a vector space X as being.an inverse system of measure spaces whose indexing, set i s the family of f i n i t e dimensional subspaces of X In t h i s paper we s h a l l consider only complex vector spaces, and we s h a l l hereafter r e f e r to them simply as vector spaces. By the term subspace we s h a l l always mean vector subspace. We note that i f F i s a f i n i t e - d i m e n s i o n a l vector space, then there i s a unique Hausdorff topology on F under which i t i s l o c a l l y convex (the Euclidean topology). Since t h i s i s the only topology on F that we s h a l l ever consider, e x p l i c i t reference to i t i s hereafter omitted. Throughout the remainder of t h i s work, we s h a l l use the following notation. For any vector space X , X i s the set of l i n e a r f unctionals on X to § , w i s the topology on X of pointwise convergence, For any A c X , A° = {f e X^ : | f ( x ) | <_ 1 for a l l x e A} . F v i s the family of f i n i t e - d i m e n s i o n a l subspaces of X A directed by C . When there can be no ambiguity we s h a l l omit the subscript X For any subspaces E and F of X with E C F > r : F -*• E i s the r e s t r i c t i o n map, E, F i . e . for a l l f e F , r E j F ( f ) = f|E . 17 In what follows, E and F w i l l always denote finite-dimen s i o n a l vector spaces. For any subspace ft of X , (X,ft) i s a dual pair i f f r Ift i s s u r j e c t i v e for every F e F Remark. With the viewpoint of inverse systems discussed i n the preceeding section, taking F v as our index set and l e t t i n g A X„ = F" for each F E F v , a A we note that, for any E , F and G i n F v with E c. F C G , A the r e s t r i c t i o n map r i s s u r j e c t i v e and continuous, E , l and rE,G rE,F ° rF,G Thus, we s h a l l make the following d e f i n i t i o n . 2.1 D e f i n i t i o n . (1) Let X be a vector space. u i s a cyl i n d e r measure over X i f f y : F e F •+ y , a Radon measure on F , F i s such that (F ,y„)„ r i s an inverse system of measure spaces r e l a t i v e F r E r to the r e s t r i c t i o n maps r . (2) p i s a c y l i n d e r measure i f f p • i s a c y l i n d e r measure over some vector space X Remark. Let X be a vector space. I f r -» * u : F e r ->• u , a f i n i t e Radon measure on' F , r then, by §0.4, p i s a c y l i n d e r measure over X i f f f o r any E and F i n F with E C F , y E = r E , F [ l J F ] • Let ft be any subspace of X . For any E and F i n with E c F , Hence, when (X,ft) i s a dual p a i r , the viewpoint of D e f i n i t i o n 1.1.2 applies, with P E = r E X ^ f ° r e a c l 1 , E e F • We s h a l l therefore make the following d e f i n i t i o n . 2.2 D e f i n i t i o n . | ft = r E,F Let X be a vector space, jn, a c y l i n d e r measure over X and ft be a subspace of X such that (X,ft) i s a dual p a i r . 19. For any outer measure £ on ft' , ^ i s a l i m i t measure of u on ft i f f (ft,5) i s a l i m i t r e l a t i v e to the r e s t r i c t i o n maps r ft of the inverse system of measure spaces (F ~, y_)_ r r ,A F Fer Remarks. From the theory of inverse systems of measure spaces we know several conditions under which we can put a l i m i t measure on the pro- j e c t i v e l i m i t set L , where L = U e n E" : £ = r w ̂ ( O , E c FV . EeF E E ) F F Since there e x i s t s a set isomorphism r : X* -> L such that r _ v ( f ) = ( r ( f ) ) _ f o r a l l f E x" and F E F , r , A r i t follows that L i s seq u e n t i a l l y maximal (Defn. 3.4). Hence, by a theorem of Bochner ([4] p. 120), we deduce that JA/ always has a l i m i t measure on X . However, l i t t l e has been said about the properties such a l i m i t measure can have. Therefore, i n the next sec t i o n , we s h a l l construct one having s p e c i a l approximation properties. v'c Unfortunately, f o r most p r a c t i c a l purposes X i s f a r too unwieldy. We s h a l l therefore be studying the problem of putting l i m i t measures on subspaces of X 20. 3. Non-topological Limit Measures. Given any c y l i n d e r measure y over a vector space X , and subspace ft of X such that ( X , f t ) i s a dual p a i r , we s h a l l determine s u f f i c i e n t conditions on y for i t to have a l i m i t measure on ft . Throughout t h i s section we s h a l l use the following notation. X i s a vector space. For any cyl i n d e r measure y over X , and subspace Q of X C y l (n) = {fi o r ^ A ] : F e F, A e Mp} , T : ftfNr"1 [A] £ Cyl (ft) -> y (A) e R + , y, ft r, A y s y^ i s the Caratheodory measure on ft generated by T n and Cyl (ft) . y ,ft y y y>x- and y = y x * • In what follows, ft w i l l denote a subspace of X such that ( X , f t ) i s a dual p a i r . From D e f i n i t i o n 1.1.2 and Remarks 1.2 we get the following assertions. 21. 3.1 Propositions. Let u be a cyl i n d e r measure over X (1) For any outer measure E, on ft, E, i s a l i m i t measure of y i f f Cyl (ft) c. M r and E, I Cyl (ft) = T 0 . y C y y,ft (2) I f there e x i s t s any l i m i t measure of y on ft , then ŷ . i s a l i m i t measure of y In view of Proposition 3.1.2, when looking for a l i m i t measure , of y on ft , we s h a l l concentrate on y^ When ft = X , we have the following r e s u l t . 3.2 Theorem For any c y l i n d e r measure y over X , y i s a l i m i t measure of y If C = { r ' ^ j K ] : F e F, K c i s compact} F,X then C i s a compact: family, and for any A e M , , y* V(A) = sup{y X(C) : C C A , C e C } . (We note that Ĉ  i s also a compact family.) 22. Proof For each F e F (1) y_ i s Radon and c r - f i n i t e . r Hence, for any A e Cyl^(X ) , (2) T*(A) = S U P { T * ( C ) : C C A , C e C} Since M X(A) 1 T*(A) for a l l A e Cyl^x*) , v?e also deduce from (1) that (3) y i s o - f i n i t e . Hence, by Thm II.2.5 of [23], the assertions of the theorem w i l l follow once we show that C i s a compact family. (Also see [24]). For any C. = r " 1 V [ K . ] £ C, j = 1,2, l e t F be the l i n e a r soan of F U F , and K = A r ^ „[K.] 1 2 j=l,2 F j ' F 1 Then K i s compact and C. <% C„ = r \ j K ] 1 2 F,X Hence, (4) C i s closed under f i n i t e i n t e r s e c t i o n s . For any C'^ C s . t . ] [ot =|= <f> for every f i n i t e a CZ C , l e t A = { f l ct : a C C* i s f i n i t e ) We note that A i s a f i l t e r b a s e ( [ 8 ] p. 211). In view of (4), for each f i n i t e a C C , l e t o P| a = r ^ ^[K ] for some F & e F and compact K c F" , and Y = U(F : a C C i s f i n i t e ) . a 23. From the remarks preceding (4),' we see that for any f i n i t e subfamilies a and 3 of C , a C 3 = > F a C F 3 ' and therefore Y i s a subspace of X n Let U be a maximal f i l t e r b a s e i n X ([8] p. 218) which i s a s u b f i l t e r b a s e of A ([8] p. 219, Thm. 7.3). Then, for each f i n i t e a C C , (r„ [U]) _T i s a maximal f i l t e r b a s e i n F ' and F ueJ ' a there e x i s t s u e U s.t. r [u] c K F a a * Since K i s compact and F i s Hausdorff, t h i s u l t r a f i l t e r a a - converges to a unique point f e K a a' We note that i f F = F„ , then f = fn . Also, f o r any f i n i t e a 3 a 3 • . subfamilies a and 3 of C with a C 3 , , r ^ ( f j = f • F ,F v 3 a ' a 3 since the r e s t r i c t i o n map i s continuous and r F ,X = r F ,F f l ° rF.,X ' a a 3 3 A Consequently, there e x i s t s a unique g e Y s.t. g i F = f for each f i n i t e a C C' 1 a a •k If f e X i s any l i n e a r extension of g , then, for each f i n i t e f e r ; \ x t r F , x ( f ) ] = ^ \ A ] C ^ \ X [ V = ^ a C C . " I r " I r r n _ " I ? , X [ f a ] C r F , a a Hence, f l ' c + • . It follows that C i s a compact family. Next, we consider the problem of fi n d i n g a l i m i t measure of y on an a r b i t r a r y ft Since y i s always a l i m i t measure of y , a p p l i c a t i o n of Lemma 1.3.1 y i e l d s the following basic r e s u l t . 3.3 Lemma. For any cyli n d e r measure y over X , y has a l i m i t measure on ft i f f (1) y*(A) = y"(Ar \ f t ) for a l l A e Cyl (x") . y However, we are int e r e s t e d i n f i n d i n g i n t r i n s i c conditions on our systems which w i l l guarantee the existence of a l i m i t measure on ft One such condition i s the following, which i s of considerabl importance i n the general theory of inverse systems of measure spaces (Bochner [4], p. 120, Choksi [6], Mallory and Sion [23]). 3.4 D e f i n i t i o n ft i s sequentially maximal i f f for any sequence {F } i n F with F c F ,- for each n neco n n+1 •k n e w , and f e F such that r_, _, (f ,-) = f , there ' n n F ,F , 1 n+1 n n n+1 ex i s t s g E ft such that r„ = f for each n e w F ,X n n 25. Remark. We note trial: X i s sequentially maximal. Consequently, A the f a c t that p i s a l i m i t measure of p follows also from a theorem of Bochner ([4], p. 120). Since p i s always a l i m i t measure of yj, , a p p l i c a t i o n of Lemmas 1.3.2 and 3.3 y i e l d s the following. 3.5 Proposition If ft i s sequentially maximal, then every cylinder measure over X has a l i m i t measure' on ft However we have the following. 3.6 Observation. If X i s a t o p o l o g i c a l vector space containing a bounded, countable, l i n e a r l y independent subset, and ft i s i t s continuous dual, then ft i s not seq u e n t i a l l y maximal, (e.g. whenever X i s an in f i n i t e - d i m e n s i o n a l , metrizable, l o c a l l y convex space). Proof Let {a : n e w} be a bounded, countable, l i n e a r l y n independent subset of X , and for each n e w l e t F n be the li n e a r span of {a ,...,an} . Then, f o r any f e X with f ( a ) = n for every n e w . 26. (1) f[{a ; n e w}] c £ i s unbounded. n Hence, there cannot e x i s t g e ft s.t. e l F = f F for 0 1 n 1 n every n e w . For i f so, then g U F i s continuous, and therefore § [ { a n : 1 1 £ ^ s bounded, which contradicts (1). "Si Since, i n the theory of c y l i n d e r measure, ft i s often the continuous dual of metrizable I.e. space ([11], [39]), i t follows that the condition of sequential maximality does not apply i n many important s i t u a t i o n s . In order that we might take f u l l e r advantage of Lemma 3.3, we therefore weaken the notion of sequential maximality. 3.7 D e f i n i t i o n Let y be a c y l i n d e r measure over X U i s y-sequentially maximal i f f for any sequence {F } i n F with F r F ,-, for every n new n n+1 n e w , and e > 0 there exxsts A e M_ for each n e w , such that n F n Z y p (A n) < e new n ' and f o r any sequence {f } with ^ n new f e F" ~ A , r (f ) = f , n n n F ,F n+1 n n there e x i s t s g e ft such that r (g) = f for each. n e w . n' . 27. The following key theorem of t h i s section i s now an immediate consequence of Lemma 1.5.2 and the above d e f i n i t i o n . 3.8 Theorem Let y be a c y l i n d e r measure over X If ft i s y-s e q u e n t i a l l y maximal, then y has a l i m i t measure on ft . We now e s t a b l i s h a condition on y which ensures that ft i s y-sequentially maximal. 3.9 D e f i n i t i o n Let y be a cylinder measure over X •k For any family H of subsets of X , y i s H-sequentially tight i f f for any sequence {F } i n F with F c. F , -, r o r each n miii n n+1 n E (a , A E M with y (A) < « t and e > 0 , 0 0 • there exists H £ H such that y p ( r ^ 1 F [A]) ~ r p X [ H ] ) < e for a l l n £ GO . n 0' n n' 3.10 Theorem Let y be a cylinder measure over X If y i s H-sequentially t i g h t for some family H of w -compact subsets of ft , then ft i s y-sequentially maximal, and therefore y has a l i m i t measure on ft 28. We point out that under c e r t a i n conditions y-sequential maximality of ft i s also a necessary condition for y to have a l i m i t measure on ft 3.11 Proposition Suppose that the Mackey topology on X induced by ft ([47] p. 369) r e s t r i c t e d to any subspace of countable dimension i s metrizable. For any c y l i n d e r measure y over X , i f y has a l i m i t measure on ft , then ft i s y-seq.uentially maximal. Proofs 3. Lemma. Let {F } be a sequence i n F with F o F for 1 n new n n+1 each n e w , * A K be a w -compact subset of X For any sequence {f } with J n n new f e r_, [K] and r (f ,,) = f . n F ,X F ,F n+i n n n n+1 there e x i s t s g e K s.t r (g) = f for a l l n e w . r , A n n Proof For each n e w , (1) r ^ 1 [ f J n K - H • n Since r F [ f n + 1 ] = f n . n n-rl 2 9 , ( 2 ) r " 1 [f ] o r 1 [f ] . F , X n F ,. n + 1 n n + 1 •k * Also, since r i s w -continuous and K i s w -compact, n - 1 * ( 3 ) K n r [f ] i s w -compact. r , A n n Since w i s a Hausdorff topology, i t follows from ( 1 ) , ( 2 ) and ( 3 ) , that rS (K r\ r " 1 x [ f n ] ) > $ , new n' and the lemma follows. Let { F } C F with F c. F f o r each n e w , n new n n + 1 {B}. C M_, , with u_ (B.) < oo for each j e w , je-w F Q F q j and Fn = U B. ,. 0 • J jew Since u i s H-sequentially t i g h t f o r some family H of k w -compact subsets of Q , given e > 0 , for each j e w choose a w -compact K c 9, s.t 8 U W F ( ^ , F [ B j ] ~ r F , X [ K j ] ) < . n 0', n J n J Let C n = . U ( I V [ B j ] ~ R F , x £ K j ] ) jew 0 ' n J n' J Ao = co V i = V i - n n - r l 30. Then, f o r any k e w , n=0 n k and k k 1+1 n=0 n j = l Hence E p„ (A ) < e F n new n I f i s a sequence s . t . f o r each n e w , f e F* ~ A , r _ _, ( f ) = f n n n F ,F n+1 n n n+1 t h e n , f o r some j e w , and hence f o r e v e r y n e w , C o n s e q u e n t l y , by the Lemma, t h e r e e x i s t s g e ft s . t . r „(g) = f f o r a l l n e w , and i t f o l l o w s t h a t ft i s / A . -r , A n n s e q u e n t i a l l y m a x i m a l . The l a s t a s s e r t i o n i s immediate f r o m Thm. 3.8. P r o o f o f 3.11 F o r any {F } c F w i t h F c F ,, f o r each n e w , l e t n new n n+1 Y = U F w i t h t h e r e s t r i c t e d t o p o l o g y , new A = { f e X : t h e r e does n o t e x i s t g e ft s . t . r F = r F X ^ f ° r a 1 1 n E ^ ' n' n' 3 1 , 8 = {r^1 X[A] : n e to, A E M } n' n be the o - f i e l d generated by B , and n' the Caratheodory measure on X generated by |B and B i Since the topology of X r e s t r i c t e d to Y i s metrizable, choose a sequence {V, }, of absolutely convex neighbourhoods k keco of the o r i g i n i n X s.t. {V, n Y}, i s a base for the neigh- K KEU) bourhoods of the o r i g i n i n Y . Using the Hahn-Banach extension theorem, one readily checks that keo) new n n I t then follows that ( 1 ) . A E © , and, since Cyl^(X ) c M , ( 2 ) A £ M , y* We note that (3) A c X* ~ fi . >v k k Since y i s o - f i n i t e and y (A) = y (A O ft) for a l l A E Cyl (X*) , from ( 2 ) and ( 3 ) i t follows that (4) y*(A) = 0 * * Since B i s a f i e l d and T i s countably additive on Cyl (X ) (Thm. 3 . 2 , Remark 1 . 2 . 5 ) , we have that n|B = T*|B and y"|8 = x*|B 1 y 1 1 y 1 •k . However, |B has a unique countably additive extension to V5 . Hence, since C M n M n y T , | $ = y * | < & , and therefore, by ( 1 ) and (4), n(A) = 0 . Consequently, given any e > 0 , there e x i s t s {B . } . C 8 -s. t. 3 JEW ' * A C U B. and Z T (B.) < e . i y i ' '• jew jew For each j e to , l e t B = r F X [B'] , B^ £ M F , J n. J n. } 3 and B. 4= r " 1 , r[B'] for any n. < n. and B' e M J n n For each n e w , l e t A n = UtB* : 3 £ a), n j = n} Then, (5) E y ( A ) < e . r n new n Further, i f {f } i s any sequence s . t . for n new each n e w , f e F ~ A , r _ _, (f ..) = f n n n F , F n+1 n n n+1 then, there e x i s t s f e x " ~ U r [A ] s . t . r , A n new n r„ „(f) = f for each n e w . F ,X n n Since A C U B. = U r " 1 V[A ] , from the d e f i n i t i o n of A 3 F ,X n jew new n follows that . (6) there e x i s t s g e Q s.t. r ? (g) = f f o r a l l n e w . n' Since the sequence {F } i n F with F tg. F f o r each n n new n n+1 n e w was a r b i t r a r y , we conclude that 0. i s y-sequentially maximal. 33. 4. Radon Limit Measures In this section we s h a l l consider the problem of f i n d i n g Radon l i m i t measures. The technique we use was communicated to us by C. Scheffer. In t h i s section we s h a l l use the following notation. X i s a vector space, ft i s a subspace of X such that (X,ft) i s a dual p a i r . For any topology G on ft and c y l i n d e r measure y over X. , • g : A C X % i n f W ( r F , X [ A ] ) : F z F } ' g A : G e G + sup {g(K) : K e K(G) , K c G} , y i s the Caratheodory measure on ft generated by g.,„ and G We s h a l l hereafter assume that y i s a fi x e d cylinder measure over X , G i s a regular Hausdorff topology on ft which i s f i n e r than w r e s t r i c t e d to ft. We have the following important assertions, 4.1 Propositions (1) y_ i s a G-Radon measure on ft , and y |G = g., b G (2) Cyl (ft) a M y y G (3) If there e x i s t s any G-Radon l i m i t measure of y on ft , then \in i s a l i m i t measure of y (7 34, In view of the above propositions, when searching f o r a' G-Radon l i m i t measure of y , we s h a l l r e s t r i c t our att e n t i o n to \in b Following Scheffer [37] we make the following d e f i n i t i o n . Our terminology i s s l i g h t l y d i f f e r e n t . 4.2 D e f i n i t i o n For any family H of subsets of X , u i s H-tight i f f for any E e F, A e with ^^.(A) < <*> , and e > 0 , there e x i s t s H e H such that u^Cr" 1 „[A] - r _ [H]) < e for a l l F e F with E C F . r Jti, r r , A We point out that the above d e f i n i t i o n i s a "uniform" version of the d e f i n i t i o n of H-sequential tightness (Defn. 3.9). We now have the following key theorem concerning the existence of a G-Radon l i m i t measure of y 4.3 Theorem u has a G-Radon l i m i t measure on 0, <=> y i s K(G)-tight. Remark The above theorem extends a r e s u l t due to Mourier [26], and Prohorov [33] (§5 Lemma 3). However, our approach i s somewhat d i f f e r e n t from t h e i r s . Theorem 4.3 has a useful c o r o l l a r y . 35. C o r o l l a r y I f X i s a metrizable, l o c a l l y convex space, a i s i t s continuous dual, and {V } i s a base for the neighbourhoods of the n new o r i g i n i n X , with v n + 1 v n f°r every n e w , then, y has a w -Radon l i m i t measure on ft <=> y has a l i m i t measure on ft <=> y i s {V°} - t i g h t , n new Proofs 4. Notation Let H = (Cyl 'CO)). A (w"|ft) , y = v G , and T = T y,ft We s h a l l need the fo l lowing lemmas. L . l For any A e'Cyl^(ft) , (1) T(A) = i n f {T(H) : A C H e H} ( 2 ) Y C A ) < I ( A ) . 1^2 For any K c K (G) , (1) r [KJ i s compact for every E e f , V t W t -wv̂ oW Û*. (2) g(K) = y (K) = inf' (T (H ) : K C H e H} (3) For any E and F i n F with E c F , and A e , K E ( R E , X [ K ] ) > ^ F C r F , X [ K ] ) • 36, Proof of L . l . l For any E e F v and B e M , since r ' i s X E E, X k w -continuous and y i s Radon, T(ft C\ r ' ^ t B ] ) = y E ( B ) = i n f {y^,(G) : B c G C E " , G i s open} > i n f ( T(H) : ft f\ ̂ " ^ [ B ] c H e H} > x(fi A r ' ^ t B ] ) . Ji, A Ji , A Proof of. L.1.2 We note that H C G , and for every H e H , g F T(H) <_ T ( H ) . Hence, y(A) = i n f { I g.,.(H) : H'c'H i s countable and A c |JH'} HeH' <_ i n f { E T(H) : H'C H i s countable and A c {JH1} HeH' < i n f {x(H) : A c H e H} = T(A) , by L . l . l . Proof of L.2.1 We only observe that for every E e F, ŷ , i s Radon, * k K(G) C f((w ) , and r„ v i s w -continuous. b, A Proof of L.2.2 For every E e F, y„ i s Radon and r i s w -continuous. ii L , X Therefore, g(K) = i n f {y w(G) : E e F, r_. V [ K ] C G <=. E " , G i s open} Ji Ji, A = i n f ( T(H) : K c H e H} . ' On the other hand, by L . l . l , y (K) = i n f { E T(H) : H'c H i s countable, K c UH'} HeH' = i n f {x(H) : K C H e H}, since, K i s G-compact, H<z G,M i s closed under f i n i t e unions, and x i s f i n i t e l y subadditive on Cyl (ft) bv 37. Proof of L.2.3 We have that [K] C r E,F E,X [K]] Hence, ^E ( rE,X [ K ] ) = ^F ( r E V r E,X [ K ] ] ) l ^ F ( r F 5 X [ K ] ) 4.1.1 To show that y i s a G-Radon measure, by Sion [44] Ch. V, Thm. 2.2, we need only show that (1) g(<j>) = 0 , g i s p o s i t i v e , monotone, subadditive and a d d i t i v e on K(G) , (2) Y ( K ) < ro f o r a 1 1 K e K(G) , Except for a d d i t i v i t y , the properties of g are immediate from L.2.2. We s h a l l now e s t a b l i s h the a d d i t i v i t y of g on K(G) Let K and K be i n K(G) with K n K = | . Since K(G) c K(w"|ft) , and • w ft i s regular and Hausdorff, there e x i s t s G. e w"|X, K.C G., i = 1,2 , with - G l f N G2 = * " However, H i s a base for w ft , and i s closed under f i n i t e unions. Consequently, since K ,K„ are w ft-compact, there e x i s t s H. e H K.C H. » 3 i = 1,2 with H n H 2 = <f> 3 8 , Then, by L.2.2, g(V + g(K 2) = y f l ( K l ) + u Q(K 2) I i n f {y-CA^) : K. c A. e M \ 3=1,2 ° ^ 2 I i n f {y„(A j) : A e M , K. c A3 <z H.} ' 3 = 1,2 " ^ J J T O * " = in f . {y (A (j A ) : A J e M , K. C A2 c H. ; j = 1,2,} , y f t 2 2 <_ y ^ f t ^ u K 2) = gC^U K 2 ) . Hence, by the s u b a d d i t i v i t y of g , gd^u K 2 ) = gO^) + g ( K 2 ) , Since and K 2 were a r b i t r a r y i t follows that g i s a d d i t i v e on K(G) . I t remains for us to prove (2). Let K e H(G) . For any F e F, since y i s Radon and r [K] r r ,X i s compact (L.2.1), •ft r\ r ~ * x [ r [ K ] ] e Cyl^(ft) and by L.1.2, Y(K) <_ y ( 0 r i . r ^ x [ r F ) X [ K ] ] ) £ T ( n ri r ^ E r [ K ] ] ) = y F ( r F ) X [ K ] ) < oo . , Hence, y(K) < oo for a l l K e K(G) . 4.1.2 Let F e F and A z V If y (A) = 0 , then, by L.1.2, Y ( f t A r ^ l A j ) = 0 . and therefore ft A r " 1 [A] e M F,X y 39, / 5 Otherwise, since u„ i s Radon, choose a Borel subset B of F r with A C B and n (B ~ A) = 0 r By the preceding observation, ft 0 r " 1 [B ~ A] e M . a, A y However, r ft i s ( j-continuous since w ft C G , F, X and'by Prop. 4.1.1, y i s G-Radon. Hence, ft f\ r"" 1 „[B] E M , F,X y and therefore, ft f\ r p ^ x [ A ] = ft ri r ^ x [ B ] ~ ft r» r ~ ^ [ B ~ A] e M . We s h a l l now e s t a b l i s h another useful lemma. L.3 For every K e K(G) , g(K) <_ Y(K) . If y i s a l i m i t measure of u , then, for every K E K(G) , . g(K) = y(K) . Proof of L.3 Let K E K(G) .. By Prop. 4.1.1, y(K) = inf',{g A(G) : K C G e G} >. g(K) . If y i s a l i m i t measure of u , then, by L.2.2 and Prop. 3.1, g(K) = i n f (T(H) : K c H e HI = i n f {y(H) : K c H e H} > y(K) , since y i s G-Rado'n and H C G 40. 4.1.3 Let £ be any G-Radon l i m i t measure of y on ft • For any K e K(G) , ?(K) = i n f U(G) : K C G e G} <_ i n f U(H) : K C H E H } since H C G , = i n f {T(H) : K c II e H} by Prop. 3.1.1 = g(K) by L.2.2 <_ Y(K) by L.3. Hence, as E, and y a r e both G-Radon measures on ft , £ (A) <_ Y (A) f o r a l l A c ft . In p a r t i c u l a r , by Prop. 3.1.1, for any A e Cyl^(fi) , T ( A ) = 5(A) < Y ( A ) , and-therefore, by L.1.2, y(A) = T ( A ) • From Props. 3.1.1 and 4.1.2 i t now follows that y i s a l i m i t measure of y 4.3 By Prop. 4.1.3, i f y has any G-Radon l i m i t measure on ft , then y i s a G-Radon l i m i t measure of y . Hence, for any E e F and A. e with P E(A) < <*> , -1 M . ..-1 Y and therefore ft n r E X [ A ] e M , y(ft r\ r ^ t A ] ) < » , y(ft O r ^ t A ] ) = sup {y(K) : K e K(G) , K c ft A r ^ t A ] : ? . Hence, for any e > 0 , there e x i s t s K e H(G) with K c ft ftr [A] £j , A and y(ft A r E 1 x [ A ] ~ K) < e In which case, for any F e F with E C F , y F ( r ^ F [ A ] - r p ) X [ K ] ) . = Y ( n ̂  r ^ x [ r ^ F [ A ] ) ~ « " r ^ r ^ K ] ]) <_ Y ( f i n r ' ^ f A ] ~ K) < e . I t follows that y i s K(G)-tight. . We now show that /<(G)-tightness of y i s a s u f f i c i e n t condition for y to have a G-Radon l i m i t measure on ft . IN view of Props. 4.1 and 3.1.1, we need only show that (1) y i s K(G)-tight => y|Cyl (ft) = T . ' I f , f o r every A e Cyl (ft) , (2) T(A) = sup (g(K) : K e K(G) , K C A} , then, f o r every A e Cyl (ft) , since y i s a - f i n i t e (L.1.2), Y(A) = sup {Y(K) :K e K(G) , K C A} >_ sup {g(K) : K e K(G) , K C A.} by L.3, = T(A) by (1). Hence, by L.1.2, Y(A) = T(A) for a l l A e Cyl (ft) , y Consequently, (1) w i l l have established when we show that, (3) y i s K(G)-tight => (2) holds for a l l A e Cyl (ft) . Suppose y i s K(G)-tight. Let E e F and B G Mg with y E ( B ) < 0 3 • Given e > 0 , since u,., i s Radon, there e x i s t s a closed C c B s y E(B ~ C) < e/2 . Since y i s «(G)-tight, there e x i s t s e K(G) s.t. for every F E F with E C F , y F ( r E y B ] - r F j X [ K ] L J ) < e/2 . Let K = K i ^ r E ! x [ c ] • Since C i s closed and r [ ft i s G-continuous, (4) K e K(G) . Further, (5) . K C ft r\ r'1 [B] . Now, for any F e F with E C F ,' ( 6 ) r F , X [ K ] = r F , X [ K l ] * r E ! F [ C ] • Hence, V r E ! F [ B ] " r F , X [ K ] ) = V ( r E > ] ~ r F , X [ K l ] ) U < F [ B ] ~*1]Y™» l V F ( r ^ p [ B ] - r ^ f ^ ] ) + v E(B - C) < e . Consequently, by L.2.3 and L.2.1, g(K) = i n f { y p ( r F X [ K ] ) : E c F e F} >, i n f { y ^ r " 1 ^ ] ) - e : E C F e F} = y E(B) - e . Since e was a r b i t r a r y i t follows that (2) holds for a l l A e Cyl («) with T ( A ) < » . However, since y i s a - f i n i t e for a l l F e F , T(A) = sup {T (A'): A ' c A , A e Cyl (ft) , x(A') < » } , Hence, (2) holds for a l l A e Cyl (ft) . Proof of Cor. 4.3. Let K = {V° : n e w } . We note that (1) ft = \JK , and (2) K c K(w*|ft) By (2) and Thm. 4.3 we need only show that (3) y i s ,.:?. /(-tight whenever y has a l i m i t measure on ft Suppose.that y has a l i m i t measure on ft . Let E e F and A e M with Vu(A) < °° * Since y i s Radon, choose a closed C C E s.t. Ji C C A and y E(C) > y^(A) - e/2 , and for each n e w , l e t * Since C i s closed and r i s w -continuous, then, by (2), h i , X (4) K e K(w"|ft) for each n e w . Further, by (1) (5) ft rs r ' ^ C ] = U K n . new Since y„ i s an outer measure, and K c K ,, for each n e w ft n n+1 we deduce from (5) that (6) y^(ft t\ r ^ t C ] ) = sup y (K ) . ' new n By Prop. 3.1.2, C7) y^ i s a l i m i t measure of y , Hence, (8) y^(ftn r ^ x [ C ] ) = y E(C) < y E(A) < » . Let e > 0 . By (6) and (8), there e x i s t s n e w s.t. 44. Then, by (7), W > V C ) ' £ / 2 > V.A ) " £ • Hence, by (4) and L.2.2, (10) g(K ) > y (A) - e . We have that, for any F e F with E e F , r F , X [ K n 1 C r ; > C l E " ! F [ A l • and therefore, = y (A) - y ( r [K ]) < y (A) - g(K ) b r r , A n — & n < e , by (10). However, K c , and therefore, for every F e F with E F , n n V ^ F ^ " r F , X ^ n ] ) < E • Since e > 0 , E e F , and A e with Vg(A) < 0 0 , xrere a l l a r b i t r a r y , i t follows that y i s K-tight. 45. 5. F i n i t e Cylinder Measures We s h a l l s p e c i a l i z e the r e s u l t s of the foregoing sections to the case of f i n i t e c y l i n d e r measures. By introducing the notion of a f i n i t e section of an a r b i t r a r y c y l i n d e r measure, we s h a l l show that, with regard to the problem of f i n d i n g l i m i t s , we can concentrate on f i n i t e cylinder measures. 5.1 D e f i n i t i o n . u i s a f i n i t e c y l i n d e r measure i f f u i s a c y l i n d e r measure over a vector space X and for some F E F v ,' X y p(F") < « . (We note that u (F ) i s independent of F E F ) r A For the rest of t h i s section we assume that X i s a vector space, ft i s a subspace of X such that (X,ft) i s a dual p a i r , 6 i s a regular, Hausdorff topology on ft which i s f i n e r than the w -topology r e s t r i c t e d to ft , p i s a c y l i n d e r measure over X r e s u l t s The following lemmas i n d i c a t e that the hypotheses of e a r l i e r can be s i m p l i f i e d when considering f i n i t e c y l i n d e r measures. 46. 5.2 Lemmas. If y i s a f i n i t e c y l i n d e r measure over X , then (1) • y X(A) = y / V(A r\ ft) for every A e Cyl (X*) <=> y'?(ft) = y"(x") . For any family fl of subsets of X , (2) y i s H-sequentially tight <=> for any sequence {F } i n F with F c F ,, for each n new n n+1 n e w , and e > 0 , there e x i s t s H e H such that y (F ~ r v[H])-< e for a l l n e w . r n r , A n n (3) If r V[H] e M for every F e F and H e H , then, r , A c y i s H-tight <=> for any e > 0 there e x i s t s H e H such that y p ( F " ~ r p X[H]) < e for a l l F e F . Proof of 5.2.1 C e r t a i n l y , i f y"(ft n A) = y"(A) for a l l A e Cyl (x") , then y*(ft) = y"(x') . On the other hand i f y (ft) = y (X ) , then, for any A e Cyyx*) , y*(X X) = y X(ft) = y" (ft A A) + y " (ft ~ A) <_ y * ( A ) + y * <x" ~ A) = y * ( X * ) -k ' * Hence y (ft r\ A) = y (A) 47. Proof of 5.2.2 We observe only that for any F e F, A e and H e H , P F(A ~ r F j X [ H ] ) < y F * ~ r [H]) . Proof of 5.2.3 Together with the observation of Proof 5.2.2 above, we note that for any E and G i n F with E c G , and H e H , V E* ~ r E , X [ H ] ) ^ G ( G * ~ r G , X [ H ] ) The assertion i s now immediate. The following theorems are now immediate consequences of, re s p e c t i v e l y , Lemma 3.3, Theorem 3.10, and Theorem 4.3 5.3 Theorems If y i s a f i n i t e c y l i n e r measure over X , then :L̂ ..>\ ] - . '! . (1) y has a l i m i t measure on ft <=> y"(ft) = y"(x") . (S i l o v [46]). (2) y has a l i m i t measure on ft i f , for any sequence (F ) i n F with F c F ,.. for each n £ 43, J H n new n . n+1 ' and e > 0 , there e x i s t s a w -compact K c ft such that y r ( F * - r [K]) < e for a l l n e w . r n r , A n n . • . 48. (3) (Mourier-Prohorov, [3 ] ̂ 5 Lemma 3) • u has a G-Radon l i m i t measure on ft <=> for any e > 0 there e x i s t s K e K(G) such that U _ ( F " ~ r V [ K ] < e for a l l F e F . r r , A Remark. We point out that Theorem 5.3.2 does not seem to have been previously stated i n the l i t e r a t u r e . We s h a l l now show that the problem of f i n d i n g l i m i t s for a r b i t r a r y c y l i n d e r measures can be recuced to that for the f i n i t e case. F i r s t , we make the following d e f i n i t i o n . 5.5 D e f i n i t i o n ^ i s a f i n i t e section of u i f f for some E e F and A e Mg with WgCA) < 0 0 , 5 i s the cylinder measure over X such that for every F e F, ^F = r F , G [ y G l r E , F " 1 [ A ] J ' for some G e F with E c G and F c G Remark. If £' i s a f i n i t e section, of u then | i s well-defined, 5 i s i n fac t a cy l i n d e r measure over X , and = u " ( r E X _ 1 [ A ] A B) for a l l B c X^ . (This remark i s proved below.) The following theorems are then r e a d i l y established. 49. 5 .6 Theorems. (1) y has a l i m i t measure on ft i f f every f i n i t e s e ction of y has a l i m i t measure on ft . (2) u i s K(G)-tight i f f every f i n i t e section of y i s K(G)-tight. Hence, y has a G-Radon l i m i t measure on ft i f f every f i n i t e s e ction of y has a G-Radon l i m i t measure on ft Proof of Remark 5.5 *• • _ i For any F e F and a c F with r [a] t M n , i n view of Remark 2.2.1 and Lemma 0.4.1, ^ G ( r F , G 1 [ a ] " r E , G 1 [ A ] ) i s independent of the choice of G e F with E c G and F C G Hence, so also i s £ F We note that for any F e F with E C F , (1) U F l r E p 1 [ A ] i s Radon. Consequently (2) ? F = y p | r E F 1 [ A ] and i s Radon. Hence, by Lemma 0.4.3, (3) E, i s Radon for every F e F. For any F and F^ i n F with F c F^ •, i f G e F with E U F u F c, G , then by Lemma 0.4.2, h = r F , G [ y G l r E , G l t A ] ] = ^ . F ^ F . F ^ G ^ E . G 1 ^ 1 1 1 50. Hence, by (3) and Lemma 0.4.1 5 i s a c y l i n d e r measure over X- We s h a l l now prove that (4) £*(B) = u*(r_. "V] n B) for a l l B c x ' L, A Let k k H = {H e Cyl (X ) : H i s w -open} , a = r^'V] . We have that H C Cyl (X*) A Cyl (X*) and by (2), T^(H) = T (H A a) f o r every H e H Hence, by Thro.. 3.1, (5) ?"(H) = T"(H A a) for a l l H e H y a Let B c X If y (B) < TO , then, since y i s an H -outer measure, for any e > 0 there e x i s t s H e H s.t. a . a B c H and y*(H) < y*(B) + e . Since a e M , we have that y« A • A A A y (B n a) + y (B ~ a) = y (B) > y (H) - e k k = y (H n a) + y (H ~ a) - e and therefore •A. a . y" ( H A a) < y" (B A a) + e . k Since y i s a - f i n i t e i t follows that f o r any B C X , (6) y*(B n a) = i n f {y"(H A <*.} : B C H e H } . a A A Since £ and y are both H -outer measures, (5) and (6) o together imply that (4) holds. Proof of 5.6.1. By Lemma 3.3 w£ need only show that (1) y X(A) = y * ( A n ft) for a l l A e Cyl (X*) i f f (2) E, (X ) = E, (ft) for every f i n i t e s e c t i o n E, of y From Remark 5.5 i t i s immediate that (1) => (2). However, (2) => y"(A) = y' (A n ft) for a l l A e' Cyl (x") with y (A) < oo . For any A e Cyl^(X ) , since y i s a - f i n i t e , choose an increasing sequence {A } i n Cyl (X ) s.t n new J y y (A ) < 0 0 for a l l n e w and A = U A new n Since y i s an outer measure, we then have that p (AO ft) = lim y * ( A A ft) = li m y (A ) = y (A) . n n new new Hence (2) =3> (1). Proof of 5.6.2. Let E, be a f i n i t e section E e F and A e M with y„(A) < E E Remark 5.5, for any F e F with V ^ F [ A ] - r F , X [ K ] ) = ~ of y determined by some oo . By (2) i n the proof of E C F , and K e K(G) , r F , X ^ } • The assertion now follows from Lemma 5.1.2, and Thm. 4.3. CHAPTER II CYLINDER MEASURES OVER TOPOLOGICAL VECTOR SPACES In t h i s chapter, we are p r i m a r i l y i n t e r e s t e d i n determining when a cylinder measure over a Hausdorff l o c a l l y convex space X w i l l have a l i m i t measure on the t o p o l o g i c a l dual X' which i s Rad^n with respect to some given topology G on X' . Since (X,X') i s a dual p a i r , the theory of the previous chapter applies with 0, = X' A Hence, i f G i s regular and f i n e r than the w -topology r e s t r i c t e d to X' , then, by Theorem 1.4.3, y w i l l have a G-Radon l i m i t measure on X' whenever y i s H-tight for some family H c K(G) . We s h a l l take G to be one of three standard topologies, and these suggest that we take f o r H the p a r t i c u l a r family E defined below. Our main concern i s then directed towards f i n d i n g conditions under which y i s E-t i g h t . 1. Notation , We point out that our t o p o l o g i c a l vector spaces are not • assumed to be n e c e s s a r i l y Hausdorff. In the rest of t h i s paper we s h a l l use the following notation. For any vector space X and V c X , V° = {f e X" : | f ( x ) | <_ 1 for a l l x e V} . 53. For any t o p o l o g i c a l vector space X , nbnd 0 i n X i s the family of'neighbourhoods of the o r i g i n i n X k k E i s the family of a l l sets K c X such that K i s w -closed and K c for some V g. nbnd 0 i n X •k X' = {f e X : f i s continuous} and for every F e F , A. r F = r F l X ' • In addition to the w -topology r e s t r i c t e d to X' , we s h a l l consider the following two topologies: c i s the topology on X' of uniform convergence on the compact subsets of X , s i s the topology on X' of uniform convergence on the bounded subsets of X Remark We note that E c K(w*) and •k | k A W X' C C C S 54. 2. E-tight Cylinder Measures. Throughout t h i s section X i s a t o p o l o g i c a l vector space and y i s a cylinder measure over X When X i s l o c a l l y convex and Hausdorff we notice that E i s nothing else but the family of w -closed equicontinuous subsets of X' Hence, f r oiu TrsvGs [47] , Props* 32.5 and 32.8, W G h.avG t h a t E c K(c*) . Consequently, a p p l i c a t i o n of Theorem 1.4.3 y i e l d s the following a s s e r t i o n . 2.1 Theorem. Let X be Hausdorff and l o c a l l y convex * • y u i s E-tight => p. has a c -Radon l i m i t measure on X' If E = K(c ) , i n p a r t i c u l a r , i f X i s b a r r e l l e d ([47] Thm. 33.1) then, y i s E-tight <=> y has a c -Radon l i m i t measure on X' Sometimes E-tightness of y can also imply the existence of a l i m i t measure on X' which i s Radon with respect to the s -topology. For example, i f X i s a Montel space ([47] p. 356) then E = K(s ) ([47], Prop. 34.5); or, i f X i s a nuclear space ([47] p. 510) then E c K(s") ([47] Prop. 50.2). Hence, on applying Theorem 1.4.3, we obtain the following theorems. 2.2 Theorem (1) If X i s a Montel space, then, y i s E-tight <=> y has an s -Radon l i m i t measure on X' (2) I f X i s a nuclear space, then, y i s E-tight => y has an s -Radon l i m i t measure on X' Even i f E £ K(s ) , E-tightness of y can s t i l l imply that y has an s -Radon l i m i t measure on X1 2.3 Theorem. Let X be Hausdorff and l o c a l l y convex. For any V e nbnd 0 i n X l e t X^ = nV® with the topology induced by the norm |* : f e X^ + s u p | f ( x ) | e R + x«=V If there i s a base \l for nbnd 0 i n X such that for each V £ 1/ , X^ i s separable. then, (1) y i s E-tight => y has an s -Radon l i m i t measure on X' (2) If X' i s a separable Banach space under s , then indee * „ k < every f i n i t e c -Radon measure on X' i s s -Radon. Proof of Theorem 2.3 We f i r s t e s t a b l i s h the following lemma. Lemma. For any V e nbnd 0 i n X s.t. X^ i s separable, i f G i s the family of a l l open subsets of X^ , then Gc cr-field generated by (c |x̂ ) 56. Proof Let ff =' { f + E V ° : e > 0 , f e X^} . For any e > 0 and f e X' , g e X' -y eg e X 1 and g e X' -> f + g e X' are homeomorphisms with respect to c , since X ! i s a t o p o l o g i c a l vector space under c 0 * * * . Hence, since V i s w -closed and w c c , H consists of c -closed subsets and therefore f/ c c r - f i e l d generated by (c |X^) However, X^ i s separable and metrizable. Consequently, G c C a - f i e l d generated by (c |X^) We now prove the theorem. (1) By Thm. 1.5.6.2, we may assume that u i s normalized. In that case, by Lemma 1.5.1.2, we need to prove that for any e > 0 ., there e x i s t s K e K(s ) s.t. u„(F ~ r_[K]) < e for every F e F . r r With the notation of 1.4, since u i s E - t i g h t , then, by Thm. 2.1 and Props. 1.4.1, ]i' ̂  i s c -Radon l i m i t measure on X' of , and by L.3 of Proofs 1.4, . (1) u c*(K) = g(K) for a l l K e K(c*) . Since u i s E-tight, for any e > 0 , there e x i s t s V e nbnd 0 i n X s.t. X^ i s separable, and for a l l F e F , y p ( F * ~ r F[V°]) < e/2 . » 57. Hence, by L.2.1 of Proofs 1.4, 1 - g(V°) < e/2 , and therefore, by (1), (2) y c > v(X' - X^) < 1 " V C*(V°) < e/2 . If 5 : A c X^ + u^(A) e R + ,. then 5 i s a c |X^-outer measure on X^ * i By the lemmas, and the f a c t that c [X^ c 6 , i t follows that £ i s a G-outer measure on X^ However, X^ i s complete ([47], Lemma 36.1, see also p. 477)- X^ i s also separable and metrizable. Hence, by Prohorov L32] Thm. 1.4, there e x i s t s K e K(G) s.t. (3) 5(X^ ~ K) < E/2 However, s |x̂  c G , hence K E K(s*) • Then, c e r t a i n l y , K E K(C ) , and by (1), (2) and (3), g(K) = u^CK) > 1 - e . From the d e f i n i t i o n of g , and L.2.1 of Proofs 1.4, u„(F" ~ r_,[K]) < s for a l l F e F . r r (2) From the lemma we have any c -Radon measure on X' i s an s -outer measure (G = s ) . The assertion now follows from Prohorov [32] Thm. 1.4 We are led by the above theorems to study conditions under which w i l l be E-tight. In view of Theorem 1.5.6.2 we s h a l l concentrate on the case when u i s f i n i t e . Conditions for u " to be E-tight w i l l then be given i n terms of the one-dimensional subspaces of X . We begin by i n d i c a t i n g a necessary such condition. 2.4 Proposition Let X be a t o p o l o g i c a l vector space, r > and p a f i n i t e c y l i n d e r measure over X If p i s E-tight, then, for any e > 0 , there * , k e x i s t s a w -Radon measure n on X with, supp n e E , such that x e X , J | f ( x ) | r d n ( f ) <_ 1 => y„ '({f e F* : | f ( x ) | >_ 1}) < F X. x where F^ i s the space spanned by x Proof. We assume that p i s normalized. With the notation of I.4, by Th. I.4.3, u , i s a w -Radon l i m i t measure on X for p ' W " Since p i s E-tight, by L.2.1. of Proofs 1.4, for any e > 0 there e x i s t s V £ nbnd 0 i n X s.t. 1 - g(V°) < E/2 , and therefore, by L.3 of Proofs 1.4, p",(X* ~ V°) < e/2 . For any x e•X , l e t I = {f e XW : |f(x)j > 1} . If r) : A C x" - y" , (A A V°) e R + , then, k £ W " k n i s a w -Radon measure on X with supp n e E and for any x e X s.t. J ] f (x) {r dn <_ 1 , P F ({f e F* : | f ( x ) | > 1}) = V**(I X> x = V ( I x ^ ° ) + V ( I x ~ - 2 N ( I -} + 2 2 $ 1 I d r' + e / 2 < f / | f ( x ) | r d n ( f ) + f i f + f = e • m The above proposition suggests the following d e f i n i t i o n s . 2-4 Notation. For any vector space X , x e X and c y l i n d e r measure y over X , F^ = space spanned by x,. D =' {f e F* : If (x) I > 1} x x 1 1 — and \ = y F 2.5 D e f i n i t i o n s . Let X be a vector space, and U be a family of subsets U of X with 0 e U (1) For any f i n i t e c y linder measure y over X , y i s U-continuous i f f for any e > 0 there e x i s t s U £ U such that x £ U = > y ( D ) < £ x x (2) For any c y l i n d e r measure y over X , y i s U-continuous i f f every f i n i t e s e ction of U i s U-continuous. (3) For any t o p o l o g i c a l space Y and T : X -> Y , T i s U-continuous i f f for every neighbourhoods V of T(0) there e x i s t s U e U such that T[U] c V . (4) When X i s a t o p o l o g i c a l space, for any f i n i t e c y l i n d e r measure y over X , y i s continuous i f f y i s f-continuous for some family V of neighbourhoods of the o r i g i n i n X only the standard The discussion of l i m i t s i n the re s t of the chapter requires above concepts. However, to explain t h e i r r e l a t i o n to notions of continuity we introduce the following d e f i n i t i o n s . 60. Let X be a vector space. For any n e w and x = -(x^,. . . ,x _^) - e X n , F = l i n e a r span of {x„,...,x , x U n-1 • Cf x J f e F* -y ( f ( x Q ) , . . . , f ( x n _ 1 ) ) e <En , and for any f i n i t e c ylinder measure y over X , X For any n e w , M(|]n) i s the family of f i n i t e Radon measures on ^ n endowed with the vague topology; i . e . for any net (n.) . , i n M((j;n) and n E M(f]n) , Tij + n i n M(fpn) i f f J fdn.. -»- / fdn for every bounded continuous f : <£n -> We note that y e M(£ n) for a l l n e OJ and x e X n X We now have the following well-known proposi t i o n (Gelfand, V i l e n k i n [11] P. 310, Fernique [9] p. 37, which shows that one can n a t u r a l l y associate c e r t a i n continuous maps with a continuous c y l i n d e r measure. (see also Appendix 1.7). 2.6 Proposition Let X be a t o p o l o g i c a l vector space. (1) y i s continuous i f f (2) y : x e X -> y^ e M($) i s continuous at 0 i f f (3) for each n e w , y 1 1 : x E X n y e M((£n) i s continuous with respect to the product topology on X 61. Remark From the proof of the above proposition one r e a d i l y checks that the following a s s e r t i o n also holds. If U i s a family of balanced, absorbent subsets of X , with tU e (J 'for every U e U and t > 0 , then, u i s (J-continuous i f f u : x e X -> u e M(C) i s (J-continuous. Hx Proof of Proposition 2.6. We show that (3) => (2) => (1) => (3) (3) => (2) take n = 1 . (2) => (1) l e t % : £ -> C be bounded and continuous, %(z) = 1 i f j i l > 1 0 i f |*| < | Let e > 0 . Choose V e nbnd 0 i n X s.t. x e V => 1/ %d~^ - / 5Cdu~| < e . Since y^ i s concentrated at the o r i g i n i n t , and has f i n i t e mass, then J = o • . Hence, for any x £ V , y (D ) = y v({z e £' : |z| L 1» 1 / *dy v < e • A . A . A . A . (1) => (3) For any F e F , and u > 0 , l e t I(u) = {(w,f) E F x F* : |f (w) | >_ u} Let n e OJ , X • ^ n ->• £ be hounded and continuous, and x e X n We s h a l l show that for any £ > 0 there e x i s t s V e nbnd 0 i n X s.t. 62, y e x + V n => |/ %dy x - / %dy^| '< e . (Note: V n e nbnd 0 i n x " ) . We assume that u i s normalized. For any z E £ n. , l e t [z| = sup {|z | : k = 0,...,n - 1} Let (i) M = sup (|%(z)| : z E £ N } , ( i i ) W be an open nbnd of the o r i g i n i n X s.t. w e W => u (D ) < e/16nM • , w w ( i i i ) t > 0 be s.t. x e tW for each k = 0 n - 1 , (iv) and l e t 6 > 0 be s.t. z j e £ U |z j | <_ t , j = 0,1 , and |z° - z 1] < 6 => \%(z°) - X i z 1 ) | < e/4 . ' Since £ - WN i s an open nbnd of 0 i n X N , there e x i s t s V e nbnd 0 i n X s.t. (v) -| V e W , (vi) x + V U c t W n . For any y e x + V n 'and F e F with F^ u F^ c F , l e t y > y F x 1 x F y • x A (5) = {f E F* : \v (f) - V (f) I > 5} y y x 1 - and Then, k=n-l B (t) = U I (t) v v * k=0 "k . y F ( A (6)) = y F ( k U 1 I (6)) J k=0 k k k=n-l. k=n-l " k=o y F ( V 6 ) ( x k - y k ) ( 1 ) ) - k ^ 0 ^ i / ^ k - y k ) ( 1 l i / 6 X x k - y k ) > < e/16M by ( i i ) and (v). From ( v i ) , we have that y e tW Hence, — v, e W f o r each k e n , and therefore, t k by reasoning as above, y (B ( t ) ) < e/I6M . b y In p a r t i c u l a r , y F ( B x ( t ) ) < e/16M . Consequently, i f B = B (t) U B (t) V A (5) , x y y then, y (B) < 3 e/16M < e/4M , and, by ( i v ) , f e F* ~ B => \X(V (f)) - ( f ) ) | < e/4 y x 1 Hence, 1/ X d y y - / X d y J = 1/ % 0 y y F - / % 0 ^d,^} < /| X , V - % 0 ¥ d y + / |X o ¥ - X 0 ¥ |d,i ~~ B y X F F*~B Y X F < 2M.e/4M + -f . U(F" ~ B) < e . — 4 F i . e . y e x + V n => | J X d y ^ - / % d y j < e . 3. Limits of Continuous Cylinder Measures Let X be a Hausdorff, l o c a l l y convex space, and y be a f i n i t e c ylinder measure .over X . In the previous section we have seen that conditions under which y i s E-tight are important for determining when y has a c - or an s -Radon l i m i t measure" on X In terms of the one-dimensional subspaces of X Proposition 2.4 gives a necessary such condition. In seeking some kind of converse to that proposition, we are l e d to introduce the concept of a weighted system i n X , which i s defined below. We s h a l l use the following notation. 3.1 Notation For any vector space X , absorbent absolutely convex V c X , and F E F , ker V = { x e X : x e t V f or every t > 0} , F v = F cv ker V , (V c\ F)° = {f e F'C : |f (x) | <_ 1 for a l l x e V f\ F} = {f E F" : f(x) = 0 for a l l x c 7^} For any t > 0 , I ( t ) = {(x,f) E F x F* : | f (x) | i.t} , I = K D • So for any x e F , f e F , I f = {x e F : |f (x) | >_ 1} , I = {f E F* : I f ( x ) | > 1} . • •v 1 — Remarks We note that (1) F v e F ; and since (x,f) e F x F f(x) e C the product topology on F x F , (2) for any t > 0 , I(t) i s closed. 3.2 D e f i n i t i o n s i s continuous with respect to c Let X be a l o c a l l y convex space. (1) (v,F»f) i s a system of S-weights i n X i f 6 > 0 ; [/ i s a family of absolutely convex neighbourhoods of the o r i g i n i n X , F c F i s directed, by C and Li F i s dense i n X and v : V e l / , F e F - > V y p , a p r o b a b i l i t y Radon measure measure on F for which f e F* ~ (V n F)° => v V j F ( I f ) >• 6 . When f i s a singleton {V} , we s h a l l write (v,F,V) instead of (v,F,l/) • (2) W i s weighted by such a system (v,F,f) i f f W i s a family of neighbourhoods of the o r i g i n i n X , for each W e W there e x i s t s V c V such that ker V c W , and v„ „(F ~ tW C\ F) -* 0 as t -> » , uniformly f o r F e F, 66. (3) (!) i s a weighted system i n X i f f W i s weighted by some system of 6-weights i n X We s h a l l now state and prove the fundamental r e s u l t s of th i s section. 3.3 Theorem Let X be a Hausdorff, l o c a l l y convex space, and fx be a f i n i t e c y l i n d e r measure over X If y i s W-continuous for some weighted system i n X , then y i s E-t i g h t . C o r o l l a r y Let X be a Hausdorff, l o c a l l y convex space, and y be an a r b i t r a r y c y l i n d e r measure over X If y i s W-continuous f o r some weighted system i n X , then y i s E-ti g h t , and therefore, y has a c -Radon l i m i t measure on X' Proof of C o r o l l a r y By Thms. 3.3, I. 5.6.2, and 2.1. We s h a l l need the following lemmas i n the proof of Theorem 3.3. They are proved at the end of the section. Lemmas (1) Let F be a f i n i t e dimensional space, and z• > 0 If £ i s a f i n i t e Radon measure on F such that x e F => ^ ( I ) < e x then £(F* ~' {0}) ±e . (2) Let X be a l o c a l l y convex space and p a continuous f i n i t e c ylinder measure over X For any dense subspace Y of X , and e > 0 , i f V i s an absolutely convex neighbourhood of. the o r i g i n i n X such that y p ( p " ~ (V A F)°) <_ e for every F e F r , then p„(F~ ~ ( V f \ F)°) < e for a l l F e F v r — A Proof of Theorem 3.3 By the Hahn-Banach extension theorem, we see that for any F e F . , and absolutely convex V e nbnd 0 i n X , (1) r p [ V 0 ] = (V A F)° . Hence, by Lemma 1.5.1.2, we need only prove that, for any e > 0 , there e x i s t s V e nbnd 0 i n X s.t. (2) y_(F* ~ ( ¥ A F)°) < e for a l l F e F . r — We assume that p i s normalized. Let W be a weighted system i n X with respect to which p i s W-continuous, and l e t (v,F,l/) be a system of 6-weights i n X by which W i s weighted. For any e > 0 , l e t 0 < e' < min(<5e,e) , W e W s.t. x e w => y (D ) < e'/4 , x x — V e V and t > 0 s.t. ker V C W and v v F ( F ~ tW r\ F) < e'/4 for every F e F Let U = V/t . Suppose that * 0 y (F ~ (U c\ F) ) <_ E for every F e F Since F i s directed by c , then U F i s a subspace of. X , and for any f i n i t e dimensional subspace E of U F there e x i s t s F e F with E c F . Hence, by (1) , (3), and L.2.3 of Proofs 1.4, y E ( E * - (U r% E)°) < £ . Since W i s a family of neighbourhoods of the o r i g i n i n y i s n e c e s s a r i l y continuous, and by hypothesis, U F i s dense i n X . Hence, from Lemma (2) and the foregoing remarks we conclude that (2) holds. It remains for us to e s t a b l i s h (3). For any F e F , y F ( F * ~ (UAF)°) = y p ( F * . ~ t ( V n F)°) • = y p ( F a ~ t(V n F)°) + y p ( F * ~ F^) since (V A F) ° c F?T . We show that each of the l a s t two terms given i n (4) i s les s than e/2 We estimate the f i r s t term. Since (v ,F,l/) i s a system of 6-weights i n X , the f £ ~ t . (V ft. F)° => f e F* ~ (Vft F)° => v V j F ( I f ( t ) ) = v y j F ( I f / t ) > 6 . Consequently, 6.u F(F^ ~ t.(V f\ F)°) f { v v > F ( I f ( t ) ) : f £ F J ~ t.(V A F)°}.p F(F^ ~ t . ( V f t F ) 0 ) <. Vy F * v F ( K t ) ) by Sion [44] Ch. I l l , Thm. 1.2.6, = J F / F ; V 1 I ^ d U p d V y F by Fubini's theorem, = / p F ( I x ( t ) ) d v V j F ( x ) = / S F ( I x ( t ) ) d v V j F = ^ F ( I x ) d \ , F ( x ) = / y ° x ) d V V , F ( x ) t t t XtW F - y _ x ( D x ) d v V , F ( x ) + f V t W F ' y x ( D x ) d V V , F ( x ) t t t t - f ~ " v V , F ( t W A F ) + 1 , V V F ( F ~ t W A F ) < f . l + l . f < 6.f . Hence, y p ( F * ~ (U ft. F)°) < e/2 . We now estimate y„(F " F ) F V We have that x e F T => x E ker V => x E W => y (D ) < e'/4 V x x => y_ (I ) < e'/4 since F c F 1 T e F . Fy x x V Hence, by Lemma (1), y (F* ~ {0}) < e'/4 . V 70. Since r p [F* ~ F J ] C F* ~{0}' , F V,F V i t therefore follows that U f ( F " ' ? - ?p < u^r'^fF* ~'{0}]j = Uv (F" ~ {0}) < e'/4 < e/4 . Then, c e r t a i n l y , (6) u F ( F * ~ F*) < £/2 . From (4), (5), and (6) we see that (3) holds. Remark. We point out that the theorem s t i l l holds when we use a somewhat weaker notion of system of 6-weights, i n which F : V e l / - > F c F d i r e c t e d by C and [J F v i s dense i n X The other d e f i n i t i o n s remain unchanged. Proofs 3. Proof of Lemma (1). For any x e F and n e w , and Consequently, f o r any x e F , . x =)= 0 , 5({f e F* : f (x) f 0}) = U I .(-)) new = lim 5(1 (-)) < e x n — new 71. Hence i f (1) there e x i s t s y e F s.t. •£({f e F* : f (y) = 0} ~ {0})- = 0 , then 5(F~ ~' {0}) = C({f e F* : f (y) ={= 0}) + ?({-f E F* : f (y) = 0} ~ {0}) We s h a l l e s t a b l i s h (1) by induction. For any subspace E of F , l e t E a = {f e F" : f(x) = 0 for a l l x e E} Let dim F = n . If n = 1 then (1) holds. We therefore assume that n >̂  2 . For any k-dimensional subspace G of F with 2 <_ k <_ n , (2) {H 3^G A : H i s a (k - 1)-dimensional subspace of G} i s an uncountable, d i s j o i n t subfamily of M Let GQ = F . Then, by (2) and the f i n i t e n e s s of F, , there e x i s t s an (n - 1)-dimensional subspace G^ of F s.t. 5(G A -' {0}') = 5(G A ~ G A) = 0 . For any 0 <_ k <_ n - 2 , i f there e x i s t s an (n - k)-dimensional subspace Ĝ . of F s.t. 5(G A - {0}) - 0 , Then, by (2) and the f i n i t e n e s s of E, , there ex i s t s an (n - k - 1)-dimensional subspace G of G1 s.t. ^ G k + i * GV =0 • Consequently, aG a + 1 - {0}) = 5 ( G A + 1 - G A) + C ( G A ~ {0}) = 0- . Hence, there e x i s t s a one-dimensional subspace G ., of F s.t. n-1 5(G A . ~ {0}) = 0 . n-1 i . e . (1) holds. 72. Proof of Lemma (2) For any F e F v , n e w, X e F n , and t > 0 , l e t A k=n-l A* (t) = A { f e F" : | f (x ) | < t} . • X k=0 k We s h a l l assume that u i s normalized. Since (V A F ) ^ = (VQ r\ F ) ^ , where i s the i n t e r i o r of V , we s h a l l further assume that V i s open. Let E e F v . A Since E i s separable there e x i s t s a countable, dense subset {x } of V A E . Then, n new (V r\ E)° = A {£ e E* : | f (x^) | £ 1} . kew Now, for any n e w , tv1^1* : |£°vi ^ Acx n.....x • k=0 mew 0 n-1 Consequently, for any 6 > 0 , there e x i s t s n e w and m e w s.t. P E ( ( V ^ E ) V V A * ( 1 + 1 ) ) < V E ( ( V A E ) ° ) + { . 0 n-1 Since y i s continuous, there e x i s t s U e nbnd 0 i n X s.t. (1) u e U => y u ( D u ) < 5/2n , and since Y i s dense i n X and V i s open there e x i s t s { y 0 ' " ' " ' y n - l } i n V S - t " (2) x, - y, e - U for a l l k = 0, . . . ,n - 1 . k k m Let F e F,, be such that E U {y„,...,y ,} C F and X 0 n-1 l e t x = ( X Q , . . . ^ ^ ) , y = (yQ.-.-.y^) Then, V F* " Ax-y(™)} = U 6 F* : |f(Xk " V 1 k=0 k=n-l - E yx,-y (Dm(x -y )} < 6 / 2 b y ( 1 ) a n d ( 2 ) * k=0 *k y k n H x k V Further, • A Y ( 1 ) C - A X ( 1 + 7 > • Hence, J , , , , . ,,F ,1s . , F „ , , , ,.F,„ N „ , F ,1 -y W 1 } ) = W y t t * A Y ( 1 ) ) + W 1 ' ~ Vy? 5 < u E((V r\ E)°) + 6 . i . e . (3) y F ( A | ( l ) ) < y E ( ( V r N E)°) + <5 . However, i f F denotes the l i n e a r soan of {y„,...,y , } y J0 Jn-1 we observe that A F(1) 3 r " 1 [ ( V A F )°] . y „ > y y Since F e F , and (V r\ F ) i s closed, we have that y Y y (V (\ F ) £ Kp and y V A y ( 1 ) ) > y F ( r F X F [ ( V A F )°]) = y p ((V A F )°) > 1 - e y' y Hence, by (3), u E((V A E)°) > 1 - £ - 6 . Since 6 was a r b i t r a r y , i t follows that u E((V A E)°) >_ 1 - £ . Consequently, since (V A E ) ^ £ M̂ , , u E(E' f ~ (VA E)°) < e . 74. 4. Induced Cylinder Measures. I t can happen that a f i n i t e c y l i n d e r measure over a Hausdorff, l o c a l l y convex space X i s given i n d i r e c t l y . For example, i t may have been induced by a f i n i t e c y l i n d e r measure u over a vector space Y and a l i n e a r map T on X to Y ([11] p. 311). In such a s i t u a t i o n we s h a l l be i n t e r e s t e d i n obtaining conditions on ]-~ and T which w i l l ensure that the induced c y l i n d e r measure over X w i l l have a l i m i t measure on X1 , Radon with respect to some given topology on X* . This kind of problem seems to have been f i r s t mentioned i n [H] Ch. IV. I t has been studied extensively, by L. Schwartz, 5. Kwapien, and others, i n a s e r i e s of papers ([19], [20], [39] - [42]). In view of the previous theory, our emphasis w i l l be on determining conditions under which the induced c y l i n d e r measure w i l l be E-tight, Using the notions of c o n t i n u i t y and weighted system we r e a d i l y obtain such conditions. 4.1 D e f i n i t i o n For any vector spaces X and Y , l i n e a r map T : X -> Y , and f i n i t e c ylinder measure y over Y , the cylinder measure £ over X induced by y and T i s defined as follows: for each F e F v > A where T p i s the adjoint of T[F , i . e . T* : f e CT[FJ)* -> f 0 (T | F) e F* . We s h a l l denote t h i s induced c y l i n d e r measure K by u D T . We prove below that 5 i s indeed a cyl i n d e r measure over X Proof For each E e F„ , T i s continuous. X E Hence, by §0.4, £ i s a f i n i t e Radon measure on E E Since a l l . the maps considered are continuous, then, by §0.4 and R.emark 1.2.1, for any E and F i n F with E c F , " X rE,F C r'F ] = r E , F t T F [ u T [ F ] ] ] = r E , F 0 V y T [ F ] ] T E 0 r T [ E ] , T [ F ] [ y T [ F ] ] = V r T [E] ,T[F] [ y T [ F ] ] ] V y T [ E ] ] = ? E ' and therefore, again by Remark 1.2.1, 5 i s a cyl i n d e r measure over X . B We now prove the following important lemma. 4.2 Lemma For any vector space X , family U of subsets U of X with 0 e U , t o p o l o g i c a l vector space Y , and l i n e a r T : X i f T i s U-continuous, then u a T i s U-continuous for every continuous f i n i t e c ylinder measure over Y 76. Proof. Let y be a continuous f i n i t e c y l i n d e r measure over Y For any x e X , by Lemma 0.4.2, <U a T ) X ( D X ) = T F [y ] ( D X ) = . T x ( T ; - 1 [ D x ] ) = y: (D ) . x :-. r, x K For any e > 0 , there e x i s t s V e nbnd 0 i n Y s.t. y e V => y (D ) < e , y y and there e x i s t s U e U s.t. T[U] C V . Then, by the f i r s t a s s e r t i o n , x e U => TJC e V => (y Q T ) X ( D X ) = P T X ( D T X ) < e • I t follows that y -,3 T i s (i-continuous. ©• Our key theorem on induced c y l i n d e r measures i s now an immediate consequence of Theorems 3.3, 2.1, and the above lemma. 4.3 Theorem. Let X be a Hausdorff, l o c a l l y convex space, Y be a top o l o g i c a l vector space, and T be a l i n e a r map on X to Y If T i s (^-continuous for some weighted system W i n X , then for every continuous f i n i t e c y l i n d e r measure over Y , y cs T i s E-tight are therefore •k y C\ T has a, c -Radon l i m i t measure over X Remark. I t i s clear that this theorem reduces to the f i n i t e case of Corollary 3.3 when X = Y and T i s the i d e n t i t y map. 77. CHAPTER III APPLICATIONS We s h a l l apply the theory of the previous chapter to a study of c y l i n d e r measures over H i l b e r t i a n and 5,^-spaces. Our r e s u l t s on c y l i n d e r measures over a r b i t r a r y Hausdorff, H i l b e r t i a n spaces generalize and c l a r i f y many known theorems (Minlos [25], Sazonov [35], Badrikian [1], Fernique [9]). In the case of £^-spaces we obtain s i g n i f i c a n t extensions of formerly known r e s u l t s (L. Schwartz [39], Kwapien [19]). Our main t o o l i s C o r o l l a r y II.3.3, which requires us to construct weighted systems i n the above spaces. In view of Proposition II.2.4, i t i s the search f o r such systems which leads us to consider the f a m i l i e s S , for r > 0 , defined below. 1. Preliminaries For any vector spaces X and Y , L[X,Y] i s the set of l i n e a r maps on X to Y For any t o p o l o g i c a l vector space X , CM(X) i s . t h e family of continuous f i n i t e c y l i n d e r measures over X 78. Remarks. From Appendix 3 . 1 . 1 and 3 . 2 . 1 , we have that for any family C of f i n i t e c y l i n d e r measures over a vector space X , there e x i s t s a coarsest topology on X under which i t i s a t o p o l o g i c a l vector space, and such that p e C => u i s continuous. This topology i s c a l l e d the C~topology. For any t o p o l o g i c a l vector space X , i f the topology of X i s the CM(X)-topology, then we c a l l X a CM-space (Appendix 3 . 1 . 2 ) . For any t o p o l o g i c a l vector space X , 0 < r < ra , and w -Radon measure n on X with supp .n e E , S = {x e X : / | f ( x ) | r d n ( f ) <1} . For each r > 0 , S r i s the family of a l l sets S c X . r, n 1 . 1 Remarks Let X be a t o p o l o g i c a l vector space. (1) For each r > 0 , there i s a unique topology on X under which X i s a t o p o l o g i c a l vector space having S as a base for i t s neighbourhoods of the o r i g i n . When r >_1 , t h i s topology i s l o c a l l y convex. We s h a l l c a l l t h i s topology the S r-topology. t r (2) If 0 < r < t , then S i s f i n e r than S , i . e . for every a e S3" there e x i s t s 3 e S*" with 3 c a (3) I f X i s l o c a l l y convex, then, for each r > 0 S*" i s a family of neighbourhoods of the o r i g i n i n X 79. We prove only 1.1.2. ' Proof of 1.1.2. For -any f i n i t e measure space (£2,n) and integrable f : 0 -> £ , i f 1 1 p = t / r and • 1- — = 1 ,' P q then, by >.L;>;H«AJ&#'s i n e q u a l i t y , /|f|rdn 1 (/;|f|r)?dn)1/p nCn> 1 / q - . Hence, ( l ) ( J|f| r d n ) 1 / r K / l f l ' d T , ) 1 7 ' n ( a ) ( t " r ) / r t • r For any S e S , r ,n n(x x) < <» , since supp n e E C K(w ') and n i s w -Radon. Consequently, by (1), i f K - n ( x A ) ( t - r ) / r t . n then S ,- C s t,5 r , n The assertion follows. To point the s i g n i f i c a n c e of the f a m i l i e s 5 we note that Proposition II.2.4 can be restated as follows. 80. 1.2 Proposition Let X be a top o l o g i c a l vector space. For any f i n i t e c y l i n d e r measure u over X , u i s E-tight => u i s S 1-continuous for every r > 0 . @ Mien X i s Hausdorff and l o c a l l y convex, the above proposi- t i o n and Theorem II.3.3 y i e l d the following a s s e r t i o n : i f u i s W-continuous for some weighted system W i n X , r then u i s S -continuous f o r each r > 0 In view of t h i s , when searching f o r weighted systems i n X we s h a l l look f o r s u i t a b l e subfamilies of S In general, S r - c o n t i n u i t y f o r some r > 0 does not imply E-tightness. (Example 1, Appendix 4). We s h a l l need the following r e s u l t on induced c y l i n d e r measures. 1.3. Proposition. Let X be a t o p o l o g i c a l vector space, Y be a vector space, and T e L[X,YJ . For any family C of f i n i t e c y l i n d e r measures over Y , i f u a. T i s E-tight for every u e C , then, for each r > 0 , T i s S -continuous with respect to the C-topology on Y 81. Proof. Let r > 0 . By Prop. 1.2, IT y e C => y a T i s S -continuous. Hence, by Appendix 3.2,1, the S -topology i s f i n e r than the (C o T)-topology on X . By Appendix 3.2.2, th i s says exactly that T i s S r-continuous with respect to the C-topology on Y . Q 2. H i l b e r t i a n Spaces. Throughout t h i s s e c t i o n , X i s a Hausdorff, H i l b e r t i a n space ([1]). i . e . X i s a Hausdorff, l o c a l l y convex space, for which there e x i s t s a family F of pseudo-inner products on X , such that nbnd 0 i n X has as a base the family of a l l sets {x e X : [x,x] <_ 1} , [.,.] e r . The fundamental theorem of t h i s section i s the following. 2.1 Theorem For each 0 < r < 0 0 , S i s a weighted system i n X The proofs of this and other assertions w i l l be given at the end of the section. Now, we concentrate on the consequences of the above theorem. 2.2 Theorems. Let y be a cy l i n d e r measure over X and .0 < r < 0 0 Then, (1) y i s E-tight <=> y i s S -continuous. (2) y i s S -continuous => y lias a c -Radon l i m i t measure on X V ' ' (3) I f K(c ) = E , i n p a r t i c u l a r , i f X i s b a r r e l l e d , then u i s 5 -continuous <=> u i s E-tight <=> \x has a c -Radon l i m i t measure on X' Using Theorems 2.2, we can now characterize c e r t a i n p o s i t i v e - d e f i n i t e functions on X (Appendix 2). 2.3 Theorem. Let i> be a p o s i t i v e - d e f i n i t e function on X and 0 < r < «> Then, i> i s ^-continuous => there e x i s t s some f i n i t e c -Radon measure E, on X' such that ij;(x) = J exp i Re f (x)dg(f) for every x e X If K(c ) = E , i n p a r t i c u l a r , i f X i s b a r r e l l e d , then ^ i s S -continuous <=> there e x i s t s some f i n i t e c -Radon measure E, on X' such that i> (x) = /exp i Re f(x)d£(f) for every x e X Remarks We note that Theorem 2.2.2 generalizes a r e s u l t of Minlos ([25] p. 303 Thm. 1). Theorem 2.3 generalizes r e s u l t s due to Minlos ([25] P. 310), and Badrilcian ([1] p. 16 Cor. 1). The s p e c i a l case when X i s a H i l b e r t space w i l l be discussed below (§2.7). 84. We point out that, with the viewpoint of §1.4,- the assertions of Theorems 2.2.2 and 2.3 for the case r = 2 can be established by using the technique of c h a r a c t e r i s t i c functionals ([1], p. 9, 2 Lemma 1, Prohorov [33]). Also, i t can be shown that the S -topology i s nothing else but the Gross-Sazonov topology on X ([35], [1], [13] p. 65). By means of Proposition 1.2 and Remark 1.1.2 we can deduce the assertions above for 0 < r < 2 from the case r = 2 . We have been unable to give a s i m i l a r deduction for the case r > 2 . However, i n t h i s context, we draw attention to §2.6 below. As consequences of Theorems 2.1, II 4.3, and Proposition 1.3, we have the following assertion concerning induced c y l i n d e r measures over X 2.4 Theorem Let Y be a vector space, T e L[X 3Y] , and 0 < r < 0 0 For any family C of f i n i t e c y l i n d e r measures over Y , y n T i s E-tight for every u e C <=> T i s S -continuous with respect to the C-topology on Y The above theorem y i e l d s immediately the c o r o l l a r i e s given below,. Corollary (2) s i g n i f i c a n t l y generalizes a r e s u l t i n [11] (p. 349).. 85. C o r o l l a r i e s Let Y be a t o p o l o g i c a l vector space, T e L[X,Y] , and r > 0 (1) If Y i s a CM-space, then y D T i s E - t i g h t for every u e CM(Y) <=> T i s S -continuous. (2) I f T i s S"-continuous, then, f o r every y e CM(Y) , y a T i s E - t i g h t , and therefore has a c -Radon l i m i t measure on . X 2.5 Remarks Under c e r t a i n circumstances one can r e a d i l y strengthen the assertions of Theorems 2.2 - 2.4. Let y be a c y l i n d e r measure over X (1) (Theorem I I . 2.3) If there e x i s t s a base (J for nbnd 0 i n X such that f o r each U e U , the Banach space X^ i s separable, then, y i s E-tight => p.-has aiv ' 5 -Radon l i m i t measure on X * y Hence, i n those theorems in v o l v i n g the existence of a c -Radon /*c k l i m i t measure on X' , we may replace _c by S (2) Let G be a regular topology on X' with w |x ' c G If E c K(G) , or E = K(G) , then the foregoing theorems may be modified as indicated by Theorem 1.4.3. 86. In p a r i t u c l a r , we note that when X i s a Montel space, E = K(s*) . (cf. Thm. II.2.2.1) The theorems above allow us to make some i n t e r e s t i n g assertions about the S -topologies. 2.6 Theorems. (1) For a l l 0 < r < oo } the fa m i l i e s of 5 -continuous c y l i n d e r measures coincide. (2) For a l l 0 < r <_ 2 , the S -topologies coincide. (3) Let Y be a to p o l o g i c a l vector space, and for each r > 0 , = {T z L[X,Y] : T i s ^-continuous} . If Y i s a CM-space, then, for a l l . 0 < r < oo , the f a m i l i e s T coincide, r Remark. In general, the S "-topologies do not coincide f o r r > 2 (Example 3, Appendix 4). S Cl e a r l y , we may i n t e r p r e t a l l of our r e s u l t s f o r the s p e c i a l case when X i s a H i l b e r t space. In p a r t i c u l a r , we have the following theorems. Theorems. Let X be a Hilbe r t space, Let 0 < r < oo . For any cylinder measure y over X , n r u i s o -continuous <=> y i s E-tight <=> y has a c -Radon l i m i t measure on X ' Let 0 < r < oo y and \p be a p o s i t i v e - d e f i n i t e function on X I/J i s S -continuous <=> * for some f i n i t e c -Radon measure £ on X ' , iKx) = / exp i Re f(x)d£(f) for a l l x e X . Let Y be a Hilbe r t space, and T e L [ X,Y] •k , y a T has a c -Radon l i m i t measure on X ' for every y e CM(Y) <=> T i s a Hilbert-Schmidt map ([36] p. 177). ([42] V I I I , Pietsch [31], Petcynski [28]). Let Y be a Hilbert space. For a l l 0 < r < 03 , {T'.E L [ X,Y] : T i s r-summable} = {T e L [ X,Y] : T i s Hilbert-Schmidt} . (For the d e f i n i t i o n of r-summability, see [31], and [42] p. VII. 3). 88. Remark By Theorem II.2.3.2, when X i s a separable H i l b e r t space, every c -Radon measure on X' i s s -Radon. Hence, i n Theorems 2.7, we can replace c by s x<rhen X i s separable. We point out that Theorems 2.7.1 and 2.7.2 are equivalent (Cor. 1.4.3, Thm. 1.5.6.2, Appendix 2.5 and 2.6). We observe that even when X i s a H i l b e r t space our work extends previously knoxm r e s u l t s . Sazonov i n [35] discusses the case xtfhen X i s separable, obtaining Theorem 2.7.2 for the case r = 2 Waldenfels i n [48] extends Sazonov's theorem to the non-separable case. Theorem 2.7.3 extends a r e s u l t given i n [11] (p. 349), where X i s assumed to be separable and r = 2 s i g n i f i c a n t l y generalizes the Pietsch-Peieynski theorem given above (Theorem 2.7.4). Proofs 2. From Appendix 3.5 and Proof 2,7.4 we see that theorem 2.6.3 We s h a l l need the following lemma. Lemma Let X be a l o c a l l y convex space, r > 0 and S = S Let P = P(M jsupp n) directed by refinement, and for each P £ P S' = {x £ X : P I i n f |f(x)| r.n(B) > 1} BeP f e B 89. Then, for any F e , Radon measure E, on F and t > 0 , £(F ~ tS) = lira £(F A tS'j . PsP Proof of Lemma We f i r s t make the following observations. (1) If P E P, Q e P, with Q f i n e r than -P , then P Q * (2) For any u > 0 , X ~ uS = U u S' P EP (3) For every P E P , Sp i s open i n X . We prove only (3). Let P E P . We have that (4) S' = U {x £ X : E i n f | f ( x ) | r . n ( B ) > 1} , B BeB feB where the union i s taken over a l l f i n i t e B c P . Hence, since supp n e E i s equicontinuous, for every B e P , x £ X •> i n f | f (x) | £ R i s continuous. feB Hence, for any f i n i t e B Q P , (5) x £ X -> E (inf | f (x) | ) r . n(B) i s continuous, B e8 feB and therefore, by (4), (3) holds. From (3) we deduce that tF A i s open for every P E P . Consequently, as P i s directed by refinement, from (2) i t follows that for any compact C i n F with C c F ~ tS , there exists P E P s.t. 90. C c tF ft. S p ' . . Hence, since £ i s Radon and F ~ tS i s open i n F , £(F ~ tS) = sup {5(G) : C C F ~ tS i s compact} = sup U ( t F ft. S p : P e P } = lim 5(tF A S ' ) . PeP 2.1 Let f be a base for nbnd 0 i n X s.t. for each V e 1/ there e x i s t s a pseudo-inner-product [.,.] on X for which V = {x e X : [ x , x ] v £ 1} . Let 2̂ r k £ a s § i v e n i n Appendix i . l . For each V'e V , l e t F = F . For each V. e f and F e F , l e t p be a p r o b a b i l i t y Radon measure on F rel a t e d to [.,.] |F x F as i n Appendix 1.3. Then, from Appendix 1.3 we see that (v,F,lO i s a system of 62~weights i n X By Remark 1.1.3, (1) S r c nbnd 0 i n X . Let S = S e S r . Since \l i s a base for nbnd 0 i n X and supp n e E , there e x i s t s V e 1/ with supp ri C " 91. Then, • x e k e r V => sup | f ( x ) | = 0 = > J |f>(x)| rdri = 0 => x e S feV° i . e . (2) k e r V c S . L e t t > 0 . F o r any B c X' , l e t f B e B and § B = p r, ( B ) 1 / r . f g . U s i n g the n o t a t i o n o f t h e above Lemma, f o r any P e P , F o t S ' C t x e F : E | f (x) | r . n(B) > t r } BeP Hence, v (F o t S l ) < v _ ( { x e F : E | g ( x ) | r > l } ) V ' V ' BeP a <_ C E sup |g ( x ) | r by A p p e n d i x 1.3.2, ' BeP xeV F < C E sup | g R ( x ) | r i ^ - C Z sup |f ( x ) | r . n ( B ) ' BeP xeV ° t ' BeP xeV < — C„ n(X') s i n c e f e f o r e v e r y B e P — r 2,r B 3 From the above lemma i t now f o l l o w s t h a t v V j p ( F ~ t S ) l ^ C 2 > r r , ( X ' ) . S i n c e C. n(X') < 0 0 , we c o n c l u d e t h a t (3) v„ _ ( F ~ t S ) -> 0 as t -* °° u n i f o r m l y f o r F e F. V ,h From ( 1 ) , ( 2 ) , and (3) we see t h a t S i s w e i g h t e d by (v,F,l/) 2.2.1 By Cor. I I . 3 . 3 , Thm. 1.5.6.2 and P r o p . 1.2 92. 2.2,2 and 2.2.3 By Thms. II.2.1 and 2.2.1. 2.3 By Remark 1.1.3 and Appendix 2.2.5, r i/j i s 5 -continuous => i s continuous at 0 = > ijj i s continuous =>(jjJF i s continuous for a l l F E p. Hence, by 2.4 and 2.5 of the Appendix, there e x i s t s a f i n i t e S -continuous c y l i n d e r measure y over X s.t. i|>(x) = J exp i Re f(x)dy (f) for a l l x e X . By Thm. 2.2.2, y has a c-.Radon l i m i t measure E, on X' Then, for every x e X , <Kx) = /F exp i Re f(x)dy (f) = J , exp i Re f(x)d£(f) . Suppose now that E = K(c ) , and for some f i n i t e Rad<5n measure E, on X' , i>(x) = / exp i Re f(x)d£(f) for a l l x e X . We note that f o r every F e F and Borel subset H of F , r F X t H ] E M . I f , for each .F £ F , ' yF = r p m > then, by Lemmas 0.4 and Remark 1.2.1, y i s a f i n i t e c y l i n d e r measure over X Further, by Lemma 0.4.2, g i s a l i m i t measure of y , and therefore i t follows that \p i s the c h a r a c t e r i s t i c f u n c t i o n a l of y 93 7S y 7\ Since E, i s c -Radon and K(c ) = E , then, from Lemma 1.5.1.2 and the d e f i n i t i o n of y we see that y i s E-tight. Hence, by Prop. 1.2, y i s S -continuous, and therefore, by Appendix 2.5, r \p i s S -continuous . 1. By Thm. 2.2.1. 2. By Thm. 2.6.1, Cor. 2 of Appendix 3.5, and Appendix 3.2.1. 3. By Cor. 1 of Thm. 2.4. 1 and 2.7.2. are consequences r e s p e c t i v e l y of Thms. 2.2.3 and 2.3, since H i l b e r t spaces are b a r r e l l e d . 3. Since X and Y are Banach spaces, by [31] p. 339, Thm. 1, 2 T i s S -continuous <=> T i s Hilbert-Schmidt. The assertion i s now a consequence of Cor. 1 of 2.4, and Cor. 3 of Appendix 3.5. 4. Since X and Y are Banach spaces, by [42] p. VII. 3, § 2 , for any r > 0 , T i s S -continuous <=> T i s r-absolutely summable. The assertion now follows from Thm. 2.6.3 and Cor. 3 of Appen dix 3.5. 94. 3. Nuclear Spaces. Nuclear spaces comprise one p a r t i c u l a r l y important family of Hausdorff, H i l b e r t i a n spaces (Grothendieck [14], see also [36] and [47]). We s h a l l therefore i n t e r p r e t the r e s u l t s of the previous s e c t i o n for the case when X i s a nuclear space. As a consequence of the s p e c i a l structure of nuclear spaces, we s h a l l be able to strengthen considerably the theorems concerning c y l i n d e r measures over a r b i t r a r y Hausdorff, H i l b e r t i a n spaces. We point out that many of the common spaces of d i s t r i b u t i o n s are i n fac t nuclear (Treves [47] Ch. 51). For our d e f i n i t i o n of a nuclear space we s h a l l use a c h a r a c t e r i - zation due to Pietsch ([29], [36] p. 178). 3.1 D e f i n i t i o n . X i s a nuclear space i f f X i s a Hausdorff, l o c a l l y convex space with the following property: for any neighbourhood U of 0 i n X , there e x i s t s another neighbourhood V of 0 i n X , and a w -Radon measure n on X with supp n C , such that {x e X : J|f (x) |dn(f) <_ 1} C U . 95. Remarks If X i s a Hausdorff, l o c a l l y convex space, then, from Remark 1.1.3 and the above d e f i n i t i o n , we see that (1) X i s nuclear i f f i s a base for nbnd 0 i n X For any nuclear space X , from (1) above, Remarks 1.1.2 and 1.1.3, i t follows that r (2) the S -topologies on X coincide f o r r >_ 1 2 In p a r t i c u l a r , taking 5 as a base f o r nbnd 0 i n X , we deduce that (3) X i s a H i l b e r t i a n space ([36] p. 102). As i n Treves [47], p. 519, we can prove that (4) E c K(s*) . Hence, i f X i s b a r r e l l e d , then (5) E = K(s*) • We point out that coincidence of a l l the S -topologies f o r r > 0 i s a consequence of ( 2 ) , ( 3 ) , and Theorem 2.6.2. The theorems given below i n 3.2 are d i r e c t consequences of the above remarks, and assertions from the previous section } s p e c i f i c a l l y , Theorems 2.2, Theorem 2.3, and Remark 2.5.2. 3.2 Theorems. Let X be a nuclear space, and u be a c y l i n d e r measure over (1) u i s continuous <=> p i s E - t i g h t . 96. (2) y i s continuous => y has an s -Radon l i m i t measure on X' k (3) I f K(s ) = E , i n p a r t i c u l a r , i f X i s b a r r e l l e d , then, y i s continuous <=> y i s E-tight <=> y has an s -Radon l i m i t measure on X' (4) Let -ty be a p o s i t i v e - d e f i n i t e function on X i> i s continuous => there e x i s t s an s -Radon measure E, on X' such that i>(x) = / exp i Re f(x)d£(f) . f or a l l x e X . If E = /((s ) , i n p a r t i c u l a r , i f X i s b a r r e l l e d , then >jj i s continuous <=> there exists a f i n i t e s -Radon measure 5 on X such that ifj(x) = / exp 1 Re f (x)d£(f) for a l l x e X . Theorem 3.2.2 extends a r e s u l t of Minlos ([25], p. 303, Thm. 1), who considered f i n i t e c y l i n d e r measures over countably normed nuclear spaces ([11] p. 56). V i l e h k i n extended that r e s u l t to the case of countable s t r i c t inductive l i m i t s of such spaces ([11] Ch. IV 2.4). Theorem 3.2.4 extends r e s u l t s due to Minlos ([25] p. 310) and Badrikian ([1] p. 17). We note that the theorems of 3.2 completely resolve a conjecture of I. Gelfand ([25] p. 310, [18], p. 222), that every f i n i t e continuous cylinder measure over a nuclear space X has a l i m i t measure on the continuous dual X' Theorem 3.2.1 has a p a r t i a l converse which extends a r e s u l t of Minlos ([25]. Thm. 4). 97. 3.3 Theorem Let X be a Hausdorff, l o c a l l y convex space. If X i s a CM-space and y E CM(X) => y i s E - t i g h t , . then X i s nuclear. Proof By Prop. 1.2, y e CM(X) => y i s S^-continuous. Hence, by Appendix 3.2.1, the S^-topology i s f i n e r than the CM(X)-topology. On the other hand, by Cor. 2 of Appendix 3.5, and Remark 1.1.3, the CM(X)-topology i s f i n e r than the S^-topology. Consequently, the CM(X)-topology = the S^-topology. Since X i s a CM-space, i t follows from Remark 3.1.1 that X i s nuclear. © Remark. . We note that a Hausdorff, l o c a l l y convex space i s not neces- s a r i l y a CM-space (Example 4.3, Appendix 4). When X i s not a CM-space we see from the above proof that the best a s s e r t i o n possible i s the following. If y e CM(X) => y i s E - t i g h t , then, the S^-topology and CM(X)-topoIogy coincide. 98. Theorems 3.3 and 3.2.1 lead to the following new c h a r a c t e r i - zation of nuclear spaces (Remark 3.1.3, Cor. 3 of Appendix 3.5). 3.4 Theorem Let X be a Hausdorff, l o c a l l y convex space. X i s nuclear i f f X i s a CM-space and y e CM(X) => y i s E - t i g h t . Concerning induced cy l i n d e r measures, Remark 3.1.4 enables us to strengthen C o r o l l a r y (2) of Theorem 2.4. In view of Remark 2.5.2, the following a s s e r t i o n i s immediate. 3.5 Theorem Let X be a nuclear space, Y be a t o p o l o g i c a l vector space, and T e L[X,Y] . If T i s continuous, then, for every y e CM(Y) , y a T i s E - t i g h t , and therefore has an s -Radon l i m i t measure on X We observe that an i n f i n i t e - d i m e n s i o n a l normed space cannot be nuclear ([47J, p. 520). As a consequence of t h i s f a c t we can assert that c e r t a i n cylinder measures over such a space X cannot have a l i m i t measure on X' 3.6 Proposition Let. X be an i n f i n i t e - d i m e n s i o n a l normed space I f u i s a f i n i t e c y l i n d e r measure over X such that the topology of X i s the {y}-topology, then y does not have a l i m i t measure on X' Proof. By Cor. 1.4.3, Prop. 1.2, and Appendix 3.2.1, y has a l i m i t measure on X* => y i s E-tight 1 => y i s 5 -continuous 1 => {y}-topology i s coarser than the S -topology => X i s nuclear, by Remarks 1.1.3 and 3.1.1. Since X i s an i n f i n i t e - d i m e n s i o n a l normed space the l a s t a s s e r t i o n cannot hold, and therefore y cannot have a l i m i t measure on X' Corol l a r y Let A be an index set. For any 1 < p < 2 , i f i s the f i n i t e cylinder measure over (A) with c h a r a c t e r i s t i c f u n c t i o n a l (Remark, Appendix 2.4) exp - ( l\ x \\ P x e £ P(A) ->  It  ) P e £ , then y does not have a l i m i t measure on (£ P(A))' 100. Proof See (1) i n Proof of Example 4.2, Appendix 4, and Proof 3.1.1 of Appendix 3. Remark For p = 2 the above c o r o l l a r y i s well known (Gross [13]). We have not seen a treatment of the case 1 <_ p < 2 i n the l i t e r a t u r e . 101. 4. & -spaces. Applied to j^-spaces, 1 <_ p <_ °° , the theory of the previous chapter y i e l d s r e s u l t s analagous to those for H i l b e r t i a n spaces. Since 2 I - i s a H i l b e r t space t h i s case has already been discussed i n §2.5. The r e s u l t s given there are stronger than those we s h a l l obtain here for an a r b i t r a r y £^-space. Notation Let A be an index set. For any 0 < r <_ °° , {x'e <?• : E |x(c0 | < °°} when r <_ a e A i £ r(A) = \ A i <• {x E C : sup lx(a) | < °°} when r = 0 0 asA We give £ (A) the usual topology, i . e . , when r < 1 , the topology generated by the quasi-norm (Appendix 3.3) b : x E £ r(A) -> E |x(a) | r e R + ; aeA when r >̂  1 , the topology generated by the norm : x e £ r(A) + (• E |x(a ) I r ) 1 / r e R + , ' aeA where we ( E | x ( a ) | r ) 1 / / r = sup |x(a) | i f r = °° . aeA. a£A For any 1 < p < 2 , U = {x E £ P(A) : .E |x(a)| 2 < 1} . P . aeA 102. For any outer measure n on a space ft £v(n) = lim I n ( B ) | l n n ( B ) | PeP(M ) B £P n where t o t C<x. ulnu = 0 when u = 0 The heart of t h i s section i s the following group of r e s u l t s , which p assert that c e r t a i n f a m i l i e s of subsets of £" (A) , 1 <_ p <_ « , are weighted systems i n £ P(A) 4.1" Theorem. Let 1 <_ p <_ oo and 1/p + 1/q = 1 For any r > 0 , l e t — r r r 5 C S consist of those sets S e S for which r ,n s a t i s f i e s the added condition £v(n) < •» , • and when 1 <_ p j<_ 2 , l e t ~ r r r S S> consist of those sets S e S for which T\ s a t i s -r,n 1 f i e s the added condition / (sup | f (x) | rdn (f) < «> . xeU P (1) If 2 < p <_ oo and 0 < r < q then S r i s a weighted system, i n £ P(A) (2) I f 2 < p < _ o o and r = q then — r p S i s a weighted system i n £ (A) (3) If 1 < p < 2 and 0 < r < oo , then S r i s a weighted system i n £ P(A) (We note that S r = S r when p = 2 .) 103. The proof of the above theorem w i l l be given at the end of the section. Now, we point out i t s immediate consequences when taken together with C o r o l l a r y II.3.3. 4.2 Theorems Let 1 < p < oo and — + — = 1 - - p q (1) If 2 < p <_ oo and 0 < r < q , then, for any c y l i n d e r measure y over £ P(A) , „r . p i s • i -continuous s=> p is E-tight •k . L\»vX p has a c -Radonimeasure on (£ P(A))' (Here, we also use Prop. 1.2. and Thm. II.2.1, noting that £ P(A) i s a Banach space and i s therefore b a r r e l l e d . ) (2) If 2 < p <_ « and r = q , then, for any cy l i n d e r measure p over £ P(A) , —x p i s S -continuous => p i s E-tight => •k _ Liy*>'̂  p p has a c -Radonimeasure on (£ (A))' (3) If 1 <_ p <_ 2 and r > 0 , then, for any c y l i n d e r measure p over £ P(A) , p i s S -continuous => p i s E-tight => •k _ p p has a c -Radonimeasure on (£ (A))' 104. Using Theorems 4.2 we can represent c e r t a i n p o s i t i v e - d e f i n i t e functions on £ P(A) as Fourier transforms of measures on (£ P(A))' The proofs of the assertions given below are s i m i l a r to the proof of Theorem 2.3, and are therefore omitted. 4.3 Theorems Let l < p < ° ° , - - + — = 1 , and \j> be a no s i t i v e - d e f i n i t e — — p q function on £ P(A) (1) _ If 2 < p <_ «> and 0 < r < q ', then, r * < i s S -continuous <=> for some f i n i t e c -Radon measure E, on (£ P(A))' , U>(x) = / exp i Re f(x) d£(f) for a l l x e £ P (A) . (2) I f '2 < p < » and r = q , then, — r * /• ^ i s S -continuous => for some f i n i t e c -Radon measure E, on (£ P(A)) ! , iKx) = / exp i Re f (x)d£(f) f or a l l x e £ P(A) . (3) If 1 <_ p <_ 2 and r > 0 , then, ~ r ^ f ^ i s 5 -continuous => for some f i n i t e c -Radon measure E, on (£ P(A))' , ip(x) = / exp i Re f(x)d£(f) for a l l x e £ P(A) . Concerning induced cylinder measures, Theorem 4.1 y i e l d s the following r e s u l t s when taken together with Theorem II.4.3. 105. 4.4 Theorems. Let 1 < p < 0 0 , — + — = 1 , Y be a vector space, C be - - P q a family of f i n i t e c y l i n d e r measures over Y , and .T e [£ P(A),Y] (1) If 2 < p <_ <*> and 0 < r < q , then, T i s S r-continuous with respect to the C-topology on Y <=> for every y e C , y u T has a c -Radon l i m i t measure on (£ P(A))' (Here, as for Thm. 4.2, we also use Prop. 1.3 and Thm. II.2.1.) (2) I f 2 < p <_ 0 0 and r = q , then, T i s ^ - c o n t i n u o u s with respect to the C-topology on Y => for every y e C , y • T i s E-tight => for every y e C , y p T has a c -Radon l i m i t measure on (I (A))' (3) I f 1 <_ p <_ 2 and r > 0 , then, T i s S 1-continuous with respect to the C-topology on Y => for every y e C , y D T i s E-tight => for every y e C , p a T has a c -Radon l i m i t measure on (& P(A))' As consequences of Theorems 4.4 we have the following extensions of r e s u l t s due L. Schwartz 139] and Kwapien [19]. They consider only the case when r = q and A i s countable. 106. C o r o l l a r i e s . (1) If 2 < p <_ «>, 0 < r < q , y e £ r (A) and T : x e £ P(A) •+ (x(a)y(a)) . e / ( A ) , aeA then, for every y £ CM(£ (A)) , u a T has a c -Radon l i m i t measure on a " (A))* (2) I f 2 < p < c o , r = q , y e £ r(A) with £ | y(ct) |11 l n | y(a) [ | < co , aeA and T : x e £ P(A) (x(a)y(a)) e £ r(A) , - x then, for every y e CM(£ (A)) , y n T has a c -Radon l i m i t measure on (£ F(A))' (3) I f 1 £ p <_ 2 , r > 0 , y e £ r(A) , and T : x e £ P(A) -»• ( x ( a ) y ( a ) ) A e £ r(A) , aeA then, for every y e CM(£ (A)) , * P y a T has a c -Radon l i m i t measure on (£ (A)) We give here the proof of onl\ T C o r o l l a r y (1) . The other proofs are s i m i l a r . Proof of Coro l l a r y (1). For each a e A , l e t e e (£ P(A)) ? : x e £ P (A) ->• x(a) e C , a and n D e the d i s c r e t e measure on (£ p(A)) r with i i r n({e )) = v(a) for each a e A a 107. Then, supp n e E , and for any x e £ P(A) , £ I (T.*)Jr = J|f 00 | rdn(f) • I t follows that T i s S -continuous, and the c o r o l l a r y i s now an immediate consequence of Thm. 4.4.1. # 4.5 Remarks. (1) " If A i s countable, then (£ P(A))' i s separable. Consequently, by Theorem II.2.3.2, every c -Radon measure on (£ ( A ) ) ! i s i n f a c t s -Radon. The foregoing theorems may then be s u i t a b l y modified. (2) We point out that C o r o l l a r y 4.4.2 i s the best r e s u l t p o s s i b l e when 2 < p <_ <» and r = q . If \ y e £ q(A) with T, | y (a) | q | In | y (a) | | = » aeA and T i s as given i n the c o r o l l a r y , then by Example 4.2 of Appendix 4, there e x i s t s p e CM(i>5(A)) such that p Q T f a i l s to be E-tight. (3) With the notation of (2) above, as i n the proof of C o r o l l a r y 4.4.1, we see that T i s S^-continuous, • and therefore, by Lemma II.4.2, p p T i s S q"-continuous. From Remark (2) above, and Theorem II.3.3, i t now follows that 1 1 for any 2 < p < ° ° , i f 1 = 1 , then - p q S q i s not a weighted system i n £ P(A) 108. However, Proposition 1.2 suggests that when searching for a weighted system i n £ P(A) we ought to look for a subfamily of S q Remark (2) then indicates that i s i n f a c t an appropriate subfamily of S q for us to consider. When 2 <_ p < °= , the construction which we use for producing a system of 5-weights i n £ P(A) depends on the f a c t that for any f i n i t e set K , x z (EK -> exp - Z |x(a) | q e ® aeK K 1 1 i s a D o s i t i v e - d e f i n i t e function on <p , where — + — = 1 p q (Remark (1) of Proofs (4), and Proof 2.2 of Appendix 2.) If 1 <_ p < 2 , then, q > 2 and the function given above i s no longer p o s i t i v e - d e f i n i t e (Schoenberg [38J p. 532). The construc- t i o n therefore breaks down when 1 < p < 2 . We can show that construction of a system of 5-weights i n £ P(A) , 1 < p < 2 , would be possible i f there were a X : R + -> £ such that for any f i n i t e set K , x e ? K X ( I |x(a) | q) e € aeK K was p o s i t i v e - d e f i n i t e on (J . If such a function X existed, then, by Appendix 2.2.4, (i) x e £ q(A) X ( Z |x(«) | q) e C aeA would be a p o s i t i v e - d e f i n i t e function on £ q(A) . However, when q > 2 , one can show as i n [5] that there does not e x i s t X : R + -> <E such that (i) holds. Nonetheless, we can s t i l l obtain a system of 6-weights i n £ P(A) , 1 <_ p < 2 , i f we use the system of 5 —weights induced by the canonical imbedding x E i P ( A ) x £ £ 2(A) . (Remark (2) of Proofs 4.) 109. (5) Remarks ( 4 ) , P r o p o s i t i o n 1.2, and t h e p r o o f of Theorem 4.1.1, ~ r r l e d us t o b e l i e v e t h a t S C S , r > 0 , would be a s u i t a b l e f a m i l y t o s t u d y when s e a r c h i n g f o r a w e i g h t e d system i n £ P(A) , 1 < p < 2 1 1 (6) We p o i n t out t h a t f o r p >_ 1 and — + — = 1 , t h e a p p e a r a n c e of q i n t h e h y p o t h e s e s a r i s e s a t t h e f i n i t e - d i m e n s i o n a l l e v e l (Remarks (4) and A p p e n d i x 1 ) . Thus, a l t h o u g h (£ P(A))' may be i d e n t i f i e d w i t h £ q(A) , we have a v o i d e d d o i n g t h i s , as c a r r y i n g out s u c h an i d e n t i f i c a t i o n m i g h t have s u g g e s t e d t h a t t h e r e l a t i o n s h i p between £ P(A) and £^(A) was c r u c i a l t o our argument. P r o o f s 4. T o g e t h e r w i t h the n o t a t i o n s o f A p p e n d i x 1.1 and P r o o f s 2, f o r any p >_ 1 , l e t p q X = £ P(A) , V =' {x e X : Ixl < 1} P P 1 'p - I . I : f e X* -> sup If(x)} . q P x e V P F o r any f i n i t e K c A , l e t | . | „ : f e (f,V + sup {|f ( x ) | : x e V ft <CK} , q ,K p r = r K c K,x P L e t ^ K F = {€ : K c A i s f i n i t e } d i r e c t e d by i n c l u s i o n , 110. For any 2 <_ p <̂  » and f i n i t e K c A , l e t K • K Yp be the product measure on £ generated by y on C , D K K and v : £ e F -> Y ? which i s Radon. Remarks (1) By Appendix 1.2, for each 2 < p < «> } (v P , F,V ) i s a system of 6 -weights i n X P P P (2) For each 1 p < 2 , since (V 2 f\ C K ) ° c (V A C K)° for every f i n i t e K c A , then, by Appendix 1.2, for every 1 <_ p < 2 , 2 (v ,F,V ) i s a system of 6 -weights i n X p 2 p 4.1. We observe that for any p 1 and r > 0 , (1) S r c nbnd 0 i n X (Remark 1.1.3). P and (2) ker V = {0} e S for every S e . V r Now, for any S =' S e S , and each B c X' , l e t ' r,n p f B e B , g B = _ n 1 / r ( B ) . f B and s = sup { | f | : f e supp n } . Then, as i n Proof 2.2.1, for any P e P , f i n i t e K c A ., and t > 0 , we have that (5) v P ( C K ^ tS') < v P({x e C K : Z \~ ZnM\V > D) , K , P ~ K BeP and, by the lemma of Proofs 2, (6) v P ( £ K ~ tS) = lim V P « E K A tS') . ' PeP(S) p I l l , Case 1. (2 < p <_ °°, 0 < r < q) . Since supp r\ z E , 0 < s < ro Then, for any t > - r/^CX') , s p B c supp n => |1 g j < 1 . Hence, by Appendix 1.2.3, the right-hand-side of (5) i s majorized by 1- r /vM j . ±- n^r ,q-r+l, eq^q/r , v, < C i n(X') + — 2^C P - ^ s V ' (x') .r p,r p fcq p v q-r p y (Since q/r > 1 , then E n q / r ( B ) = n q / r ( x ' ) E [ p ( B ) / n ( x ' ) ] q / r BeP p BeP P < n q / r ( x ' ) E n(B)/r,(x') = n q / r ( x ' ) . ) _ BeP P • P ' Whence, by (5) and (6), (7) Y K « E K ~ tS) < ± - C S rn(X')'+ i - 2TTC ( 4 ^ ± I ) S V / r ( X ' ) . P - fcr p,r p t q p q-r p Since the c o e f f i c i e n t s of 1/t and 1/t are f i n i t e and independent of K i t follows that K K (8) Y P ( C ~ tS) -> 0 as t ->• °° uniformly for a l l f i n i t e K c A . By (1), (2), and (8), 5 i s wei ghted by (v P,F,V ) . 112. —V Case 2 (2 < p < », r = q, S e 5 ) . • If c = sup u In u 0<u<s then 0 < C < oo 1 1/r For any t > — n (X') ., by Appendix 1.2.3, the right-hand-side s p of (5) i s majorized by (9) C E |r, r(f g ) | q „ + 2TTC E | r_,£ g) | q' | In | r ( i g ) | P ' q B £P q ' p BeP q ' q ' The f i r s t term of (9) i s majorized by • — C s qri(X') . t q p sq p The second term of (9) i s majorized by 2^Cp ^ E ^ ( ! f B | q ) q n ( B ) [ | l n t | + l | l n n ( B ) | + | In | f B | q | ] l i n t ! 2TTC S q n(X') + — 2TTC S q - E n ( B ) | l n n ( B ) | t q . P P t q P q BeP ' + — 2irC c n(X') . t q P P Hence, by (5), (6) and (9), Y (C tS) P 2TTC < — [c s qn(x') + — E q£v(n) + 2,TC C n ( X ' ) ] - t q p.q P q P P + J i ^ l [ 2 n C q t q p P By the hypotheses, the c o e f f i c i e n t s of l / t q and | l n t | / t q are f i n i t e and independent of K . Hence, (10) v P ( C K ~ tS) -> 0 as t -> oo uniformly for a l l f i n i t e K C A K By (1), (2) and (10) S r i s weighted by (v P,F,V ) P 113. Case 3 (1 <_ p <_ 2 , r > 0 , S e S r) By Appendix 1.2.3, for any t > 0 , the right-hand-side of (5) i s majorized by °2,r B E £p lrK(K>l2,K ' Hence, by (5) and (6), v 2 ( £ K ~ tS) = lim Y o ( C K ^ t S ' ) K ' PeP(S) 2 P = C . l i m E sup „ |f (x)|) r.r,(B) t PeP BeP xeUr\C P 1 ~ c 2 r l i m 1 ( s u p ( s u p I f ( x ) l ) 1 ) n(B) • t 5 PeP BeP feB xeU p = 7 C / * ( s u p | f ( x ) | ) r d n ( f ) • t Z ' r xeU P By hypothesis, the c o e f f i c i e n t .of 1/t i s f i n i t e and independent of K . Hence 2 K (11) v„(C 1 ~ tS) -> 0 as t -> °° uniformly for a l l f i n i t e K C A .' K By (1), (2) and (11), ~r 2 " S i s weighted by (v ,F,V ) 114. APPENDIX • In t h i s Appendix we e s t a b l i s h a number of r e s u l t s and construc- tions which are necessary for the discussions of Chapter I I I . In the l a s t section we give some counterexamples which complement the considera- tions of Chapter I I I . 1. "Special Measures, on Finite-Dimensional Spaces. In t h i s section we s h a l l construct s p e c i a l measures on finite-d i m e n s i o n a l spaces. The existence of these measures enables us to produce systems of <5-Weights i n H i l b e r t i a n spaces and i n & P-spaces, P L 1 . Notation K i s a f i n i t e set. For any 1 5 . p l 0 3 , p q (x e c K : z n r v l x ( a ) l P 5 l ) when p < °°, P (x e C : sup |x(a)| < 1} when p = » aeK For any f e ( C K ) " , q xeV sup |f(x) P (We note that V° = {f e (t K )" : If.l < 1}) p ' 'q ~ 115. A i s the Lebesque measure on <C ' . For any f i n i t e dimensional space F , I = {(x,f) e F x F* : |f ( x) | >_ 1} . The constructions of t h i s s ection w i l l be based on the asser- tions given below. 1.1 Lemmas Let 2 <_ p <_ «> and r > 0 (1) There e x i s t s a s t r i c t l y - p o s i t i v e , continuous (9 : £ + R + P such that exp - |w|q = J (exp i Rewz)(9 (z)dw(z) for a l l w e £ (2) When 2 < p <_ <»• , there e x i s t s (i) 0 < C < co P such that ( i i ) 0 (z) < C / | z l 2 + q for a l l z e t . p p 1 Hence, when r < q , for any u > 0 , 2TTC 2 f l + l / u q r ] i f r < q , ( m ) q-r n z| r Q (z)dA(z) < <u ? 1< z 2TTC lnu i f r = q , and (iv) f Q (z)dX(z) < 2TTC Jul q u<|z| P - P1 1 (3) When p = 2 , 1 I z i 2 0n(z) = -— exp L T j — for every z e £ 2 4T( 4 1 1 6 . Notation and Remarks For each 2 <_ p <_ m , l e t Y : B c C + f 1 J dA £ R + , 'p ' B u ' 6 = Y ({z e € : Izl > 1 } ) . P P i i - For any r > 0 , l e t C 9 r = H Z | r <M Z) <*X(z) • For each 2 < p <_ °o > and any 0 < r < q , l e t C be the constant of Lemma 1 . 1 . 2 , P and 2TTC + J i ' |z| ^ ( z ) d A ( z ) if r < q q-r C  |z|<l P P>r 2TTC + / | z | r 0 (z)dA(z) i f r = q P |z|<l ' P We note that i n view of Lemmas 1 . 1 . 1 and 1 . 1 . 2 , Y i s a p r o b a b i l i t y Radon measure on € , 5 > 0 p 0 < C < co, p 0 < C < CO . 117, 1.2 Lemmas Let 2 <_ p <_ oo and K K Yp be the product measure on € generated by the measure y on € , P K K (1) Yp i s a p r o b a b i l i t y Radon measure on C K « 0 K f (2) f e («T) ~ V => Y (I ) > <5 P P - P (3) For any sequence {f } C (<Ĉ ) , and r > 0 , l e t n new B = {x e C K : -E If (x) 1 r > 1} 1 n 1 new (i) I f p = 2 , then q = 2 and new ( i i ) If p > 2 , r <_ q , and l ^ n l q — ^ ^ o r e v e r y n e w , then, C E If | r + 2^C ( q " r + 1 ) E If I2 i f r < q , p,r 1 n'q p q-r 'n'q new new C E If l q + 2TTC E If | q | l n | f I I if r = q p,q 'n'q p '.n 1q 1 1 n 1q 1 new ^ r new n n 1.3. Lemma. Let F be a fin i t e - d i m e n s i o n a l vector space. I f [.,.] i s a pseudo-inner product on F and V = {x e F : [x,x] <_ 1} , then, there ex i s t s a p r o b a b i l i t y Radon measure £ on F such that (1) f e (ker V) 3' ~ V° => £(I £) 1 6 2 • 118. (2) For any sequence {f } i n F , n neoo £({x e F : E | f ( x ) | r > 1}) < C. E ( s u p | f ( x ) | ) r nea) new xeV Proofs 1. 2 1.1 Let \ be the Lebesque measure on R From Blumenthal and S e t o o r [3] p. 263, we have the following facts, (See also Levy [21] Ch. VII.) For any 0 < q <_ 2 , there e x i s t s a s t r i c t l y p o s i t i v e continuous 2 + 9 : R R P s.t, ( i ) exp - | t | = /[exp i ( t . u ) ] 0 (u)dA(u) f or a l l t e R 2 , 2 2 where, for any t e R , u z R , t.u = t Q u 0 + and | t | = / ( t 2 + t 2 ) . If q < 2 , there e x i s t s 0 < c < <» s.t. q l i m | t | 2 + q 9 (t) = c i • i P q Hence, i f ( i i ) C = sup |t| 2 + q0 (t) , P xeR P then, ( i i i ) 0 < C < P and (iv) 6 (t) < C / | t | 2 + q for a l l & P — P 119. Consequently, f o r any 0 < r <_ q , and u > 0 , / | t | r 6i (t)dx(t) < c / ! t | r . — 1 — d x e t ) p • ~ p i< t!<u • |t! 2 + q = 2TTC j • ( p ^ + q ~) ''"dp using polar coordinates. P l£p£u By i n t e g r a t i n g the l a s t term i t follox-Js that (v) 2TTC 2TTC £ [ l - l / u q r ] < ^[l+l/uq r ] , i f r < q , 1< t q-r - q-r T\Q (t)dA(t) < <U P ~ | | — 2TTC lnu i f r = q P For q = 2 , using the fa c t that 2 , 2 ( l / / 2 r r ) | R (exp i xy)exp - — dx = exp - ~ - , a d i r e c t computation shows that (vi) <92<t) = ^ e x p Hence, for any r > 0 , •r" ( v i i ) / | t | 0 2(t)dA(t) < » . Let 2 T : z e <C -> (Re z, Im z) e R , and for each 0 < q <_ 2 •, & = 6 o T . P P The assertions of Lemmas 1.1 now follow from (i) - ( v i i ) above, and the properties of the map T , namely, for any z and w i n (C , Re wz = (Tw).(Tz) ' , and T i s an isometric, measure preserving, homeomorphism. 120. 1.2. K * N o t a t i o n F o r any f e (<S ) , l e t - f / | f | q > <?f : x e ffK -> f (x) e £ "K,f f r K and l e t Y p - f [Y p] ~ f — ~K f T : w e £ -> (exp i Re wz)dy ' ( z ) e £ 1 .'2.1 ' T h i s f o l l o w s from t h e f a c t t h a t Yp l s a p r o b a b i l i t y Radon measure on £ 1.2.2 F o r each a e K , l e t e = 1 r , e £ a {a} K * Then, f o r any f e (£ ) > and w e £ , - ̂  " K f (1) \p (w) = J e x p i Re wz dy ' ( z ) = /exp i Re f ( w x ) d y (x) = II [/(exp i Re w f ( e ) x ( a ) ) d y ( x ( a ) ) ] aeK a P = exp - 1 Iw f ( e ) | q by Lemma 1.1.1, ' aeK a = exp - l w ! q | f | q . 1 q S i n c e a Radon measure on a f i n i t e - d i m e n s i o n a l space i s u n i q u e l y d e t e r m i n e d by i t s F o u r i e r t r a n s f o r m ( B o c h n e r ' s Theorem, A p p e n d i x 2. 3 ) , i t f o l l o w s t h a t (2) I f l = 1 => Y K ' f = Y 121. However, and f = 1 . K * 0 I I f e (C ) - V U => f > 1 P 1 'q - Hence, = 6 , by (2) above. P 1.2.3. (I) Since E f (x) i s a s e r i e s of p o s i t i v e Y o ~ m e a s u r a D l e • n ' - 2 new K functions on C , ^(B) = / l B . d Y ^ < / ( Z | f n ( x ) r ) d Y ^ ( x ) new = E l f n i 2 /|f;W| rdY|(x) new = E \ f j z 2 j\z\X d Y 2 n ( . Z ) new = C„ E If I* by (2) of Proof 1.2.2. 2, r 1 n '2 new ( i i ) Let and H = {x e C K : |f (x)I < 1} f o r each n e w , n 1 n — H = C\ H n new K K h :' x e € -> 1 i f x e C ~ H E |f ( x ) | r i f x e H . n 1 new 122. We have that = I Y„({x e £ : |f'(x)| > l/.|f I }) new P n n q -K,f = Z Y_ n'({z e C : |z| > 1 / I f n I a > > new (2) < 2TTC £ If | q by (2) of Proof 1.2.2. — p 1 n'q J new and Lemma 1.1.2.(iv). f'(x) I i s a series of p o s i t i v e v -measurable n 1 • 'p new r functions on , we also have that / i H h d Y p w i H [ £ igx«r^>> new = £ new < I |£ i „ / 1„ | f ' ( x ) | r d T (x) — ' n ' q ' H 1 n 1 p new n ~K,f' = Z |f | r / | z | r d Y n ( z ) Hence n'q ' ' ' 'p ,'z|<l/|f j ' 1 — 1 n'q Z I f | r / | z | r 6 (z)dA(z) 'n'q ' i i p v ' M - ^ ^ J q by (2) of Proof 1.2.2. new new / l 1 7 h d Y < I If T [ / | z | r 6 (z)dA(z) + / | z | r 9 (z)dA(z)] ' H p — ' n ' q - ' p / I I p X J z <1 1< z <1/ f n'q L e t t i n g a = J | z | r 6 (z)dA(z) , from P > r p | z| <1 Lemma 1.1.2 ( i i i ) , we have that 2TTC Z |f | r ( a + P-[l + If i q r ] ) i f r < 1 n 1q p,r q-r 'n'q neoj (3) / < E If | r ( a + 2TTC | l n | f I I) i f r = q 'n'q p,r p 1 'n'q' new ^ Consequently, i f r < q , then, from (1), (2) and (3), 2TTC Y ( B ) < 2TTC Z | f | q + Z If | r ( a + P - [ l + | f | q 1 ] ) P - P new n q new n q P ' r q " r n q = C Z |f | r + 2TTC (~ q^) Z |f | r , p,r 'n'q p q-r 'n'q new new since a l l terms are p o s i t i v e . The case r = q i s established from (1), (2) and (3) s i m i l a r l y . 1.3 We f i r s t suppose that [.,.] i s non-degenerate. If so choose a [.,.]-orthonormal basis K f o r F . Let T : z e ( C -> Z z . a e F . „ a aeK and .Then, * , K * f e F •+ f = f o T" e (C ) T i s a homeomorphism. K K ^ Further, f or any x' e C , y' e C , and f e F , (1) [Tx'jTy'] = <x',y'> , where <.,.> denotes the inner product in C , and (2) sup | f ( x ) | = |f| x eV Whence, i f h ~ Y2 ° T 124. then and 5 l ( I f ) = Y 2 < L F ) C(B) = y^ax 1 E C K : Z | f n ( x ' ) | r > 1}) new The assertions now follow from (3) and Lemmas 1.2. When [.,.] i s degenerate, l e t F^ be any subspace of F s.t. F i s the direct sum of F^ and ker V (Possibly, F = {0}) . We have that [.,.]|F x F^ i s non-degenerate, since F̂ /TN ker V = {0} Let £ be the probability Radon measure on F^ determined as above, and E : H c F + 5 ( H / > F 1 ) e R + . Then, (3) ^ i s a prob a b i l i t y Radon measure on F Since F i s the direct sum of F^ and ker V , for any x e X there exists a unique representation x = x^ + , with x^ e F^ and x^ E ker V Consequently, (4) f e (ker V ) 3 => sup |f ( x ) | = sup | f ( x ) | xeV xeVKF and therefore, from the non-degenerate case above, (5) f e (ker V ) a ~ V° => r p (f) e F* ~ (V A F^)^ r F (f) f => e;(i r) = ^ ( I ) > 5 2 . » a I f f e F (ker V) f o r any n e w , then n sup If (x) | • = » xeV and therefore (2) of the lemma holds. If f e (ker V) f o r every n e w , then from the non-degenerate case above, 5(B) = £ '({x e F. : Z | f ( x ) | r > 1}) 1 1 n 1 new < C 2 r Z sup | f n ( * ) | r ' new XEVAF^ = C Z sup • | f ( x ) | r by (4). new xeV 126. 2. • P o s i t i v e - d e f i n i t e Functions on Vector Spaces. In this section we give a number of useful r e s u l t s concerning p o s i t i v e - d e f i n i t e functions on vector spaces. 2.1 D e f i n i t i o n Let X be a commutative group. ip i s a p o s i t i v e - d e f i n i t e function on X i f f if) : X -> C , and f or any n e to, {x^,. . . >xn_^} £ X , {z , . . ., <Z C , n-1 _ E Zk Z£ ^ ( xk ~ X £ } - ° • k,£=0 We s h a l l need the following elementary assertions about p o s i t i v e - d e f i n i t e functions on groups. 2.2 Propositions Let X be a commutative group. (1) If if; i s a p o s i t i v e - d e f i n i t e function on X , then 0 <_ CO) < °° . (2) Let if) be a p o s i t i v e - d e f i n i t e function on X , and Y be a commutative group. If T : Y -> X i s a homomorphism, then if) 0 T i s a p o s i t i v e - definite function on X (3) I f cf and if) are p o s i t i v e - d e f i n i t e functions on X , then q»ip i s a p o s i t i v e - d e f i n i t e function on X (4) If (ilO. , i s a net of p o s i t i v e d e f i n i t e functions on J Jed X , and ip : X C i s such that 4(x) = lira I|J . (x) f o r a l l x e X , j e J 3 then, i s a p o s i t i v e - d e f i n i t e function on X (5) If X i s a t o p o l o g i c a l group, and i s p o s i t i v e - d e f i n i t e function on X , then \p i s continuous on X <=> \p i s continuous at 0 e X For f i n i t e - d i m e n s i o n a l spaces we have the following version of a w e l l known, representation theorem (Rudin [34] p. 19 1.4.3, Bochner [4] p. 58). 2.3 Theorem Let F be a f i n i t e - d i m e n s i o n a l vector space. i s a continuous p o s i t i v e - d e f i n i t e function on F i f f there e x i s t s a unique f i n i t e Radon measure £ on F such that 4<(x) = / exp i Re f(x)d£(f) f o r a l l x'e F Using the above theorem, as i n [11] (p. 349) one r e a d i l y establishes i t s following i n f i n i t e - d i m e n s i o n a l analogue. We omit the proof. (The theorem given i n [11] i s formulated only f o r r e a l vector spaces. See also [48]). 128. 2.4 Theorem Let X be a vector space. ij) i s a p o s i t i v e - d e f i n i t e function on X with I^|F continuous f o r every F e F , i f f there e x i s t s a unique f i n i t e c y l i n d e r measure u over X such that I|J(X) = / exp i Re f(x)du (f) f o r a l l x £ X . Remark When u and i|; are re l a t e d as i n the foregoing theorem, we c a l l i|; the c h a r a c t e r i s t i c f u n c t i o n a l of u (Prohorov [33]). The f i n a l theorem of th i s s e c t i o n i s useful f o r determining continuity properties of cy l i n d e r measures. As an adjurxctto Proposition II.2.6, i t further motivates the terminology "continuous cyl i n d e r measure", introduced i n D e f i n i t i o n s II.2.5.1. 2.5 Theorem Let X be a vector space, u be a f i n i t e c y l i n d e r measure over X , and V be a family of balanced, absorbent subsets V of X with uV e V f o r every u > 0 If i|) i s the c h a r a c t e r i s t i c f u n c t i o n a l of u , then u i s l/-continuous <=> <jJ 1 S ^-continuous. C o r o l l a r y Let X be a t o p o l o g i c a l space, and p be a f i n i t e c y l i n d e r measure over X If ii) i s the c h a r a c t e r i s t i c f u n c t i o n a l of p. , then p i s continuous <=> iji i s continuous. Proofs 2 2.2.1 Taking n = 1, x^ = 0 , and = 1 , the assertion follows immediately from Defn. 2.1, 2.2.2 For any n e w , {x^,...,xn_^}C X , and {z^,..., z n_ n} ^ ® > n-1 _ n-1 1 Z k Z £ * 0 T ( X1< " V = E Z k Z £ * ( T x " T x } -̂ ° • k,£=0 k 1 l< £ k,£=0 k * X k X £ 2.2.3 From [48] we have that Cf(x) = Cp (-x) f o r a l l x e X Hence, by Defn. 2.1, f o r any ne u>, {x A,...,x . } c X , and 0 n-1 { V \ ' Z n - l } C S ' M = (cpCx, - x ) ) " }_n i s a p o s i t i v e - d e f i n i t e * k £ k, £—U Hermitian matrix. Hence, there ex i s t s an n x n-matrix T s . t . M = TT* , where T = (t ) , T* = ( t * ) , and t * = I , • 130. Consequently, n-1 _ . I n V ^ k " V ^ k ~ V . n-1 _ n-1 1 V * ( E \ sK ?H'(xk _X£} k,£=0 * s=0 ' b'y' 1 n-1 n-1 E 1 {W s)(zkt£ s^ ( xk " V -° > s=0 k,£=0 K K ' S K l , S K 1 and therefore \p i s p o s i t i v e - d e f i n i t e . 2.2.4 For any n e w , {x^, . . . ,x , } c X , and {z r t..... z , } C u n-1 U n-1 n-1 _ n-1 _ . E n V * * ( xk " X£ } = ^ ^ ̂  r ZkZ£ V X k " V - ° k,£=0 . j e J k,£=0 J 2.2.5 From Rudin [34], p. 18, 1.4.1 (4), we have that for any x and y i n X , | * ( x ) - i K y ) | 1 2<JJ(0) Re (^(0) - if»(x - y)) . The assertion follows. 2•3 This theorem i s a s p e c i a l case of a general theorem i n Harmonic Analysis ([34], p. 19 1.4.3). However, i t i s r e a d i l y derived from the r e a l case treated by Bochner ([4] p. 58). 2.5 Together with the notations of II.2.4 and II.2.6, for each x e X , l e t ijj (w) = f exp i Re wzdu (z) for every z e € We note that (1) i|> (w) = IJJ(WX) for every z e (C Suppose that u i s (/-continuous. Since z e £ -> exp i Re z e (C i s bounded and continuous, by (1) and Prop. II.2.6 we have that (2) <|) i s [/-continuous Suppose that I}J i s (/-continuous. Since ip/c i s p o s i t i v e - d e f i n i t e ' f o r every c > 0 , by Prop. 2.2.1 we may assume that * ( 0 ) = 1 • Given any e > 0 , choose o < e« < f<^=h , t > 2//e* , and V e f s.t. x e V => 1 - Re IJJ(X) < e' . By (1) and the fac t that V i s balanced, (3) x e V , z e <C , |z| <_ 1 => 1 - ^ (z) < e T . i 12 2 2 If z = u^ + i u 2 e ' <C ., then |z| = + u 2 , and therefore, by (3), for any x e V , (4) u 2 + u 2 <_ 1 => 1 - i|> (V) < e' . Hence, by the lemma given by Kolmogorov i n [17], for any x e V (5) U ( D ) = y ({z e « : |z| > 1}) < (e' + ~ ) X X ~ ^ - l t2 < e Since e was a r b i t r a r y , i t follows that u i s (/-continuous Proof of Cor o l l a r y 2.5 We note that nbnd 0 i n X has a base balanced, absorbent sets V , with e.V £ V Hence, by the above theorem and Prop. 2.2.5, /A i s continuous <=> u i s (/-continuous <=> iff i s (/-continuous <=> ij> i s continuous. 132. V c o n s i s t i n g of for every E > 0 3. CM-spaces. For any family C of f i n i t e c y l i n d e r measures over a vector- space X , we s h a l l define the C-topology on X and give some of i t properties. We s h a l l e s t a b l i s h examples of t o p o l o g i c a l vector spaces whose topologies are exactly those determined by the f a m i l i e s of continuous f i n i t e c y l i n d e r measures. We note that' for sets X and Y , topology G on Y , and T : X -> Y , '.{T _ 1[G] : G e G} i s a topology on X . We s h a l l r e f e r to i t as the-topology on X induced by G and T 3.1 D e f i n i t i o n Let X be a vector space. (1) . For any family C of f i n i t e c ylinder measures over X , the C-topology i s the topology on X having for a base a l l subsets V of X with V = x + e A {y e X : y (D ) < e} yeH y y for some x e X , f i n i t e H C C , and e > 0 (2) X i s a CM-space i f f X i s a t o p o l o g i c a l vector space whose topology i s the CM(X)-topology. Concerning C-topologies we have the following assertions. 134. 3 • 2 Propositions Let Y be a vector space, and C be a family of f i n i t e cylinder measures over Y (1) Y i s a topological vector space under the C-topology, which i s the coarsest such topology with respect to which C i s a family of continuous cylinder measures. In p a r t i c u l a r , when Y i s a topological vector space, and C = CM(Y) , the C-topology i s coarser than the o r i g i n a l topology of Y (2) For any vector space X and T e L[X,Y] i f C o T = {y a T : y e C> • , then, the C Q T-topology i s the topology on X induced by the C-topology and T We s h a l l now show that the class of CM-spaces contains many interesting topological vector spaces. However, not a l l topological vector spaces are CM-spaces. In Appendix 4 we give an example of a Banach space which i s not a CM-space (Example 4.2). 135. 3.3 D e f i n i t i o n s (1) Let X be a vector space b : X -> R"*~ i s a pseudo-quasi-norm on X i f f b(0) = 0 , for any x and y i n X , b(-x) = b(x) , b(x + y) < b(x) + b(y) , and z e € -»- b(zx) e R~*~ i s continuous at 0 E. (D (For any family {b.}. T of pseudo-quasi-norm on X , as i n Yosida [49] p. 31, one can show that X i s a t o p o l o g i c a l vector space under the coarsest topology on X making b^ continuous f o r every j e J ). (2) For any measure space (A,n) , and r > 0 , r,. „A „ . , , r i ,-1 r L (A,n) — {f G £ '• f i s n -measurable, / |f| dr\ < 00}' , b r : f e i r(A, n) ^ / | f ! r d n e R + , and when r >_ 1 , | - I : f e Lr(A,n) - (/ | f . | r d n ) 1 / r e R + . Remarks When 0 < r < 1 , b^ i s a pseudo-quasi-norm on L 1(A,n) , which i s therefore a t o p o l o g i c a l vector space under the coarsest topology making b^ continuous. 136. When r >_ 1 , |•| i s a pseudo-norm on L~(A,n) > which i s therefore a l o c a l l y convex space under the coarsest topology making |'| continuous. We s h a l l hereafter assume that L (A,n) , r > 0 , car r i e s . t h e appropriate topology indicated by the foregoing observations. We s h a l l need the following lemmas, which are. of independent i n t e r e s t . 3.4 Lemmas (1) Let X be a vector space, and b be a pseudo-quasi-norm on X . If \p is a p o s i t i v e - d e f i n i t e function on X such that the coarsest topology on X making IIJ continuous coincides with the coarsest topology making b continuous, then, there e x i s t s a f i n i t e c y l i n d e r measure y over X whose c h a r a c t e r i s t i c function i s , and, the {y}-topology on X i s the coarsest topology on X with respect to which b i s continuous. (2) Let X be a vector space. If {bv,}T7 .. i s a family of V Vey pseudo-quasi-norms on X such that f or each V e I/, there e x i s t s a p o s i t i v e - d e f i n i t e function I|J on X s a t i s f y i n g the hypothesis given i n (1) above, then, X i s a CM-space under the coarsest topology making b^ continuous for each V c V (3) Let (A,n) be a measure space. For any 0 < r <_ 2 , f e L r(A,n) exp - b (f) e £ i s a p o s i t i v e - d e f i n i t e function on Lr(A,n) 137. The following theorem and i t s c o r o l l a r i e s i n d i c a t e that many of the t o p o l o g i c a l vector spaces considered i n this paper are i n f a c t CM-spaces. 3.5 Theorem X i s a CM-space whenever X i s a t o p o l o g i c a l vector space having a family f of neighbourhoods of 0 which s a t i s f i e s the following conditions: ( i ) {eV : V e V, c > 0} i s a base for nbnd 0 i n X . ( i i ) For each V e C , there ex i s t s a measure space (A^,!"^) , rV 0 < r v <_ 2 , and T y e L[X,L (^,r\ )] ,.such that r V = {x e X : J | T v ( x ) | d n v < 1) . C o r o l l a r i e s . (1) Let (A,n) be a measure space. For any 0 < r <_ 2 , IT L (A,ri) i s a CM-space. In p a r t i c u l a r , (A) i s a CM-space. (2) Let X be a t o p o l o g i c a l vector space. For any 0 < r <_ 2 , X with the S -topology i s a CM-space. (3) Every H i l b e r t i a n space i s ' a CM-space. 138. Proofs 3 Notation For any vector space X , e > 0 , y e X , K e ) = {(x,f) e X x x* : |f( x ) | >_ e } , D y > c = (f e F j = | f ( y ) | > , and f o r any family C of f i n i t e c y l i n d e r measures over X V ( C , E ) = H {X e X : p ( D ) < e } UEC X X , £ Remark Since D = D for a l l x e X and e > 0 , x, e x e i t follows that V(C,e) = E A {x e X : y (D ) < e} yeC x x 3.2,1 We s h a l l only prove the f i r s t a s s e r t i o n . The second then follows immediately from the d e f i n i t i o n . Let V = {V(H,E) : H c C i s f i n i t e , e > 0} . In view of the remark above, i t w i l l be s u f f i c i e n t i f we show that 1/ has the following properties. (i) 0 e V for every V e 1/ ( i i ) \J i s a f i l t e r b a s e . For each V e t / , ( i i i ) there e x i s t s U e l' s . t . U + U c V . (iv) V i s absorbent, (v) V i s balanced. (Treves [47] p. 21) 139. Proofs of ( i ) - (v) (i) For any e > 0 , and therefore 0 e V for every V e V ( i i ) For any 0 < 6 < E , y e C , and y e X , y (D J > y (D ) . y y>o - y y,e Hence, i f V(e_.,C'J e V , j = 0,1 , and e = min {e^e.^ ' then. V ( e,C„U C\) C H V(e.,C.) . 0 1 j-0,1 2 2 ( i i i ) Let V = V(e,fO , and U = V(e/2,H) . For any x e X , y e Y , and f e F , , (II.2.6), (x,y) |f(x) + f ( y ) | < | f ( x ) | + | f ( y ) | , and therefore, I x + y ( e ) C I x(e/2) U I t(e/2) . Consequently, for any x e U , y e U , and y e H , y (D ) = y^ (I (e)) x+y x+y,e F, . x+y (x,y) < y F ( I x ( e / 2 ) ) + y p (I (e/2)) (x,y) (x,y) = y (D , ) + y (D . ) < e . x x,e/2 y y,e/2 i . e . . U + U C V . (iv) For any x e X , e > 0 , t > 0 , and y e C , and 0 < u < t => D , C B , x,e/u x,e/t 140. Consequently, since y^ i s f i n i t e , lim y , (D . ) = l i m y (D ) x/n x/n,e x x,ne rv-K° n-x" = y ( A D ) = y (0) = 0 . new x,ne x Hence, for any V e 1/ , there ex i s t s n e w s.t. x/n e V i . e . V i s absorbent. (v) For any y e C , x e X , e > 0 and z e C . with | z | <_ 1 , y (D ) = y (D, , ) = y (D , .,) z x- zx,e x |z|x,e x x,e/|z| < y (D ) since D , > . C D — x x,e x,e/|z| x,e Hence, for any V e 1/ , zV C V . 3.2.2. As i n Lemma II.4.2, for any x e X , e > 0 , and . y e C , Hence, for any e > 0 and f i n i t e subfamily H of C , T _ 1 [ A ' (y e Y : y (D ) < e}] yeH y y ' £ = C\ {x e X : y (D T ) < e) yeh x » = A {x e X : (y a T) (D ) < e} . yeH x X ' C I t follows that '{T - 1[V] : V e V} i s a base f o r the C a T-topology neighbourhoods of 0 i n X , where V i s as defined i n Proof 3.2.1. However, from Proof 3.2.1 we see that \J i s a base for the C-topology neighbourhoods of 0 i n Y The as s e r t i o n now follows from Prop. 3.2.. 1 and the l i n e a r i t y of T Lemma Let F be a fini t e - d i m e n s i o n a l space. If b i s a pseudo- quasi-norm on F , then b i s continuous on F Proof of Lemma. Let K be a basis of F Every x e F has a unique representation £ z (x)a , aeK a and the norm x e F -> E I z (x) I e R + „ a • aeK generates the topology of F For any net (x.). _ i n F , J J £ J x. -> 0 => E I z (x.) I -> 0 => 3 aeK a 3 z (x.) -> 0 for each a e K => a J b(z (x.)a) -> 0 for each a e K => a 3 E b(z (x.)a) -> 0 => b(x.) -> 0 , since aeK a 3 3 b(x.) < E b(z (x.)a) . J aeK J Hence b i s continuous at 0 e F . However, for any x and y i n F , |b(x) - b (y) | £ b ( x - y) , and therefore c o n t i n u i t y of b at 0 e F implies continuity of b on F 142. 3.4.1 By the above Lemma, 4)|F i s continuous f o r every F e F , and therefore, by Thm. 2.4, there i s a cyli n d e r measure y over X whose c h a r a c t e r i s t i c f u n c t i o n a l i s ip . From the hypothesis, X i s a to p o l o g i c a l vector space under the coarsest topology making continuous. Hence, by Cor. 2.5 and Prop. 3.1.1, the {y}-topology = coarsest topology making y continuous = coarsest topology making \p continuous = coarsest topology making b continuous. 3.4.2 By Prop. 3.1.1 and Lemma 3.4.1. 3.4.3 Let a : B E M •> a E B . n • B For any P e P(M ) , l e t d(P) be the family of f i n i t e subsets of P directed by i n c l u s i o n . Then, f o r any f E L (A,n) , b (f) = lim lim E |f(a )| r * n C B ) . r PeP(M ) Ked(P) BeK n Consequently, since t E R exp - t £ R i s continuous, we have that i 1/r , r exp - b (f) = lim lim II exp - |n(B) f (ct ) | r PeP(M ) Q£d(P) BEQ r n 1/r Since f E L (A,n) -> n(B) f ( a j j ) i s l i n e a r for every B £ M , we deduce from Lemma 1.1.1 and Props. 2.2.2 - 2.2.4 that f E L 1(A,n) exp - b (f) E £ i s p o s i t i v e - d e f i n i t e . 143, 3.5 For each V e f , l e t = 1 when r ^ < 1 1/r^ when r ^ > 1 b v : x e X ->• (/|T v(x) | d ^ and \J> : x e X exp - b (x) e C For each V e 1/ , we have that by i s a pseudo-quasi-norm on X , and for every t > 0 , r v e V tV = {x e X : b (x) <_ t } Since the topology of X i s completely determined by i t s neighbourhoods of 0 , i t follows that the topology of X i s the coarsest topology making b^ continuous f o r every V e V . However, by Lemma 3.4.3, ifj i s p o s i t i v e - d e f i n i t e , and since t e R + •+ exp - t e (0,1] i s a homeomorphism, i t follows that the coarsest topology on X making ^ continuous = the coarsest topology on X making b^ continuous. The theorem i s now a consequence of Lemma 3.4.2. C o r o l l a r i e s (1) and (2) are. immediate consequences of the theorem. Proof of Corollary (3) Re c a l l i n g the d e f i n i t i o n of a H i l b e r t i a n space (§111.2), we need only make the following observation. Let X be a vector space. For any pseudo-inner-product [.,.] on X , there exists a measure space (A,n) (A i s an 2 index set and n i s counting measure on A ), and T e L[X,L (A,n) ] , such that 0 [x,x] =/ |T(x)| idn f o r a l l x e X . (Treves [47], p. 115 - 116.) 144. 4. Examples 4.1 Example There ex i s t s a Banach space X and f i n i t e c y l i n d e r measure u over X such that y i s S^-continuous but i s not E-tight. Proof Let A be a set. Together with Notation 1.1, l e t X = £^~(A) with the usual topology (Notation I I I . 4 ) , [•',.] : ( x , y ) e X x X + Z x(a)y(a) e £ aeA 4 : x e X -»- exp - [x,x] e C For any f i n i t e K C A , K K * T ; w e £ -> f e (C ) , where K. w K f (x) = Z x(a)w(a) for a l l x e C aeK Since [x,x] = Z |x(a) | 2 f o r a l l x e £ 1(A) , aeA as i n the proof of Lemma 3.4.3, we deduce that \p i s a p o s i t i v e - d e f i n i t e function on £~^(A) . Since x e X ->- /[x,x] i s a norm on £^(A) , we furt h e r deduce that \{J|F i s continuous for every F e r By Thm. 2.4, there e x i s t s a cylinder measure y over £^~(A) whose c h a r a c t e r i s t i c function i s \> . Then, for any f i n i t e K K C A , and x e C , as i n Proof 1.2.2, / exp i Re f ( x ) d ( Y 2 o T^) (f) .= <Kx) / exp i Re f ( x ) d u ( f ) = / exp i Re f(x)dy (f) . (Note that T i s a homeomorphism, and therefore y 0 r i s Radon.) Hence, by Thm. 2.3, (1) V-Y^,!- 1 • Consequently, f o r any t > 0 , with the notation of Proofs III.4, y c K ( r R [ t v J ] ) = y ^ K C t O ^ n £ K ) ° ) ' = Y^Qw e c K : S U P ! w ( a ) | £ t}) aeK = n / e 2 (w(a))dX(w(a)) aeK | / \ i |w(a) I<t However, / 6 2(z)dX(z) < 1 , | z|<t ' since / 6 2(z)dA(z) = 1 and i s s t r i c t l y p o s i t i v e (Lemmas 1.1). I t follows that i n f y £ K ( r K [ t v J ] ) = 0 , f i n i t e K c A and therefore, by Lemma 1.5.1.2, . y cannot be E - t i g h t . On the other hand, by Pietsch [30] p. 82, Prop. 4, there e x i s t s S_ e- s.t. 1, u [x,x] <_/| f(x)|d n ( f ) f o r a l l x e X , and therefore x e X ->- [x,x] e R + i s .^-continuous. Hence <JJ i s S^-continuous. Consequently, by Thm. 2.5, y i s S^~-continuous. 4.2 Example Let A be a set, 2 < p < 0 0 , and 1 = 1 - P q I f y e £ q(A) i s such that E | y ( a ) | q | l n | y ( a ) | | = - aeA and T : x e £ P(A) ( x ( a ) y ( o ) ) e £ q(A) , then there exists y e CM(£ q(A)) such that y p T i s not E-tig h t . Notation. Together with the notations of §2.1 and Proofs III.4, f o r any t _> 0, , l e t b(t) = t q | l n t | , t > 0 , and b(0) = 0 . We s h a l l need the following lemma ([41] Lemma2].) Lemma. Let w : A -> <L with |w(a)| 1 f o r a l l a e A There exists a constant 0 < C < <» such that fo r every f i n i t e K A , Y K ( { z e £ K : E |z ( a ) w ( a ) | q > 1}) >_ e" 1 - exp - E b(|w(a)|) P aeK aeK Proof of Lemma Let 6 be the function of Lemma 1.1.1. p From [3] p. 263, 0 < lim | v | q + 2 6 (v) < |v|-*° p 147. Hence, there e x i s t s 0 < C' <; «> s . t . (1) 0 (v) > C 7 | v | ^ + 2 f o r a l l v E C with Ivl > 1 . By Taylor's theorem, for any 0 < t < 1 , 1 - exp - t = t exp - t' f o r some 0 <_ t' < 1 , and therefore (2) 1 - exp - t >_ t e 1 f o r a l l 0 £ t <_ 1 . Let (3) C = 27re _ 1C' . Then, f o r each a e A , (4) f(l - exp -- | v w ( a ) | q ) 0 ( v ) d A ( v ) >_ e" 1 /|vw( a) | q6 ( v ) d A ( v ) by (2), 0<_| w(a) | <1 i e ' V /|w( a)| q • 1 q + 2 dA(v) by (1) l£|v|<_ l/|w(a) | ' V' = 27re~1C ,K*»lfy 1/p dp l<p<_l/|w(a) | = C b(|w( a)|) by (3). For any f i n i t e K C A , i f B R = {z e C K : Z | z ( a ) w ( a ) | q > 1} , aeK then, Y D ( V l / 1P ( z ) ( 1 " 6 X P " S l z ( a ) w ( a ) | q ) d Y ^ ( z ) P K aeK P = / ( l - exp - Z | z ( a ) w ( a ) | q ) d Y K ( z ) aeK P - / l ( z ) ( l - exp - Z | z ( a ) w ( a ) | q ) d Y K ( z ) £ ~B aeK P > / ( I - exp - Z |z(a)w(a)| q)d Y K(.z) - (1 - e" 1) aeK P = e 1 - II f exp - | z(a)w(a) | q d Y (z(a)) . aeK P 148. However, for each a e K , / exp - | z ( a ) w ( a ) | q d Y p ( z ( a ) ) = 1 - / ( l - exp - | z ( a ) w ( a ) | q ) d Y ? ( z ( a ) ) <_ 1 - Cb(w( a)) by (4) above. Hence, Y K 0 O > e" 1 - n [1 - Cb(w(a))] P K _ aeK > e ^ II exp - Cb(w(a)) since 1 - u < e U f o r a l l u > 0 . , aeK -1 exp - C E b(w(a)) aeK Proof of Example 4.2 If h : x e £.q(A) -> exp - £ | x ( a ) | q e £ , aeA then h i s continuous. By Lemma 3.4.3 and the lemma of Proofs 3., h i s p o s i t i v e d e f i n i t e and h|F i s continuous for every f i n i t e dimensional subspace F of £ q(A) . Hence, by Thms. 2.4 and 2.5, (1) there e x i s t s a continuous f i n i t e c y l i n d e r measure y over £ q(A) with c h a r a c t e r i s t i c f u n c t i o n a l h Clear l y , (2) y e CM(£ q(A)) . Choose (3) t > 0 s.t. |y(a)|/t <_ 1 for a l l a e A . Let 0 < 6 < e" 1 . For any f i n i t e subfamily K of A , l e t K K * h : z e € h (z) e (€ ) , with K K K l i K ( z ) ( x ) = E x(a)z(a) f o r a l l x e C aeK 149, then, as i n e a r l i e r proofs (Proof 1.2.2 (1), Proof 4.1(1), (4) u - Yp o \ • Hence, for any f i n i t e K c A s . t . a e K => y(a) ̂  0 , (y a I) K ( ( ( C K ) * - t(V rv £ K)°) £ K . P = y „ ( ( £ K ) A ~ tT*"" 1^. r\ C K)°) by Lemma 0.4.2, and the fact £ K <EK P that £ K = T[£ K] ; = yh{z e £ K : Z | ^ z ( a ) | q > 1}) P aeK >_ e ̂  - exp - C E b(|y(a)|/t) by the Lemma. aeK Now, £ b(|y(a)|/t) = °° f o r any t > 0 , aeA Therefore there e x i s t s f i n i t e J c A s . t . a e J = > y(a) =j= 0, and e" 1 - exp - C E b(|y(a)|/t) > 6 . ae J Hence, (y G T) T ( ( £ J ) A - t(V A C J)°) > 6 . • £ J P Since t > sup |y(a)[ , and 0 < <5 < e ^ were a r b i t r a r y , aeA and with the notation of Proofs 111,4, r j (tV p°) = t ( V p A £ J)° , i t follows from Lemma 1.5.1.2 that y • T i s not E- t i g h t . 4.3 Example There e x i s t s a Banach space which i s not a C M-space. Proof. Let and CQ = {x e € : lira x(n) = 0} , I I | . . , + * : x e c„ -> sup x(n) e R , £ = £ (CO) 2 T : £ CQ be the canonical imbedding As i s w e l l known, c^ i s a Banach space under the topology generated by the norm | "| From Pietsch [30] p. 83, Remark 2.2, (1) T i s not S^-continuous. From Kwapien [19] we have that 2 u e O I ( C Q ) => y D T has a l i m i t measure on (£ )' , and therefore, by Cor. 1.4.3, u e CM(c Q) => y Q T i s E - t i g h t . Hence, by Prop. I I I . 1 . 3 , (2) T i s S"*"-continuous with respect to the CM(cQ)-topology on c Q . From ( 1 ) and (2) i t follows that the CM(CQ)-topology does not coincide with the norm topology, i . e . CQ i s not a CM-space. 4.4 Example r 2 For any r >_ 4 , the S -topology on £ (w) does not 2 coincide with the S -topology. 2 2 oof We s h a l l construct a T : £ (to) -> £ («#) which, w i l l be r 2 S -continuous but not S -continuous, from which i t follows r 2 that the S -and S -topologies do not coincide. For each n e w , l e t n {n} -2/r a = n , n 2 r T : x e £ (w) -> (a x ) e £ (w) . n n new As i n the proof of Cor. III.4.4.1, we conclude that (1) T i s S -continuous. 2 I f T were also S -continuous, then, there would exist a * 2 * w -Radon measure n on (£ (w)) with supp n e E , s.t. ( / | f ( x ) | 2 d n ( x ) ) 1 / 2 < 1 => |Tx|r < 1 . Hence, l T x l r 1 / | f ( x ) | 2 d n for a l l x e £2(w) . Consequently, for any k e w I a 2 = E | T e J 2 </ E | f ( e ^ | 2 d n ( f ) n<k n<k n<k <_ f ( sup |f( x ) | ) 2 d n ( f ) , | x | 2 l l since {e } i s a orthonormal basis of the Hil b e r t space n new £ (w) . Since supp n e E , and k e w was a r b i t r a r y , i t follows that v 2 L a < 0 0 n new 152, However, this i s impossible, since 2 -4/r ^ -1 , a = n > n , and Y, 1/n n — new Hence m • c2 . T i s not o -continuous, Remark. In view of Theorem I I I . 2.6.3, from the above example we see that f o r every r >_ 4 , I (to) i s not a CM-space. 153. BIBLIOGRAPHY 1. A. Badrikian: Remarques sur l e s theoremes de Bochner et P. Levy. Symp. on Prob. Methods i n Anal. Lect. notes (31) Springer (1967). 2. : Seminaire sur l e s fonctions a l e a t o i r e s l i n e a i r e s . Lecture notes i n Math. 139 Springer-Verlag. 3. R.M. Blumenthal and R.K. 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