UBC Theses and Dissertations

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UBC Theses and Dissertations

Cylinder measures over vector spaces Millington, Hugh Gladstone Roy 1971

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-CYLINDER MEASURES OVER VECTOR SPACES  by  HUGH GLADSTONE ROY MILLINGTON B.Sc,  University  o f West  Indies,  Jamaica, 1966  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  i n t h e Department of MATHEMATICS  We a c c e p t t h i s t o the r e q u i r e d  t h e s i s as conforming standard.  The U n i v e r s i t y o f B r i t i s h March 1971  Columbia  In p r e s e n t i n g t h i s  thesis  an advanced degree at the L i b r a r y s h a l l I  f u r t h e r agree  in p a r t i a l  fulfilment of  the U n i v e r s i t y of  British  make i t f r e e l y a v a i l a b l e  that permission  for  the requirements f o r  Columbia,  I agree  r e f e r e n c e and  for extensive copying o f  this  that  study. thesis  f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department o r by h i s of  this  written  representatives. thesis  for  It  financial  i s understood that c o p y i n g o r p u b l i c a t i o n gain s h a l l  permission.  Department  of  The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada  Columbia  not be allowed without my  ii.  Supervisor:  P r o f e s s o r M.  Sion  ABSTRACT  In  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 m e a s u r e s  from t h e v i e w p o i n t o f i n v e r s e systems we  consider  the problem of f i n d i n g  o f measure spaces.  l i m i t s f o r the i n v e r s e system o f  measure spaces d e t e r m i n e d by a c y l i n d e r measure space  Specifically,  y  over a  vector  X  * For (X,n) the  i s a dual pair,  on  of the algebraic dual  establish  any r e g u l a r t o p o l o g y  p o i n t w i s e c o n v e r g e n c e , we y  c o n d i t i o n s on  convex space.  introduce  G  When  X  We  such  which  that  ensure  . on  £2  , finer  than the topology  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 £2  condition  Radon w i t h r e s p e c t t o  X  , we y  use t h i s  concept i n d e r i v i n g  w i l l h a v e a l i m i t measure on  £2  G  a p p l y our t h e o r y to the study o f c y l i n d e r measures over  H i l b e r t i a n spaces and  G  i s a H a u s d o r f f , l o c a l l y c o n v e x s p a c e , a n d . £2  c o n d i t i o n s under which  Radon w i t h r e s p e c t t o  u  X  the concept o f a weighted system i n a l o c a l l y  the topological dual of  further  Q  f o r i t t o h a v e a l i m i t m e a s u r e on We  is  we  Q  e x i s t e n c e o f a l i m i t measure on For  of  any s u b s p a c e  Jl^-spaces, o b t a i n i n g  a n d 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  significant extensions results.  iii.  TABLE OF CONTENTS Pages INTRODUCTION CHAPTER 0:  1 PRELIMINARIES  3  1.  Set-theoretic Notation  3  2. 3. 4.  Outer Measures and I n t e g r a l s Radon M e a s u r e s I n d u c e d Radon M e a s u r e s  4 6 6  CHAPTER I : 1. 2. 3. 4. 5.  I n v e r s e Systems o f Measure Spaces C y l i n d e r Measures o v e r V e c t o r Spaces N o n - t o p o l o g i c a l L i m i t Measures Radon L i m i t M e a s u r e s F i n i t e C y l i n d e r Measures  CHAPTER I I : 1. 2. 3. 4.  CYLINDER MEASURES OVER VECTOR SPACES  CYLINDER MEASURES OVER TOPOLOGICAL VECTOR SPACES  Notation E - t i g h t C y l i n d e r Measures L i m i t s o f Continuous C y l i n d e r Measures Induced C y l i n d e r Measures  CHAPTER I I I : A P P L I C A T I O N S 1. 2. 3. 5.  Preliminaries H i l b e r t i a n Spaces N u c l e a r Spaces £P-spaces.  APPENDIX: 1. 2. 3. 4.  S p e c i a l M e a s u r e s on F i n i t e - d i m e n s i o n a l s p a c e s 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 CM-spaces Examples  BIBLIOGRAPHY:  9 9 16 20 33 45  . 52 52 54 64 74  77 77 82 94 101  114 114 126 133 144  153  ACKNOWLEDGEMENTS  I wish throughout Dr.  thew r i t i n g  of this thesis.  I wish  like  thesis.  t o thank M i s s Barbara K i l b r a y  guidance  t o thank  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  I would this  t o thank Dr. S i o n f o r h i s i n v a l u a b l e  also  suggestions.  Finally,  for her accurate typing of  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 I.M.  Gelfand  ( G e n e r a l i z e d random p r o c e s s e s ,  tionary  random d i s t r i b u t i o n s  process  ( J . Doob  one  defines a stochastic integral  quantum f i e l d  [ 1 0 ] ) a n d K. I t o ( S t a -  [ 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  [ 7 ] ) , and a r i s e n a t u r a l l y  On t h e o t h e r h a n d ,  i n probability  theory  (in particular,  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 the theory o f i n t e g r a t i o n over f u n c t i o n spaces I.M.  Gelfand  considered in  L. G r o s s  a n d , A.M. Y a g l o m  [12], I . Segal  are defined with respect [13] In  the study  o n some p r o b a b i l i t y  space  (Gelfand  p.  the space o f measurable  [10], I t o [15], Gelfand  functions  and V i l e n k i n  [ 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  and V i l e n k i n  [39]).  concen-  i n one, a c y l i n d e r measure i s viewed  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  inverse  ( e . g . as  o f c y l i n d e r measures r e s e a r c h e r s have  approaches:  [ 1 1 ] Ch. I V , F e r n i q u e  L. S c h w a r t z  [ 4 3 ] ) , where t h e i n t e g r a l s  t o some c y l i n d e r m e a s u r e  a s a l i n e a r map o n a v e c t o r s p a c e i n t o  Gelfand  ([13],  p. 5 3 - 5 4 ) .  t r a t e d o n two m a i n  [25],  when  ( [ 4 ] p. 1 3 7 , [ 7 ] p. 4 2 6 , [ 1 5 ] p . 2 1 1 ) .  t h e demands o f t h e o r e t i c a l p h y s i c s  theory  by  s e t s o f a v e c t o r space  ft  (Minlos  [ 1 1 ] Ch. I V , P r o h o r o v [ 3 3 ] , B a d r i k i a n [ 1 ] ,  Inherent  i n both approaches  ( o r p r o j e c t i v e ) system o f measure spaces  i s t h e n o t i o n o f an ( [ 2 5 ] p. 293, [11]  3 0 9 , [ 3 3 ] p . 4 0 9 , [ 1 ] p. 2, [3.9] p. 8 3 2 , [ 9 ] p . 3 4 ) .  In of  t h e s i s we v i e w a c y l i n d e r m e a s u r e a s a n i n v e r s e  measure spaces  vector the  this  space  X  "target"  i n d e x e d by t h e f i n i t e .  space  Moreover, ft  on w h i c h  b a s i c problem of f i n d i n g functionals  we  dimensional subspaces  do s o w i t h o u t any a p r i o r i  o f t h e s y s t e m on a s p a c e  i s then a n a l y z e d w i t h v a r i a b l e  ft  to the a l g e b r a i c  choice of  dual  X  .  The  ft  of  i n Chapter I .  key i d e a t h e r e i s to examine t h e measure t h e o r e t i c s i z e relation  of a  the l i m i t measure i s t o l i v e .  a limit  To t h i s  of  system  linear  The  ft  in  end t h e n o t i o n o f  "almost" sequential maximality i s introduced. Next, finding  a R a d o n l i m i t m e a s u r e on  space and locally we  ft  consider ft  when  i s i t s topological dual.  convex, by i n t r o d u c i n g  establish  terms  i n C h a p t e r I I , we  When  C h a p t e r I I I we  X  vector  i s H a u s d o r f f and X  r e s p e c t to a weighted  t h e a p p e n d i x we  system.  apply the theory of Chapter I I t o the  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  in  i s a topological  of  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  study o f c y l i n d e r measures over H i l b e r t i a n ,  In  X  the concept of a weighted system i n  of the n o t i o n of c o n t i n u i t y w i t h In  t h e more s t a n d a r d p r o b l e m  nuclear,  and  s e v e r a l p r e v i o u s l y known  £^-spaces, results.  establish mainly technical results  t h e p r o o f s o f 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  used  counter-examples.  ,  3.  CHAPTER 0  PRELIMINARIES  1.  Set-theoretic  In  (•1)  Notation.  t h i s w o r k we  s h a l l use the f o l l o w i n g  0 i s t h e empty s e t . For  any s e t s  A  and  B  ,  A ~ B = { x £ A : x i B} w i s the s e t of f i n i t e R R  i s the f i e l d +  <£  i s the f i e l d we  shall  F o r any s e t  - LJH  X  =  For  any  H|A  c  =  {HA  X  A  c  h "  "such  H  that"  to " s . t . "  of subsets of  X  ,  H,  n HeH  H  : H'  i s countable,  disjoint,  and  = [_|H}  ,  A :H  (if i s a c o m p a c t f a m i l y if  .  and f a m i l y  = {H'C. A  numbers.  abbreviate  HeH P(H)  ordinals.  o f complex numbers.  H , ]~\H =  u  .  of real  = { t e R : t _> 0}  In proofs  (2)  notation.  E  ff} .  i f f f o r any  i s finite  =>  n" c  (I H  H  ,  , then  [\H* ^ 0  .  .  4. F o r any  topology  For any  on  set  X  and  A  For any  f [A] _ 1  F o r any I I  2.  x  Y  c  if  x e A  OeS  if  x e X ~ A  c  : x e  Y  X  = (Y e Y = {x  and  ,  B}  Y , I c X x Y , x e X , y e Y  : (x,y)  e  1}  ,  ,  e l } .  Outer Measures and I n t e g r a l s .  Our m e a s u r e - t h e o r e t i c Caratheodory,  (1)  ,  A}  X : (x,y)  £  ,  ,  [ B ] = {x e X : f ( x ) e sets  G}  >  x  f(x) e Y  = {f (x)  compact i n  1 e <E  f : X -> Y , B  f|A : x E A +  ,  i s c l o s e d and  A  1, : x e X ->  f  X  = {K c X : K  K(G)  (3)  G  For any M  as g i v e n by M.  set  X  and  S i o n i n [44]  Caratheodory  i s the f a m i l y of  n  i s an  approach i s e s s e n t i a l l y and  [45].  measure  n  on  X  ,  n~measurable s e t s .  A-outer measure i f f A c.  ^(A)  t h a t of  = inf{n(A')  : A d  n i s an o u t e r measure i f f n  A'  M  , and  n  f o r any  A c  X  ,  e rt}  i s an M - o u t e r measure.  n  Throughout t h i s work a l l measures c o n s i d e r e d w i l l be o u t e r measures,  n i s t h e C a r a t h e o d o r y m e a s u r e on  X  g e n e r a t e d by  and A  T  iff A i s a family of subsets  of  X  with  0 e A  ,  + T : A n(B)  -> R n  , and f o r any A i s countable,  X  ,  Bcr|jH}  i s a s e t and  n  .  i s an  outer  X  observe that  f o r any measure s p a c e  (X,n)  ,  P(M )  i s directed  refinement. 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  complex-valued  therefore also complex-valued i n t e g r a l s .  f o r any  n-measurable  '  / fd  f : X -> R  =  ( i n f f[B])*n(B)  n  lim E' ( s u p f [ B ] ) - n ( B ) P e P ( M ) BeP  Further,  f o r . any  f : X ->• R  /*fd  = n  , the outer  +  lim Z (sup P e P ( M ) BeP n  a well-defined point i n  R  U  {°°}  functions  H o w e v e r , we  ,  +  lim I P e P ( M ) BeP  n  is  BC  Integration.  We  and  = 0  i s a measure space i f f X  m e a s u r e on  by  T(0)  T(H) : f/c  = i f{ E HeH  (X,n)  (2)  with  .  integral  f[B])'n(B)  point  on  out  X that  6.  (3)  Radon M e a s u r e s  I n t h i s p a p e r , many o f t h e m e a s u r e s we c o n s i d e r f a c t be Radon m e a s u r e s . For  and  any s e t X  and t o p o l o g y  ,  (i)  n  i sa  G-outer measure on  X  (ii)  K e K(G) => n ( K ) < ». , G e G  ,  ,  n ( G ) = sup'{n(K) : K  G-Radon m e a s u r e  n  on  X  C  G  , K e K(G)\  .  , n  I n d u c e d Radon M e a s u r e s  (X,n)  and  Y  be an a b s t r a c t s p a c e .  T : X -> Y T[n]  0  X  Xi f f  Let  n  on  G-Radon m e a s u r e o n  supp n = s u p p o r t o f  (4)  G  d e f i n i t i o n s below.  i sa  f o revery  any  the relevant  ri  (iii) For  We g i v e  T  - 1  F o r any f i n i t e measure space  ,  i s t h e C a r a t h e o d o r y measure on  Y  generated by  a n d { A C Y : T [ A ] e M^} _ 1  We s h a l l u s e t h e f o l l o w i n g  lemmas.  Lemmas  (1)  will i n  F o r any  A e M^-j  T [A] 1  e  M  >  and t h e r e f o r e  T [ n ] ( A ) = n(T [ A ] ) 1  7.  (2)  For any  space  Z  and  U  : Y -> Z  ,  U [ T [ ] ] = (U n  (3)  If  X  and  Y  are t o p o l o g i c a l spaces,  i s Radon, then  T[n]  Proof of Lemma 4.1.  Let &  : T  _ 1  _ 1  n  j  If (1)  c  A'  r\  •  A  Since  is a  and  T[n](A) = 0  A'  T[n](A) =  .  ,  [A])  A-outer measure, t h e r e e x i s t s A  i s c o n t i n u o u s , and  [A] E M } n  that f o r a l l A e A T[nJ(A) = ( T  A e Hpj-  T  i s Radon.  = (Ac !  F i r s t we note  Let  T)[n]  0  . a-field e A  and  T[nJ  i s an .  s.t.  T[r|](A')  .  then  0 < n ( T [ A ] ) <_ n ( T [ A ' J ) = T [ ] ( A ' ) = 0 _ 1  _ 1  n  In g e n e r a l , s i n c e  T[n](Y) <  .  ,  00  T [ n ] ( A ' ~ A) = 0 and  t h e r e f o r e by ( 1 ) , n(T~ [A'  ~ A]) = 0  1  and  T  _ 1  [ A ' ~ A] e M n  Hence, s i n c e T  _ 1  T ^[A'] e M  ,  JA] = T [A'] - T _ 1  _ 1  [ A ' ~ A] e M n  The  second  a s s e r t i o n now  f o l l o w s immediately  T[n]|A = n o T "  1  .  from 1  the f a c t t h a t  Proof o f Lemma 4.2.  We need o n l y observe  '{BCZ  : (U T )  and  f o r any  _ 1  0  [B]  t h a t , by 4.1 above,  e M^} = {B C Z : U [ B ] e _ 1  B CZ Z  s . t . (U T )  _ 1  0  [B]  n((U o T ) [ B ] ) = T [ ] ( U _ 1  _ 1  n  P r o o f of Lemma 4.3.  Let A e T  and  _ 1  M T  n  ,  [B])  £ > 0  a n c i  n  [A] e  therefore, since  e x i s t s a compact  r ]  e .  .  I  By Lemma 4.1,  ,  i s a f i n i t e Radon measure, t h e r e  K c T "*"[A] s . t .  n(T [A.]) - n(K) < e . ' • _1  Then  T[K] c A  T[n](A) Hence, s i n c e (1)  i s compact and  - T[ ](K) n  e > 0  <.n(T [A]) - n(K) _1  was a r b i t r a r y ,  T[n](A) = sup{T[ri](C)  Since  T[n] i s f i n i t e  :CC A  since  .  i s open}  ,  T [ n ] i s an o u t e r measure we t h e r e f o r e conclude  that f o r a l l (2)  i s compact}  i t then a l s o f o l l o w s t h a t  T[n](A) = i n f {T[n] (G) : G ^ A and  <e  BC Y  ,  T[n](B) = i n f { T [ n ] ( G ) : G O B  consequently  T [ n ] i s Radon.  i s open}  . ®  CHAPTER I  CYLINDER MEASURES OVER VECTOR SPACES  As  indicated  i n the i n t r o d u c t i o n , we s h a l l  measures as b e i n g s p e c i a l i n v e r s e [6]).  In the f o l l o w i n g  results  1.  systems of measure spaces  s e c t i o n we i n t r o d u c e  t h a t we s h a l l r e q u i r e  treat  about such systems.  section,  F i s an index s e t d i r e c t e d 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  in  r_, „ : X_, L, r r  F X„ L  with  E < F  i s surjective,  with b e i n g the i d e n t i t y map.  (Choksi  the b a s i c n o t i o n s and  I n v e r s e Systems of Measure Spaces.  Throughout t h i s  cylinder  <  10. Definitions  (X  ,p ) r c£ r  to the maps iff,  r  f o r any r  and,  i s an i n v e r s e sj'Stem of measure spaces  E  r  =  r  , and  G  E,F ° F,G  in  F  £ M  F  (XLpjPp)^,^^  be a n i n v e r s e  F  t h e maps  ,  ,  = u (A)  Let  E < F < G  ,  y (r^ [A]) F  with  '  r  for a l l A e M r~^[A]  to  ,  E, t E,F  E,G  relative  E  system o f measure spaces  relative  r E, b  If  ft  i s a s e t , and f o r each  p_ r  limit  : ft -> X „ r  relative  i s surjective,  t o t h e maps  p  measure spaces, i f f f o r each P  E  =  r  E,F ° F  F e F  ,  then,  of the given  F E  and  F  i n  we c a l l  (ft,0  inverse F  a  system of  with  E < F  '  P  and, F  i s an outer  measure on  ft  such that  f o ra l l A e M  , E  p'V] e M  ,  ap'V]) = U (A) . E  For  the rest of t h i s  s e c t i o n we a s s u m e  that  (X  ,p ) r  i s an i n v e r s e  system o f measure spaces r e l a t i v e  r  r  r £r  t o t h e maps  r  b,.b  .  1.2  Definitions  For  any s e t ft , and s u r j e c t i v e maps  that f o r any P  '(1) (2) (3)  E  E =  r  and  F  E,F ° F P  C y l ( f t , ) = {P~V]  in  F  with  1  +  p  n„ i s the Caratheodory measure on ft, p ii,  _ p  and  When there ft and  such  ,  ,  ( f t , p ) : p ' ^ ] e Cyi(n,p) -»• u ( A ) e R  T  : ft ->• X F  F  '  : F e F, A e Mp}  P  T  E < F  p  Cyl(ft,p)  Q  , generated by  .  can be no ambiguity we s h a l l omit the s u b s c r i p t s  p  Remarks  The f o l l o w i n g a s s e r t i o n s [6], M a l l o r y  and Sion  are r e a d i l y established.  (Choksi  [23]).  (1)  Cyl(ft,p)  is a field.  (2)  T i s well-defined  (3)  Cyl(ft, ) c M  (4)  (ft,n)  P  r  and i s f i n i t e l y a d d i t i v e on  Cyl(ft,p)  .  i s a l i m i t r e l a t i v e to the maps  p  of the g i v e n r  inverse  system of measure spaces i f f n|Cyl(ft,p) = x  (5)  .  There e x i s t s an o u t e r measure is a limit  r e l a t i v e to the maps  of measure spaces  <=>  p  E, on ft such t h a t  r  o f the g i v e n  inverse  (ft,£) system  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 t h a t (ft>n— —) i s a l i m i t ft,p  r e l a t i v e t o t h e maps  p  ,  F  ft C ft , and  f o r each  F e F  ,  Pp = Pp|ft  i s surjective.  We s h a l l be i n t e r e s t e d  i n d e t e r m i n i n g when  (ft,n  n  "  i s a l i m i t r e l a t i v e to the maps  1.3  p  5  ) P  itself  F  Lemmas.  With the above n o t a t i o n and hypotheses, (1)  ) ft, p  i s a limit  i n v e r s e system  r e l a t i v e to the maps  p  F  o f the g i v e n  of measure spaces i f f  r r r - ( A ) = n— — ( A f\ ft) f o r a l l A e C v l (ft, p) Si,p ft,P v ' (2)  F o r any  F'c F  A(F') not e x i s t  , let  be the s e t o f a l l  f e  ft  such t h a t t h e r e does  g e ft w i t h Pp(g)  If,  .  = P ( )  f o r every  f  F  f o r every  {F } C n new  n--(A({F } )) = 0 ft,p n new  F  F e F'  with  F  n  . < F  n+1  f o r each  ,  then n - - ( A ) = r n r - ( A O ft) f o r a l l  A e Cyl(ft,p)  new  P r o o f s 1.  Let C y l ( f l . p ) = C y l , Cyl(n,p) = C y l  T  T  ft,p  '  T  T  ft,p  ft, p  1.3.1 —  —  ,  '  ft,p  Since t h e maps  p  are surjective,  —  f o r each  A  in  Cyl ,  Q  t h e r e 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 C y l  i s bijective  and T(A) = 7 ( A ) B o  Hence, f o r any  for a l l A e Cyl  fi  ,  n(Bnfi) = inf{_E_ H  = inf{ E T(H) : HeH = n(B Consequently, any  n  n)  if  .  e  7(H) :  He  i s countable,  B n f t c j j H }  H  H e  Cyl  Cyl  i s countable,  B  (ft,n)  i s a limit,  .  On the other hand, i f = n ( A f\ ft) f o r a l l A e C y l ,  then, a g a i n by Remark 1.2.4., f o r any  A e Cyl ,  n (A) = "7(1 A 0) = 7 ( A ) = 7 ( A ) = x (A) therefore  c j j H }  then, by Remark 1.2.4., f o r  7(Ac\ ft) = n ( A ) = T ( A ) = 7 ( A ) = 7 ( A )  and  ft  .  A e Cyl ,  7(A)  A  (ft,n)  is a limit.  ,  14.  2.  F'  F o r any s u b f a m i l y  F  of  ;l e t  ,—1.  Cyl(F') = { We  shall  [ B ] : F e F' , B e M } p  show t h a t f o r a n y rf(A') = 0  In which  P p  and  A e Cyl  A'c  ft  s.t.  r f ( A ~ A') = rf (A (\ ft) .  case, rf(A) = r f ( A ~ A') + rf(A  a n d t h e lemma  n e w  , let  r\ ft C I l H  A  a  A') = Tf(A ~ A') = rf(A f\ ft)  follows.  For each  For each  there exists  n e w  and  , choose  Cyl £ Heff  be c o u n t a b l e  with  < 7f(A f\ ft) + 1/n  7(H)  .  n F  countable  F  c  with  n  {A} u H c Cyl(F )  (1)  n  Since  (X^ ^ p ) p p  ^  e  t o t h e maps (2)  F  for  each  r  s  a  n  i n v e r s e system o f measure spaces  , we may  i s a sequence  n  .  n  {F  further  .}. n , j jew  assume in  F  relative  that with  F  . < F . n,j n,j+l  j e w  Let A  n  = A(F  n  )  and A' = Since  Tl(A ) = 0  (3) Let  U A n new f o r every  n(A') = 0 n e w  .  7  n e w  ,  then  .'  F o r each  (g) = ? ( f )  f e A ~ for a l l  there e x i s t s F e  F  .  g e ft s . t .  15,  In p a r t i c u l a r , by (1),. g e A  .  Hence, f o r some f o r some  G e F n g e H  H e H n ,  , with  H = p (J  [p-''"[H] ] " G  ,  and c o n s e q u e n t l y , f e p^tpgCf)] = P It follows  [p (g)] c G  p/tPgtH]] = H  .  that  A ~ A c UH n  and  1 G  n  ,  therefore  7 ( A ~ A ) < n ( A aft)+ 1/n  (4)  n  Since  —  A c ft ~ ft f o r each n  .  n e w ,  A' C ft ~ ft • Hence  A <Aftc A ~ A' and  ,  t h e r e f o r e , by ( 4 ) , f o r 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.  C y l i n d e r Measures over V e c t o r  We  Spaces.-  s h a l l view a c y l i n d e r measure over  a v e c t o r space  X  as  being.an i n v e r s e system of measure spaces whose indexing, s e t i s the f a m i l y of  f i n i t e dimensional In t h i s  we  subspaces of  paper we  s h a l l c o n s i d e r o n l y complex v e c t o r spaces,  s h a l l h e r e a f t e r r e f e r to them simply  subspace we  s h a l l always mean v e c t o r  We  note that i f  F  t h e r e i s a unique H a u s d o r f f convex F  X  s h a l l ever  By  the  term  subspace.  i s a f i n i t e - d i m e n s i o n a l v e c t o r space, topology  (the E u c l i d e a n t o p o l o g y ) .  t h a t we  as v e c t o r spaces.  and  on  F  Since  then  under which i t i s l o c a l l y  this  i s the o n l y topology  on  c o n s i d e r , e x p l i c i t r e f e r e n c e to i t i s h e r e a f t e r  omitted. Throughout the remainder of t h i s work, we  s h a l l use  the f o l l o w i n g  notation. For any v e c t o r space  ,  X  i s the s e t of l i n e a r  w  i s the topology  For any  A c X  F  d i r e c t e d by shall  r  C  subspaces  E, F  E j F  : |f(x)|  of p o i n t w i s e  <_ 1  .  to  §  ,  convergence,  for a l l  x e A}  .  When t h e r e can be no ambiguity  omit the s u b s c r i p t  : F  for a l l r  X  X  i s the f a m i l y of f i n i t e - d i m e n s i o n a l subspaces of  v  A  For any  on  f u n c t i o n a l s on  ,  A ° = {f e X^  i.e.  X  (f)  E  and  -*• E  of  X  with  i s the r e s t r i c t i o n  f e F = f|E  F  X  , .  E C map,  F  >  we  X  17  In what f o l l o w s ,  E  and  F  will  always denote f i n i t e - d i m e n  s i o n a l v e c t o r spaces. For any subspace (X,ft)  ft  of  X  , Ift  i s a dual pair i f f r  i s surjective for  F e F  every  Remark.  With the v i e w p o i n t of i n v e r s e systems d i s c u s s e d i n the F as our index s e t and l e t t i n g A f o r each F F , A  preceeding s e c t i o n , taking X„ = F"  a  we note  t h a t , f o r any  v  v  E  E , F  the r e s t r i c t i o n map  and  G  r E,l  in  F A v  with  E c. F C G  ,  i s s u r j e c t i v e and c o n t i n u o u s ,  and r  E,G  r  E,F ° F,G r  Thus, we s h a l l make the f o l l o w i n g  2.1  Definition.  (1)  Let X  be a v e c t o r  definition.  space.  u i s a c y l i n d e r measure over  X  iff y : F e F •+ y F is  such  , a Radon measure on  F  ,  that  (F ,y„)„ i s an i n v e r s e system F rEr r  to the r e s t r i c t i o n maps  r  .  o f measure spaces  relative  (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 v e c t o r space  X  Remark.  Let  X  be a v e c t o r space. r : F e r ->• u r  u  If  -» * , a f i n i t e Radon measure on' F  ,  then, by §0.4, p i s a c y l i n d e r measure over E  and  F y  Let with  E c  F  in E  =  ft  r  with  E,F  [ l J  F  EC F  i f f f o r any  ,  •  ]  be any subspace o f  F  X  X  .  F o r any  E  and  F  in  , |ft= r E,F  Hence, when  (X,ft)  i s a d u a l p a i r , the v i e w p o i n t o f D e f i n i t i o n  1.1.2 a p p l i e s , w i t h  P  E  =  r  E X^  f  °  r  e a c l 1 ,  E  e  F •  We s h a l l t h e r e f o r e make the f o l l o w i n g  2.2  Definition.  Let and  definition.  ft  X  be a v e c t o r space, jn, a c y l i n d e r measure over  be a subspace  of  X  such that  (X,ft)  i s a dual pair.  X  19.  For  any o u t e r measure  £  on ft' ,  ^ i s a l i m i t measure of (ft,5) r  ft r ,A  i s a limit  u  on ft i f f  r e l a t i v e t o the r e s t r i c t i o n  of the i n v e r s e system of measure spaces  maps  (F ~, y _ ) _ F Fer r  Remarks.  From the theory o f i n v e r s e systems o f measure spaces we know s e v e r a l c o n d i t i o n s under which we can put a l i m i t measure on the p r o jective limit L =  set L  U  e  , where  n  E" : £  EeF  E  = r E  )  w  F  ^(O  , E  c  FV  .  F  S i n c e t h e r e e x i s t s a s e t isomorphism  r : X* -> L such  that r_ ( f ) = ( r ( f ) ) _ fora l l r ,A r v  it  follows  f E x"  and  F E F  ,  that L  i s s e q u e n t i a l l y maximal (Defn. 3.4).  Hence, by a theorem o f Bochner ([4] p. 120), we deduce that has  a l i m i t measure on  X  .  However, l i t t l e  p r o p e r t i e s such a l i m i t measure can have. we s h a l l  JA/ always  has been s a i d about the  T h e r e f o r e , i n the next  section,  c o n s t r u c t one h a v i n g s p e c i a l a p p r o x i m a t i o n p r o p e r t i e s . v'c  U n f o r t u n a t e l y , f o r most p r a c t i c a l purposes unwieldy.  We s h a l l  X  i s f a r too  t h e r e f o r e be s t u d y i n g the problem of p u t t i n g  measures on subspaces of  X  limit  20.  3.  Non-topological  Limit  Measures.  Given any c y l i n d e r measure subspace  ft  of  X  determine s u f f i c i e n t  y  over a v e c t o r  space  X  , and  such t h a t  (X,ft)  i s a d u a l p a i r , we  conditions  on  f o r i t to have a l i m i t measure on  y  shall  ft . Throughout X For  t h i s s e c t i o n we s h a l l use the f o l l o w i n g  i s a vector  space.  any c y l i n d e r measure  Cyl T  :  ft  y^ T  y  (n) = {fi o r ^ A ]  y,  ftfNr"  1  r, A  [A]  £  Cyl  over  X  , and subspace  : F e F, A (ft) -> y  y  s  e  (A)  i s the Caratheodory measure on  y  Mp} e R  ft  Q  of  , +  ,  generated by  Cyl (ft) .  and  n  y ,ft  notation.  y  y>x-  and y  In what  = y * x  •  follows, ft w i l l denote a subspace of dual  X  such t h a t  (X,ft)  is a  pair.  From D e f i n i t i o n 1.1.2  and Remarks 1.2 we get the f o l l o w i n g  assertions.  X  21.  3.1  Propositions.  Let (1)  u  be a c y l i n d e r measure over  For any outer measure  X  E, on ft, E, i s a l i m i t measure of  y  iff  (ft) c. M  Cyl  r  y (2)  E, I Cyl y  and  C  I f there  (ft) = T  e x i s t s any l i m i t measure of  a l i m i t measure of  .  0  y,ft y  on ft , then  y^. i s  y  In view of P r o p o s i t i o n 3.1.2, when l o o k i n g f o r a l i m i t measure , of  y  on  ft  , we s h a l l c o n c e n t r a t e  When ft = X  3.2  on  y^  , we have the f o l l o w i n g  result.  Theorem  For any c y l i n d e r measure y  y  over  i s a l i m i t measure of  X  ,  y  If C = {r'^jK] F,X  : F e F, K c  i s compact}  then C i s a compact: f a m i l y , and  f o r any  A e M , y*  V(A)  ,  = sup{y (C)  (We note t h a t  X  C^  : C CA  , C e C }  i s a l s o a compact  .  family.)  22.  Proof  For each (1)  y_  F e F i s Radon and  cr-finite.  r  Hence, f o r any (2) Since  A e Cyl^(X )  T*(A) = S U { T * ( C )  : C C A , C e C}  P  M (A) 1 T * ( A )  ,  for a l l A e Cyl^x*)  X  ,  v?e a l s o deduce from (1) t h a t (3)  y  Hence, by Thm  is  o-finite.  II.2.5 of [23], the a s s e r t i o n s of the theorem w i l l  f o l l o w once we show t h a t For any  C. = r "  the l i n e a r soan of  F  1 V  C  [ K . ] £ C, j = 1,2,  U F 1  Then  K  i s a compact f a m i l y .  i s compact and  , and  K =  2  (Also see [ 2 4 ] ) .  let F  A j=l,2  r F  be  ^ „[K.] j ' F 1  C. <% C„ = r \ j K ] 1  2  F,X  Hence, (4)  C i s c l o s e d under f i n i t e For any  C'^  Cs.t.  A = { f l ct : a C  C*  ] [ot =|= <f>  intersections. f o r every f i n i t e  i s finite)  We note t h a t A i s a f i l t e r b a s e ( [ 8 ] p. 211). (4), f o r each f i n i t e a C C ,let o P| a = r ^ ^[K ] K  c  F"  f o r some  F  &  e F  , and Y = U(F  a  : a C C  a CZ C  i  s  finite)  .  In view of  and compact  , let  23.  From the remarks p r e c e d i n g subfamilies a  and  C  3  a  =  >  and a  F  therefore  C  of  3  F  Y  3  (4),' we  C  see  t h a t f o r any  finite  ,  '  i s a subspace of  X n  Let  U  be a maximal f i l t e r b a s e i n  i s a s u b f i l t e r b a s e of  A  ([8] p. 219,  X  ([8] p.  Thm.  7.3).  finite a C C , (r„ [U]) _ i s a maximal f i l t e r b a s e i n F ueJ T  F  exists  u e U  s.t.  r  [u] c F  Since  a  K  i s compact and a converges to a unique p o i n t f We  e K  a  note t h a t i f  subfamilies r  = F„  K  a  *  i s Hausdorff, t h i s -  a  , then  3  a and  3  of  ^ ( f j = f • F ,F 3 a ' a 3 s i n c e the r e s t r i c t i o n map  ultrafilter  f  = f  C  .  n  A l s o , f o r any  3  a with  finite  • .  a C  3  ,  ,  v  F  ,X  g iF 1  f e X  a C  C  r  = f  a  If  =  F  i s continuous  ,F ° F.,X ' a a 3 3 Consequently, t h e r e e x i s t s a unique r  and  a' F  a  F  which  Then, f o r each  ' '  a  there  218)  •k  and  r  fl  a C  f o r each f i n i t e  a  i s any  A  g e Y  l i n e a r extension  s.t.  C' of  g  , then, f o r each  . " f  e  r  r  I  "  ;\ t F r  x  I  ( f ) ]  C  i s a compact  =  r r  ? A ,X ^\ a  ,x  [ ]f  n  C a ^ ]  C  _  "  r  Hence,  fl'c + • It follows  that  I  \FX ,a V  . family.  [  =  ^  finite  Next, we c o n s i d e r of  y  the problem of f i n d i n g a l i m i t  measure  on an a r b i t r a r y ft y  Since  y  i s always a l i m i t measure o f  , a p p l i c a t i o n of  Lemma 1.3.1 y i e l d s the f o l l o w i n g b a s i c r e s u l t .  3.3  Lemma.  For  any c y l i n d e r measure  y  over  X  ,  y has a l i m i t measure on ft i f f  y*(A) = y " ( A r \ f t )  (1)  f o r a l l A e C y l (x")  .  y However, we a r e i n t 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 on our systems which w i l l guarantee the e x i s t e n c e  conditions  of a l i m i t  measure  on ft One  such c o n d i t i o n  importance i n t h e g e n e r a l  i s the f o l l o w i n g , which i s o f c o n s i d e r a b l  theory of inverse  (Bochner [ 4 ] , p. 120, C h o k s i [ 6 ] , M a l l o r y  3.4  systems of measure spaces  and S i o n  [23]).  Definition  ft i s for  s e q u e n t i a l l y maximal i f f  any sequence  new  , and '  exists  f  n  {F } n neco e F  •k  n  in F  with  such t h a t  =  f  n  f o r each  new  f o r each  r_, _, ( f ,-) = f F ,F , n+1 n n n+1 1  g E ft such that  r„ F ,X n  F c F ,n n+1  , there  25.  Remark.  We note  trial:  X  i s s e q u e n t i a l l y maximal.  Consequently,  A  the f a c t  p  that  i s a l i m i t measure of  theorem o f Bochner  follows  a l s o from a  ( [ 4 ] , p. 120).  p  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 f o l l o w i n g .  3.5  Proposition  If over  ft  X  i s s e q u e n t i a l l y maximal, then every c y l i n d e r measure  has a l i m i t measure' on ft  However we have the f o l l o w i n g .  3.6  Observation.  If  X  i s a t o p o l o g i c a l v e c t o r space  countable,  linearly  dual,  ft  then  independent  containing  s u b s e t , and ft i s i t s c o n t i n u o u s  i s n o t s e q u e n t i a l l y maximal,  (e.g. whenever  i s an i n f i n i t e - d i m e n s i o n a l , m e t r i z a b l e , l o c a l l y  Proof  Let  {a : n e w} be a bounded, c o u n t a b l e , n  independent linear f(a  subset o f  span of  ) = n  X  , and f o r each  {a ,...,a }  f o r every  n  n e w .  .  a bounded,  new  Then, f o r any  X  convex s p a c e ) .  linearly let F f e X  n  be the  with  26.  (1)  f[{a  ; n e w}] c  n  Hence, t h e r e  £  i s unbounded.  cannot e x i s t  g e ft s . t . e l F = f F n n 0 1  every  new.  therefore  Since,  i n the  For  §[{  a  :  then ^  £  n  theory  d u a l of m e t r i z a b l e  11  i f so,  of c y l i n d e r measure,  I.e.  space ( [ 1 1 ] ,  t h a t we  i s continuous,  ft  i s often  [39]), i t follows  apply  the  and (1).  t h a t the  condition  i n many important s i t u a t i o n s . we  of s e q u e n t i a l m a x i m a l i t y .  Definition  Let U  y  be a c y l i n d e r measure over  sequence  new,  and there  {F  } n new  exxsts  f o r any f  n  in  F  with  F  n  r F  ,-, n+1  f o r every  e > 0 A  n  e M_ F  Z new  ' and  X  i s y - s e q u e n t i a l l y maximal i f f  f o r any  sequence ^  e F" ~ A n n  there e x i s t s  , r  {f  n  F ,F n  f o r each  new,  such  that  n  y  (A )  p  n  < e  n  }  new  with  (f ) = f n+1 n  g e ft such t h a t  "Si  continuous  might take f u l l e r advantage of Lemma 3.3,  t h e r e f o r e weaken the n o t i o n  3.7  F  bounded, which c o n t r a d i c t s  s  of s e q u e n t i a l m a x i m a l i t y does not In order  g U  for  1  r  , (g) = f  n' .  f o r each.  new.  27.  The f o l l o w i n g key theorem of t h i s consequence of Lemma 1.5.2 and the above  3.8  definition.  Theorem  Let If  y  be a c y l i n d e r measure over  ft  is  measure on  ft  We now is  3.9  s e c t i o n i s now an immediate  X  y - s e q u e n t i a l l y maximal, then  y  has a l i m i t  .  e s t a b l i s h a c o n d i t i o n on  y  which ensures t h a t ft  y - s e q u e n t i a l l y maximal.  Definition  Let  y  be a c y l i n d e r measure over  X •k  For any f a m i l y y is  H  of s u b s e t s of  H-sequentially  f o r any sequence n E (a , A E M  {F } n miii with y  F  in  X  tight i f f F c. , -, n n+1 and e > 0 ,  with  (A) < « 0 0 • there e x i s t s H £ H such t h a t y (r^ [A]) ~ r [H]) < e n 0' n n' t  1  p  3.10  F  p  ,  X  F  r  o  r  each  f o r a l l n £ GO  .  Theorem  Let If  y y  be a c y l i n d e r measure over is  H-sequentially  w -compact subsets o f ft i s and  therefore  ft  X  t i g h t f o r some f a m i l y  , then  y - s e q u e n t i a l l y maximal, y  has a l i m i t measure on ft  H  of  28. We  point  m a x i m a l i t y of measure on  3.11  ft  out  t h a t under c e r t a i n c o n d i t i o n s  i s also a necessary condition  y-sequential  for  y  to have a  ft  Proposition  Suppose t h a t the Mackey t o p o l o g y on ([47] p.  369)  restricted  is metrizable. if  y  For any  has  to any  X  induced by  ft  subspace of c o u n t a b l e dimension  c y l i n d e r measure  a l i m i t measure on  ft  y  , then  over ft  X is  , y-seq.uentially  maximal.  Proofs  Lemma.  3.  Let {F 1  } n new  each  F  a sequence i n  with  there  w -compact subset of  sequence  J  f  *  be a  For any  e r_, [K] F ,X n  n  r  {f } n new  n  exists r  For  be  F  n  o  F  new, K  Proof  limit  n  each  ,A  and  g e K  (g) = f  new  (1)  r^  Since  r  n  n  F ,F n n+1  n-rl  ( f ,,) n+i  s.t for a l l  [f n  r  new  [fJnK-H  F  n + 1  ]  A  with  , 1  X  = f  • n  .  .  = f  n  .  n+1  for  29, r" [f ] o r [f Fn , X n n +F 1,.  (2)  1  Also, since  1  ]  .  n + 1  •k  r  i s w -continuous  and  *  K  i s w -compact,  n  - 1  (3)  K n r  r  Since  w  n  *  [f ] ,A n  i s w -compact.  i s a Hausdorff  t o p o l o g y , i t f o l l o w s from  ( 1 ) , ( 2 ) and ( 3 ) , t h a t rS new and  (K r\ r " [ f ] ) > n'  $  1  x  n  ,  the lemma f o l l o w s .  Let {F  F with  } C n new  {B}.  C  M_, ,  F  je-w and  F  =  n  f o r each  1  new,  u_ (B.) < oo f o r each  with  F  Q  j e w  ,  j  q  U B. ,.  • J  0  Since  F c. F n n +  jew i s H - s e q u e n t i a l l y t i g h t f o r some f a m i l y  u  H of  k  w -compact subsets o f choose a 8  w -compact  W F  U  Q c  K  ^ , F 0', n  (  [  n  , given  B  j  e > 0  9, s . t ~  ]  J  ,X n  r F  [ K  j  J  Let C  A  n  .  =  U  jew  (  I  V  [  0'n  B J  j  ]  ~ F R  o =o c  Vi Vi =  n  , f o r each  n-rl  ,x£ j  n'  K  J  ] )  ] )  <  .  j e w  30.  Then, f o r any  n=0  k e w  n  ,  k  and k  n=0 Hence  n  E new  p„ F  j=l  n  (A ) < e n  If  i s a sequence f  then,  n  e F* ~ A n n  f o r some  and hence  r  n  „(g) ,A  = f  sequentially The  Proof  of  by  s . t . f o r each  , r _ _, (f ) = f F ,F n+1 n n n+1  ,  new,  t h e Lemma, t h e r e e x i s t s  for a l l n e w ,  n  n e w ,  g e  ft  s.t.  and i t f o l l o w s t h a t  ft  maximal.  last  a s s e r t i o n i s i m m e d i a t e f r o m Thm.  3.8.  3.11  {F } c n new  F o r any Y =  j e w  f o r every  Consequently, r  1+1  k  U new  F  F  with  F  n  c  F  w i t h the r e s t r i c t e d  A = {f e X  ,, n+1  topology,  : t h e r e does n o t e x i s t r  F  =  n'  r  F  X ^ n'  f  f o r each  °  r  a  g e ft s . t . 1  1  n  E  ^  '  n e w ,  let  i s /A.-  31, 8 = {r^  1  X  : n e to, A E M  [A]  }  n'  n  be the  o - f i e l d generated by  B  , and  n' the Caratheodory measure on  X  B i  r e s t r i c t e d to  Since the topology  choose a sequence  of  X  generated by Y  |B  and  i s metrizable,  {V, }, of a b s o l u t e l y convex neighbourhoods k keco X s . t . {V, n Y}, i s a base f o r the neigh-  of the o r i g i n i n  K  bourhoods of the o r i g i n i n  KEU)  Y  .  Using the Hahn-Banach extension  theorem, one r e a d i l y checks that keo)  new  n  n  I t then follows that  (1)  .A E ©  ,  Cyl^(X ) c M  and, since  ,  (2)  A £ M , y* We note that (3)  Since  A  c y  X* ~ fi .  >v  is  A E Cyl (X*) (4) Since (Thm.  k  o - f i n i t e and  k  y (A) = y (A O ft) f o r a l l  , from ( 2 ) and ( 3 ) i t f o l l o w s that  y*(A) = 0 B  i s a f i e l d and  T  *  i s countably a d d i t i v e on  * Cyl (X )  3 . 2 , Remark 1 . 2 . 5 ) , we have that n|B1  =  and  T*|B 1  y  y"|8 1  =  x*|B y 1  •k .  However, V5 .  |B  has a unique countably  Hence, since  n  CM  M  n T,|$=y*|<&,  and therefore, by ( 1 ) and (4), n(A) = 0  .  y  a d d i t i v e extension to  Consequently, g i v e n any  8  {B . } . C 3 JEW A C  B  =r  '  and  j e to  *  Z T (B.) < e y i jew  exists  .  '  '•  , let [B'] , B^  X F  n. }  J  , there  -s. t .  U B. i jew  For each  e > 0  £  M  ,  F  n. 3  J  and B. 4= r " , [B'] n  f o r any  1  r  n. < n.  and  B' e M n  J  For each  n e w  , let  = UtB*  A n  : 3 £ a),  = n}  n j  Then, (5)  y  E  new Further, i f each  (A ) < e .  n  n  {f } i s any sequence s . t . f o r n new  new, f  then,  r  e F  n  ~ A  n  n  , r _ _, ( f ..) = f F ,F n+1 n n n+1  there e x i s t s  r„ „(f) = f F ,X n n Since  A C  U jew  B. = 3  f e x" ~  U new  f o r each U new  r  r  n  [A ] ,A n  s.t.  new.  r " [A F ,X n 1  V  ] n  , from the d e f i n i t i o n of  A  f o l l o w s that . (6)  there e x i s t s  S i n c e the sequence n  new  was  maximal.  g e Q s.t. r {F } n new  a r b i t r a r y , we  ?  n' in F  (g) = f with  conclude that  for a l l F tg. F n n+1  0. i s  n e w . f o r each  y-sequentially  33.  4.  Radon L i m i t Measures  In t h i s s e c t i o n we s h a l l c o n s i d e r Radon l i m i t measures.  the problem  of f i n d i n g  The t e c h n i q u e we use was communicated to us by  C. S c h e f f e r . In t h i s s e c t i o n we s h a l l use the f o l l o w i n g X  i s a vector  space,  ft i s a subspace For any t o p o l o g y  G  • g : A C X % i n f g  : G  A  y  G  e  notation.  of  X  (X,ft)  such t h a t  i s a dual p a i r .  on ft and c y l i n d e r measure W  ( r  F,X  + sup {g(K) : K  [ A ] )  :  F  z  F  }  over  '  K(G) , K c G}  e  y  ,  i s the Caratheodory measure on ft generated by g.,„  and  G  We s h a l l h e r e a f t e r assume t h a t y i s a f i x e d c y l i n d e r measure over  X  ,  G i s a r e g u l a r H a u s d o r f f t o p o l o g y on ft which i s f i n e r than  w  restricted  We have the f o l l o w i n g important  4.1  Propositions  (1)  y_  isa  assertions,  G-Radon measure on ft , and  b (2)  C y l (ft) a M y y I f there e x i s t s any  to ft.  y |G = g., G  G  (3)  then  \i  n  (7  G-Radon l i m i t measure o f  i s a l i m i t measure of  y  y  on ft ,  X. ,  34,  In view of the above p r o p o s i t i o n s , when s e a r c h i n g G-Radon l i m i t measure o f  y  , we s h a l l r e s t r i c t  f o r a'  our a t t e n t i o n to  \i  n  b Following  Scheffer  [37] we make the f o l l o w i n g  definition.  Our t e r m i n o l o g y i s s l i g h t l y d i f f e r e n t .  4.2  Definition  For any f a m i l y  u is f o r any there  E e F, A e  u^Cr"  o f s u b s e t s of  X  ,  H-tight i f f  exists  r  H  1  with  HeH  „[A]  such  - r_  Jti, r  ^^.(A) < <*> , and  e > 0  ,  that  [H]) < e f o r a l l  F e  F  with  E CF  .  r ,A  We p o i n t out t h a t the above d e f i n i t i o n i s a " u n i f o r m " v e r s i o n of the d e f i n i t i o n of  H-sequential  tightness  (Defn. 3.9).  We now have the f o l l o w i n g key theorem c o n c e r n i n g of a  4.3  G-Radon l i m i t measure of  y  Theorem  u has a <=>  y  G-Radon l i m i t measure on 0, is  K(G)-tight.  Remark  The above theorem extends a r e s u l t due to M o u r i e r  Prohorov  [33] (§5 Lemma 3 ) .  from  the e x i s t e n c e  [ 2 6 ] , and  However, our approach i s somewhat d i f f e r e n t  theirs.  Theorem 4.3 has a u s e f u l c o r o l l a r y .  35. Corollary  If  continuous  X  i s a metrizable, l o c a l l y  d u a l , and  {V  } n  origin  in  X  , with  y has  a  y has  a limit  is its  i s a base f o r the neighbourhoods of  f°r  v +  a  the  new  v n  convex space,  1  n  w -Radon l i m i t  every  new  measure on  measure on  ft  , then,  ft  <=>  <=>  y i s {V°} -tight, 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 f o l l o w i n g lemmas.  L.l  For any  (1)  T(A) = i n f {T(H) : A  (2)  Y  CA)  A e'Cyl^(ft)  < I ( A )  1^2  For any  (1)  r  (2)  g(K)  (3)  For any  K  [KJ  ,  C  H e  .  c K (G)  ,  i s compact f o r every  = y (K) = i n f ' ( T ( H )  KE  H}  E  ( R  and  E,X  [ K ] )  F  in  > ^F  C r  : K C F  F,X  E e f e  H  with  [ K ] )  VtWt  -wv^oW^U*.  H}  E c  •  ,  F  , and  A e  ,  36, Proof of L . l . l  F o r any  (ft  and  C\ r ' ^ t B ] )  y  , since  E  Ji, A  We note t h a t  g ( H ) <_ ( H )  r  E  = y ( B ) = i n f {y^,(G) : B c  T  F T  B e M  ' E, X  is  i s Radon,  > i n f ( ( H ) : ft f\ ^ " ^ [ B ] c  P r o o f of. L.1.2  and  v  X  k w -continuous T  E e F  G C E" , G  H e H} > x(fi A r ' ^ t B ] ) Ji , A  H C  G  , and f o r every  i s open} .  H e H ,  .  T  Hence, y(A) = i n f { I g.,.(H) : H ' c ' H HeH' <_ i n f { E T ( H ) : H'C HeH'  i s c o u n t a b l e and  H i s c o u n t a b l e and  < i n f {x(H) : A c H e H} = T ( A )  Proof o f L.2.1 K(G)  C  We o n l y observe  * f((w ) , and  P r o o f of L.2.2  r„ b, A v  F o r every  , by  t h a t f o r every  is  Ac  |JH'}  {JH }  A c  1  L.l.l.  E e F, y^,  i s Radon,  k w -continuous.  E e F, y„ ii  i s Radon and  r  is  w  -continuous.  L, X  Therefore, g(K)  = i n f {y (G) w  Ji  : E e F, r_. [ K ] C Ji, A V  = i n f ( (H) : K T  c  On the o t h e r hand, by  y (K) = i n f { E  H e H}  G <=. E " , G  i s open}  . '  L.l.l, T(H) : H ' c  H i s countable,  Kc  U' H  }  HeH'  = i n f {x(H)  : K  C H e H},  since,  and  K  is  G-compact,  x  is finitely  H<z G,M  s u b a d d i t i v e on  i s c l o s e d under f i n i t e C y l (ft)  unions,  bv  Proof  37.  of L.2.3  We have t h a t  C  [K]  r  E,F  E,X  [K]]  Hence,  ^E E,X (r  4.1.1  [K])  =  ^F EV E,X ( r  To show t h a t  r  y  is a  [ K ] ] )  l ^  ( F  r  F X  [  K  ]  )  5  G-Radon measure, by S i o n  [44] Ch. V,  Thm. 2.2, we need o n l y show that (1) on (2)  g(<j>) = 0 , g  K(G)  ,  ( ) <  ro  K  Y  Except L.2.2.  f  o  r  1  e K(G)  K  1  ,  f o r a d d i t i v i t y , the p r o p e r t i e s o f  K  and  K(G) c  G  However,  l  f  K  be i n  K(w"|ft)  G. e w"|X,  unions.  a  g  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  Let Since  i s p o s i t i v e , monotone, s u b a d d i t i v e and a d d i t i v e  N  K.C G  2  with  K n  K  g  on  = |  , with  "  Consequently,  since  K ,K„  are  exists K.C  H  n  H  2  H. » 3i =  .  -  H i s a base f o r w ft , and i s c l o s e d under  H. e H  K(G)  , and • w ft i s r e g u l a r and H a u s d o r f f ,  G., i = 1,2  *  =  K(G)  a r e immediate from  <f>  = 1,2  with  w  ft-compact,  finite there  there  exists  38, Then, by L.2.2, g(V  + g(K ) = 2  I  y f l  (  K l  ) +  u (K ) Q  i n f {y-CA^) : K. c A.  3=1,2  °  ^  I i n f {y„(A ) : A 3 = 1,2 " T  O  M  e  3  ^  J  = gC^U  e  K )  gO^)  K ) = 2  Since  J  "  M  J  ft  , K. C A2  and  K  c  H.  2  ; j = 1,2,}  + g(K ) 2  g  , ,  were a r b i t r a r y i t f o l l o w s t h a t  2  It  i s additive  remains f o r us to prove ( 2 ) .  K e H(G)  Let  .  For any  F e F, s i n c e  y  i s Radon and r  is  compact  (L.2.1),  •ft r\ r ~ * [ r  [ K ] ] e Cyl^(ft)  x  and  by L.1.2,  (K)  <_ ( 0 r i . r ^ [ r y  = y (r F  F ) X  [K])  x  F ) X  < oo  .  [ K ] ] ) £ ( n ri r ^ E r [ K ] ] ) T  ,  Hence, y(K)  4.1.2  g  K(G) .  on  Y  ,  2  .  2  Hence, by the s u b a d d i t i v i t y of  gd^u  <z H.} '  , K. c A  y  2  \  *  (j A ) : A  <_ y ^ f t ^ u K )  M  2  j  = i n f . {y (A  e  2  Let If  oo  for a l l  F e F and  y (A) = 0 Y  and  <  K(G) .  A z V  , then, by L.1.2,  (ft A r ^ l A j )  therefore ft A r " [A] e M F,X y 1  K e  = 0  .  r  [K] r ,X  39,  Otherwise, s i n c e  /  u„  5  i s Radon, choose  r  a B o r e l subset  B  of  F  with A  B  C  and  n (B ~ A) = 0 r  By the p r e c e d i n g o b s e r v a t i o n , ft 0 r " [B ~ A] a, A  M  1  However,  r  ft F, X  e  i s (j-continuous  and'by Prop. 4.1.1, y ft and  f\  .  y  is  r"" „[B] E M F,X y  G-Radon.  since  w ft C G  ,  Hence,  ,  1  therefore,  ft f\ r ^ [ A ] = ft ri r ^ [ B ] ~ ft r» r ~ ^ [ B ~ A] p  x  x  e  M  .  We s h a l l now e s t a b l i s h another u s e f u l lemma.  L.3  For every g(K)  K e K(G) <_ ( K ) Y  If  y  K E  K(G) ,  .  g ( K ) = y(K)  Let  A  >. g ( K )  u  ..  By Prop.  4.1.1,  : K C G e G}  .  i s a l i m i t measure of  g(K) = i n f (T(H) : K c H e > y(K)  , then, f o r every  .  K E K(G)  y(K) = i n f ' , { g ( G )  y  .  i s a l i m i t measure of  Proof of L.3  If  ,  , since  y  u  HI  is  , then, by L.2.2 and Prop.  = i n f {y(H) : K c H e  G-Rado'n and  H C G  H}  3.1,  40.  4.1.3 •  Let  £  be any  G-Radon l i m i t measure of  y  on ft  K e K(G) ,  For any  ?(K) = i n f U ( G ) : K C G e G} <_ i n f U ( H )  : K C H E H }  = i n f {T(H) : K c = g(K)  H C G  ,  by Prop. 3.1.1  by L.2.2  <_ ( K )  by L.3.  Y  Hence, as  II e H}  since  E, and  y  a  r  both  e  G-Radon measures on ft ,  f o r a l l A c ft .  £ (A) <_ (A) Y  In p a r t i c u l a r , by Prop. 3.1.1, f o r any T ( A ) = 5(A) < ( A )  A  e Cyl^(fi) ,  ,  Y  a n d - t h e r e f o r e , by L.1.2, y(A) =  T(A)  •  From Props. 3.1.1 and 4.1.2 i t now f o l l o w s t h a t measure of  4.3  A. e  i s a limit  y  By Prop. 4.1.3, i f y is a  y  has any  G-Radon l i m i t measure of with  G-Radon l i m i t measure on ft , then y y  .  Hence, f o r any  E e F  and  P ( A ) < <*> , E  -1 ft n r [ A ] E X  M M  e  Y  . ..-1 , y(ft r\ r ^ t A ] ) < » ,  and t h e r e f o r e y(ft O r ^ t A ] ) = sup {y(K) : K e K(G) , K c ft A r ^ t A ] : ? Hence, f o r any  e > 0  , t h e r e e x i s t s K e H(G)  with  .  K c ft ftr [A]  £j , A and y(ft A r  1 E  x  [A]  ~ K) < e  In which case, y (r^ [A] F  f o r any  - r  F  [K])  p ) X  <_ ( f i n  It  ~ « " r ^ r ^ K ] ])  F  r'^fA]  Y  ~ K) < e  follows that  . We now  y  .  y  show that  condition for  ,  .  = Y(n ^ r ^ [ r ^ [ A ] ) x  E C F  F e F with  is  K(G)-tight.  /<(G)-tightness  to have a  of  y  G-Radon l i m i t  is a  sufficient  measure on  ft  .  IN  view of Props. 4.1 and 3.1.1, we need o n l y show that (1) 'If,  y is  K(G)-tight  f o r every  (2)  T  A  => y | C y l  f o r every Y  (A)  K(G)  e  A e C y l (ft)  >_ sup {g(K) : K  K(G) ,  e  , KC  A}  , since  K(G)  = sup {Y(K) :K e  = (A)  .  C y l (ft) ,  e  ( A ) = sup (g(K) : K  then,  (ft) = T  K  y  A.}  is  a-finite  (L.1.2),  A}  , K C  C  ,  by L.3,  by ( 1 ) .  T  Hence, by L.1.2, Y  (A)  = (A)  Consequently, (3)  y is  Suppose Let  E  Given  is  F and e > 0  y (B E  y  F E F  with  is  E  =>  B  G  Mg  (2) h o l d s  with  , since  E C B  ,  for a l l A  u,.,  y ( ) < B  03  E  e  C y l (ft) .  F  - r  •  i s Radon, t h e r e e x i s t s a c l o s e d  C c  B s  .  «(G)-tight, there e x i s t s  y (r y ] F  C y l (ft) y  e  K(G)-tight.  ~ C) < e/2  Since  A  (1) w i l l have e s t a b l i s h e d when we show t h a t ,  K(G)-tight  y  e  for a l l  T  , F j X  [  K ] L  J ) < /2 e  .  e  K(G)  s . t . f o r every  Let K  =  Since  K  i ^  C  ! x  r E  c  •  ]  i s closed  K e K(G)  (4)  [  and  r  [ ft i s  G-continuous,  .  Further, (5) . K C ft Now, ( 6 )  r\ r'  F e F  f o r any F,X  r  [B]  1  = F,X  [ K ]  r  . with  [ K  l  *  ]  E C E!  r  [  C  F  ,' •  ]  F  Hence,  V ! r  E  = V l V  (  F  r  ( r ^  F  [ B ]  "  E > p  F,X  r  [ K ] )  ~ F,X  ]  r  [ B ]  [  K  l  ]  - r ^ f ^ ] )  )  < F  U  [  B  ~*1]Y™»  ]  + v ( B - C) < E  .  e  Consequently, by L.2.3 and L.2.1, g(K)  = inf {y (r p  F  >, i n f { y ^ r " ^ ] ) 1  = y (B) - e E  Since  e  [K]) : E c  .  with  However, s i n c e  y  i t follows  T(A) < » is  = sup { T ( A ' ) : A ' c  Hence,  F e F}  - e : E C F e F}  was a r b i t r a r y  A e C y l («)  T(A)  X  that  (2) h o l d s f o r a l l  .  a-finite  for a l l  F e F  ,  A , A e C y l (ft) , x ( A ' ) < » }  (2) h o l d s f o r a l l  A e C y l (ft)  .  ,  Proof of Cor. 4.3.  Let  K = {V° : n e w }  We note  that  , and  (1) ft = \JK (2)  K c K(w*|ft)  By (2) and Thm. (3)  .  4.3 we need o n l y show t h a t  y i s ,.:?. /(-tight whenever  Suppose.that E e F and  y  y  C C A  and  Ji  and f o r each  has a l i m i t measure on ft  has a l i m i t measure on  A e M  Since  y  with  ft  Vu(A) < °°  i s Radon, choose a c l o s e d y ( C ) > y^(A) - e/2  C C E  *  s.t.  ,  E  new  . Let  , let  * Since (4)  C K  i s c l o s e d and e K(w"|ft)  r  hi,  is  X  f o r each  w - c o n t i n u o u s , then, by ( 2 ) ,  n e w .  F u r t h e r , by (1) (5) ft rs r ' ^ C ] = Since  y„ ft  U K . new i s an o u t e r measure, and  we deduce from  n  K c n  K  ,, n+1  (5) t h a t  (6)  y^(ft t\ r ^ t C ] ) = sup y (K ) ' new By Prop. 3.1.2,  .  C7)  ,  n  y^ i s a l i m i t measure of  y  Hence,  (8) Let  y^(ftn r ^ [ C ] ) = y ( C ) < y ( A ) < » . x  e > 0  E  E  .  By (6) and ( 8 ) , t h e r e e x i s t s  new  s.t.  f o r each  new  44.  Then, by ( 7 ) ,  W  > V  '  C )  > V. " •  £/2  £  A)  Hence, by (4) and L.2.2, (10)  g(K ) > y (A) - e  F e F with  We have t h a t , f o r any r  and  F , X  [  K  n  1  C  r  .  ; >  C  l  E " ! F  [  A  E e F ,  l  •  therefore,  = y (A) - y ( r b  r  r,A  [K ]) < y (A) - g(K ) n — & n  < e , by (10). However,  K  n  c  V ^ F ^ Since  e > 0  , and t h e r e f o r e ,  n  " F,X^n r  , E e F  xrere a l l a r b i t r a r y ,  ] )  <  E  , and  i t follows  f o r every  F e F  with  • A e that  with y  is  Vg(A) <  00  K-tight.  ,  E  F  ,  45.  5.  F i n i t e Cylinder  We the  case of  finite  5.1  s h a l l s p e c i a l i z e the  f i n i t e c y l i n d e r measures.  By  the  foregoing  introducing  to the problem of f i n d i n g l i m i t s , we  can  sections  the n o t i o n  of  to a  s h a l l show t h a t , concentrate  on  c y l i n d e r measures.  Definition.  u is a finite a vector  space  X  c y l i n d e r measure i f f and  y (F") p  (We  r e s u l t s of  s e c t i o n of an a r b i t r a r y c y l i n d e r measure, we  with regard finite  Measures  note t h a t  f o r some  u  E F X  F  i s a c y l i n d e r measure over  ,'  v  < « .  u (F )  F E F  i s independent of  r  For  the  r e s t of t h i s s e c t i o n we  X  i s a vector  6 i s a regular,  X  such t h a t  r e s u l t s can  be  (X,ft)  H a u s d o r f f topology on  w - t o p o l o g y r e s t r i c t e d to  p i s a c y l i n d e r measure over  The  that  space,  ft i s a subspace of  than the  assume  following  ) A  i s a dual p a i r ,  ft ft  which i s f i n e r ,  X  lemmas i n d i c a t e that  s i m p l i f i e d when c o n s i d e r i n g  the hypotheses of  finite  earlier  c y l i n d e r measures.  46.  5.2  Lemmas.  If (1)  •  y  is a finite  y (A)  c y l i n d e r measure over  = y ( A r\ ft) f o r every  X  (2)  y is for  family  fl  of s u b s e t s of  H-sequentially tight any sequence  new,  and  {F } n new  e > 0  y  <=>  .  ?  For any  , then  A e C y l (X*)  /V  y' (ft) = y"(x")  X  X  ,  <=> in  F  with  c  F  F  n  , there e x i s t s  (F ~ r [H])-< e n r ,A n n r [H] e M f o r every r ,A c  HeH  for a l l  v  ,, n+1  such  f o r each  that  new.  r  (3)  If  y is for  any  H-tight e > 0  y (F"  ~ r  p  P r o o f of  F e F  V  X  HeH  , then,  <=>  there e x i s t s p  and  [H])  < e  HeH  such  for a l l F e  F  that  .  5.2.1  Certainly, i f  y"(ft n A) = y"(A)  y*(ft) = y"(x') On  y (ft) = y (X )  X  = -k  Hence  , then, f o r any  ,  y * ( X ) = y ( f t ) = y" (ft A A) + y " (ft ~ A) X  , then  .  the o t h e r hand i f  A e Cyyx*)  f o r a l l A e C y l (x")  y*(X*)  '  y (ft r\ A) = y  *  (A)  <_ y * ( A )  + y * <x"  ~ A)  47.  P r o o f o f 5.2.2  We observe HeH  o n l y t h a t f o r any  F e F, A e  and  , P (A F  ~ r  F j X  [H])  < y *  ~ r  F  [H])  .  P r o o f o f 5.2.3  Together note  with  the o b s e r v a t i o n o f Proof  t h a t f o r any  E  and  G  in  F  5.2.2 above, we  with  E c G  , and  H e H , V * E  ~ E,X r  [ H ] )  ^ G  (  G  *  ~ G,X r  [ H ] )  The  a s s e r t i o n i s now immediate.  The  f o l l o w i n g theorems a r e now immediate consequences o f ,  r e s p e c t i v e l y , Lemma 3.3, Theorem 3.10, and Theorem 4.3  5.3  Theorems  If (1)  y  i s a f i n i t e c y l i n e r measure over  y has a l i m i t measure on y"(ft) = y"(x")  (2)  ft  X  , then  :L^..>\ ] - . '! .  <=>  .  (Silov [46]).  y has a l i m i t measure on ft i f , for and  any sequence J  H  e > 0  (F ) n new  in  , there e x i s t s a  y (F* - r [K]) < e r n r ,A n n r  F  with  F c F ,.. n . n+1  w -compact for a l l  f o r each  K c ft such  n £ 43, '  that  n e w . .  •  .  48.  (3)  (Mourier-Prohorov, • u has a  G-Radon l i m i t measure on ft <=>  f o r any  e > 0  r  V  [K]  < e  stated  t h e problem of f i n d i n g l i m i t s f o r  c y l i n d e r measures can be recuced  Definition  A e Mg  with  f o r every  .  i n the l i t e r a t u r e .  F i r s t , we make the f o l l o w i n g  5.5  F e F  that  out t h a t Theorem 5.3.2 does n o t seem to have been  We s h a l l now show t h a t arbitrary  for a l l  such  r,A  We p o i n t  previously  K e K(G)  there e x i s t s  U_(F" ~ r  Remark.  [3 ] ^5 Lemma 3)  ^ i sa finite  WgCA) <  0 0  ,  to t h a t  f o r the f i n i t e  case.  definition.  section of  u  i f f f o r some E e F and  5 i s the c y l i n d e r measure over  X  such  that  F e F, ^F  =  r  F,G  [ y  f o r some  G e F with  Remark.  If  l E,F" r  G  E c G  £' i s a f i n i t e  1 [ A ] J  and  ' F c G  section, of  5 i s i n f a c t a c y l i n d e r measure over = u"(r ( T h i s remark i s proved  The f o l l o w i n g  _ 1 EX  [ A ] A B)  X  u  then  |  i s well-defined,  , and  f o r a l l B c X^  .  below.)  theorems a r e then r e a d i l y  established.  49.  5 .6  Theorems.  (1)  y has a l i m i t a limit  (2)  measure on ft i f f every f i n i t e  measure on  u i s K(G)-tight  ft  s e c t i o n of  y has  .  i f f every f i n i t e  section  of  y  is  K(G)-tight.  Hence, y has a of  y  G-Radon l i m i t  has a  measure on ft i f f every  G-Radon l i m i t  finite  section  measure on ft  P r o o f o f Remark 5.5  *• • For any F e F and  a c F  Remark 2.2.1 and Lemma ^G is  (  F,G  r  1  [  a  independent  "  ]  r  _ i with  r  [a] t M  , i n view of  n  0.4.1,  E,G  1  [  A  ]  )  of the c h o i c e o f  G e F with  E c G  and  F C G  Hence, so a l s o i s £ F We note (1)  U l F  r  that  p [ ] 1  E  A  F e F with  f o r any i  s  E C F  ,  Radon.  Consequently (2)  ?  = y |r  F  1  p  E  F  [A]  and i s Radon.  Hence, by Lemma 0.4.3, (3)  E,  i s Radon f o r every  For any E U F u F h  =  r  F  c, G F,G  [  y  and  F^  in  F e F. F  with  F c F^  •, i f G e F w i t h  , then by Lemma 0.4.2, Gl E,G r  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 We  s h a l l now prove t h a t  £*(B)  (4)  X-  = u*(r_. "V] L, A  n B)  for a l l  x'  B c  Let k  k  H = {H e C y l (X ) : H  is  w -open}  ,  r^'V] .  a = We have t h a t  H C C y l (X*) A C y l (X*) and by ( 2 ) , T^(H) = T (H A a)  f o r every  HeH  Hence, by Thro.. 3.1, (5)  ?"(H) = T " ( H A a) f o r a l l y Let B c X I f y (B) <  H - o u t e r measure, f o r any a . B c H Since  and  HeH TO  e > 0  y*(H) < y*(B) + e  a e M , y«  a , then, s i n c e there e x i s t s  y HeH  i s an  a  s.t.  .  we have t h a t  A • A A A y (B n a) + y (B ~ a) = y (B) > y (H) - e k  k  = y (H and  n a) + y (H ~ a) - e  therefore a.  •A.  y" ( H A a) < y" (B A a) + e  .  k  Since y (6) Since  is  y*(B £  A  a-finite  i t f o l l o w s t h a t f o r any  B  C  X  ,  n a) = i n f {y"(H A <*.} : B C H e H } . and  t o g e t h e r imply  A y  a a r e both  t h a t (4) h o l d s .  H - o u t e r measures, (5) and (6) o  Proof o f 5.6.1.  By Lemma 3.3 w£ need o n l y show t h a t (1)  y ( A ) = y * ( A n ft) f o r a l l X  A e C y l (X*)  iff (2)  E, (X ) = E, (ft) From  f o r every f i n i t e  section  E, o f y  Remark 5.5 i t i s immediate t h a t (1) => (2).  However, (2) => y"(A) = y' ( A n ft) f o r a l l A e' Cyl y (A) <  oo  For any  )  i n c r e a s i n g sequence  , since  {A } n new  y  i s a - f i n i t e , choose an  i n C y l (X ) s . t y J  f o r a l l n e w and  00  A  Since  with  .  A e Cyl^(X  y (A ) <  (x")  y  = U A new n i s an o u t e r measure, we then have t h a t  p ( A Oft)= l i m y * ( A A ft) = l i 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  E e F and  A e M  E  with  Remark 5.5, f o r any  s e c t i o n of y„(A) < oo E  F e F  The  [  A  ]  - F,X r  [ K ] )  =  ~  .  determined  by some  By (2) i n the p r o o f of  E C F  with r  V ^ F  y  F,X^  , and }  K e K(G)  •  a s s e r t i o n now f o l l o w s from Lemma 5.1.2, and Thm. 4.3.  ,  CHAPTER I I  CYLINDER MEASURES OVER TOPOLOGICAL VECTOR SPACES  In t h i s c h a p t e r , we  are p r i m a r i l y i n t e r e s t e d i n d e t e r m i n i n g  when a c y l i n d e r measure over a H a u s d o r f f l o c a l l y convex space have a l i m i t measure on the t o p o l o g i c a l d u a l  X'  r e s p e c t to some g i v e n t o p o l o g y  Since  G  on  X'  .  X  will  which i s Rad^n w i t h (X,X')  p a i r , the theory of the p r e v i o u s c h a p t e r a p p l i e s w i t h  i s a dual  0, = X'  A Hence, i f  G  i s r e g u l a r and  f i n e r than the  to  X'  , then, by Theorem 1.4.3,  on  X'  whenever  take we  G  y  is  y  w -topology  w i l l have a  H - t i g h t f o r some f a m i l y  G-Radon l i m i t measure H c  to be one of t h r e e standard t o p o l o g i e s , and  take f o r  H  the p a r t i c u l a r f a m i l y  concern i s then d i r e c t e d  E  restricted  K(G)  .  We  shall  these suggest t h a t  d e f i n e d below.  Our main  towards f i n d i n g c o n d i t i o n s under which  y  is  E-tight.  1.  Notation  We  ,  p o i n t out t h a t our t o p o l o g i c a l v e c t o r spaces a r e not •  assumed to be n e c e s s a r i l y  Hausdorff.  In the r e s t of t h i s paper we For any v e c t o r space  X  and  V ° = {f e X"  V c X : |f(x)|  s h a l l use  the f o l l o w i n g n o t a t i o n .  , <_ 1  f o r a l l x e V}  .  53. For any t o p o l o g i c a l v e c t o r nbnd 0  in  X  space  X  ,  i s the f a m i l y of'neighbourhoods  o f the o r i g i n i n X  k  K c X  E i s the f a m i l y o f a l l s e t s K c  and  f o r some  k  such  V g. nbnd 0  that  in  K  is  w -closed  X  •k  X' = {f e X and  : f  i s continuous}  F e F , A. F = F l ' •  f o r every r  r  X  In a d d i t i o n to the f o l l o w i n g two c  w -topology r e s t r i c t e d  compact s u b s e t s o f i s the topology on bounded s u b s e t s o f  Remark  We note  that  E c K(w*) and •k | W  X'  , we s h a l l c o n s i d e r the  topologies: i s the t o p o l o g y on  s  to  X' C  k C  C  A S  X' X X' X  of u n i f o r m convergence  on the  , of u n i f o r m convergence  on the  54.  2.  E - t i g h t C y l i n d e r Measures.  Throughout  this section  y i s a c y l i n d e r measure over When  X  X  i s a t o p o l o g i c a l v e c t o r space and  X  is locally  convex and H a u s d o r f f we n o t i c e that  E  n o t h i n g e l s e but the f a m i l y of  w - c l o s e d e q u i c o n t i n u o u s s u b s e t s of  Hence, f r oiu TrsvGs  32.5 a n d 32.8, W G h.avG  [47] , Props*  is X'  that  E c K(c*) . Consequently,  2.1  a p p l i c a t i o n of Theorem 1.4.3 y i e l d s the f o l l o w i n g  Theorem.  Let  X  be H a u s d o r f f and l o c a l l y *  u is  E - t i g h t => If  Thm. y is  E = K(c  )  y  c -Radon l i m i t measure on  , i n particular, i f  E - t i g h t <=>  y has a  s -topology. E = K(s  X'  i s b a r r e l l e d ([47]  )  ( [ 4 7 ] , Prop. Ec  y  X'  can a l s o imply the e x i s t e n c e  which i s Radon w i t h r e s p e c t to the  F o r example, i f  ([47] p. 510) then  K(s")  X  i s a Montel  34.5); o r , i f  space X  ([47] p. 356)  i s a n u c l e a r space  ([47] Prop. 50.2).  Theorem 1.4.3, we o b t a i n the f o l l o w i n g  2.2  X  X'  c -Radon l i m i t measure on  E - t i g h t n e s s of  a l i m i t measure on  then  convex  33.1) then,  Sometimes of  p. has a  •  assertion.  Hence, on a p p l y i n g  theorems.  Theorem  (1)  If y is  X  i s a Montel space,  E - t i g h t <=>  y  has an  then,  s -Radon l i m i t measure on  X'  (2)  If y is  X  E - t i g h t => y  Even i f that  2.3  y  i s a n u c l e a r space,  E £ K(s  has an  Theorem.  Let  V e nbnd 0  in  X^ =  ,  y  s -Radon l i m i t measure on  X  be H a u s d o r f f and l o c a l l y  can s t i l l  X'  imply  1  convex.  For any  X let  nV®  w i t h the t o p o l o g y induced by the norm  |* : f e X^ + s u p | f ( x ) | x«=V If  t h e r e i s a base  V  1/  £  s -Radon l i m i t measure on  E - t i g h t n e s s of  X  )  has an  then,  e R  \l f o r nbnd 0  +  in  X  such t h a t f o r each  , X^  i s separable.  then, (1)  y  (2)  If  is  E - t i g h t => y  X'  every f i n i t e  We Lemma. G  „  , then indee k  c -Radon measure on  X'  establish V e nbnd 0  the f o l l o w i n g  is  <  s -Radon.  in  X  s.t.  lemma. X^  i s the f a m i l y of a l l open subsets of  generated by  s  X'  2.3  first  For any  s -Radon l i m i t measure on  i s a s e p a r a b l e Banach space under *  P r o o f of Theorem  has an  (c  |x^)  i s separable, i f X^  , then  Gc c r - f i e l d  56.  Proof  Let  ff =' { f + V °  : e > 0 , f e X^}  E  For any g e X'  e > 0 eg  -y  and  X  e  f e X'  and  1  g  ,  X'  e  f + g  ->  are homeomorphisms w i t h r e s p e c t a topological vector . Hence, s i n c e  V  0  and  to  space under  c -closed  X'  e  c  , since  X  is  !  c  * w - c l o s e d and  is  H c o n s i s t s of  .  w  *  c  c  *  ,  subsets  therefore c  f/  c r - f i e l d generated by  However,  X^  i s separable  and  (c  |X^)  metrizable.  Consequently, G c We (1)  now  C  prove the  By Thm.  a-field  g e n e r a t e d by  1.5.6.2, we  may  assume t h a t u  e > 0  ., t h e r e  exists  u„(F  ~ r_[K])  < e  r  2.1 ]i' ^  and  of 1.4,  u  of P r o o f s  is  is  y (F*  1.4,  .  u  is  X  E - t i g h t , then,  X'  of  ,  .  for a l l  K e K(c*)  E - t i g h t , f o r any in  e > 0  . , there  exists  s.t.  separable,  for a l l p  since  c -Radon l i m i t measure on  V e nbnd 0  and  s.t.  F e F  is  c  X^  )  Props. 1.4.1,  u * ( K ) = g(K) Since  f o r every  need to prove t h a t f o r  and  by L.3  (1)  K e K(s  i s normalized.  r  With the n o t a t i o n Thm.  |X^)  theorem.  In t h a t case, by Lemma 1.5.1.2, we any  (c  F e  F  ,  ~ r [ V ° ] ) < e/2 F  .  »  by  57.  Hence, by L.2.1 of P r o o f s 1.4, 1 - g(V°) < e/2 , and (2)  y  c > v  t h e r e f o r e , by ( 1 ) , ( X ' - X^) < 1 " V * ( V ° ) < e/2  .  C  If 5 : A c X^ then  5  is a  u^(A) e R  +  ,.  +  c |X^-outer measure on  X^  * i  By the lemmas, and the f a c t it  follows that  However, X^ L32] (3)  X^  £  is a  c [X^ c 6  G-outer measure on  i s complete  , X^  ( [ 4 7 ] , Lemma 36.1, see a l s o p. 477)-  i s a l s o s e p a r a b l e and m e t r i z a b l e . K e K(G)  Thm. 1.4, there e x i s t s  Hence, by Prohorov s.t.  5(X^ ~ K) < E/2 However, s |x^ c G K E K(s*) • Then, c e r t a i n l y , g(K)  , hence  K E K(C )  = u^CK) > 1 - e  From the d e f i n i t i o n o f u„(F"  g  ~ r_,[K]) < s  r  (2)  that  , and by ( 1 ) , (2) and ( 3 ) , . , and L.2.1 of P r o o f s 1.4,  for a l l  From the lemma we have any an  F e F  .  r  s - o u t e r measure  from Prohorov  c -Radon measure on  (G = s )  .  X'  is  The a s s e r t i o n now f o l l o w s  [32] Thm. 1.4  We a r e l e d by the above theorems to study c o n d i t i o n s under which w i l l be case when  E-tight. u  i s finite.  Conditions of  In view of Theorem 1.5.6.2 we s h a l l c o n c e n t r a t e on the  for  u " to be  the o n e - d i m e n s i o n a l subspaces of  n e c e s s a r y such c o n d i t i o n .  E-tight w i l l X  .  then be g i v e n i n terms  We b e g i n by i n d i c a t i n g a  2.4  Proposition and  p  Let  X  be  a topological vector  a f i n i t e c y l i n d e r measure over  If  p  is  E-tight,  *  exists a  r >  X  then, f o r any  ,  space,  e > 0  , there  k  w -Radon measure  n  on  X  with, supp  n e E  ,  such t h a t x e X  , J | f ( x ) | d n ( f ) <_ 1 =>  where Proof.  y„ '({f e F* : | f ( x ) | F X. x i s the space spanned by x  F^  We  assume t h a t  p  u , '  is a  p  is  w -Radon l i m i t measure on  there e x i s t s  1 - g(V°)  I  x e•X = {f e X  n is a  p  f o r any  in  1.4, X  for  s.t.  1.4,  .  : |f(x)j  W  > 1}  £  . e R  x e X  , then,  +  "  W  k  w -Radon measure on  ({f e F*  F  < e/2  - y" , (A A V°) k  P  for  , let  r) : A C x"  and  X  ,  of P r o o f s  p",(X* ~ V°) any  V £ nbnd 0  < E/2  t h e r e f o r e , by L.3  If  I.4.3,  E - t i g h t , by L.2.1. of P r o o f s  e > 0  For  by Th.  W "  Since  and  <  i s normalized.  With the n o t a t i o n of I.4,  any  >_ 1})  r  w i t h supp  J ] f (x) { dn  s.t.  : |f(x)|  X r  > 1}) =  <_  1  n e E  ,  V**(I > X  x  = V -  2  (  N  I  ( I  x ^ ° -  }  +  )  +  2  V 2 $  (  I  1  < f /|f(x)| d (f) +f r  n  The  x I  ~ dr  '  +  e / 2  i f + f = e  •  m  above p r o p o s i t i o n suggests the f o l l o w i n g d e f i n i t i o n s .  2-4  Notation. y  over  For any v e c t o r space X  X , x e X  and c y l i n d e r measure  , F^ = space spanned by D  ='  x  x,.  {f e F* : If (x) I > 1} x  1  —  1  and \  2.5  Definitions. subsets  (1)  =  U  F  Let  X  X  with  of  For any f i n i t e y is U £ U  y  be a v e c t o r space, and  U  be a f a m i l y o f  0 e U  c y l i n d e r measure  y over  U - c o n t i n u o u s i f f f o r any  X  e > 0  , there  exists  such t h a t  x £ U = > y ( D ) < £ x x (2)  F o r any c y l i n d e r measure y is  y  over  U - c o n t i n u o u s i f f every  X  ,  finite  s e c t i o n of  U  is  U-continuous. (3)  F o r any t o p o l o g i c a l space T T(0)  is  U-continuous  there e x i s t s T[U] c  (4)  When  X  y over  V  Y  and  T : X -> Y  i f f f o r every  U e U  neighbourhoods  X  of  . c y l i n d e r measure  , iff  y  is  f-continuous  V of neighbourhoods o f the o r i g i n i n  The d i s c u s s i o n o f l i m i t s only the above concepts. notions  V  such t h a t  i s a t o p o l o g i c a l space, f o r any f i n i t e  y i s continuous  standard  ,  However,  of c o n t i n u i t y we  f o r some f a m i l y  X  i n the r e s t o f the c h a p t e r to e x p l a i n t h e i r introduce  requires  r e l a t i o n to  the f o l l o w i n g d e f i n i t i o n s .  60.  Let  X  be a v e c t o r  For any n e w F •  x  Cf  and  and  space.  x = -(x^,. . . ,x _^) - e X  = l i n e a r span of  {x„,...,x U n-1  ,  n  ,  J f e F* -y ( f ( x ) , . . . , f ( x _ ) ) e <E  ,  n  x  Q  f o r any  finite  n  1  c y l i n d e r measure  y  over  X  ,  X new,  For any M(|] )  i s the f a m i l y of f i n i t e  n  Radon measures on  ^  n  endowed w i t h the vague t o p o l o g y ; i.e.  f o r any net  Tij for  (n.) . ,  + n i n M(fp )  in  note t h a t  y  n  n E M(f] )  and  ,  n  i f f J fdn.. -»- / fdn  n  every bounded continuous We  M((j; )  e M(£ ) n  f : <£ -> n  f o r a l l n e OJ  and  x e  X  n  X We Vilenkin  now  [11] P.  have the f o l l o w i n g well-known p r o p o s i t i o n 310,  naturally associate measure.  Fernique  (Gelfand,  [9] p. 37, which shows t h a t one  c e r t a i n continuous maps w i t h a c o n t i n u o u s  (see a l s o Appendix  2.6  Proposition  Let  (1)  y i s continuous  X  can cylinder  1.7).  be a t o p o l o g i c a l v e c t o r  space.  iff (2)  y : x e X -> y^ e M($)  i s continuous a t  0  iff (3)  f o r each y  11  : x E X  n e w , y  n  topology on  X  e M((£ ) n  i s continuous with respect  to the p r o d u c t  61.  Remark  From the p r o o f of the above p r o p o s i t i o n one r e a d i l y  checks  t h a t the f o l l o w i n g a s s e r t i o n a l s o h o l d s . If with  U  i s a f a m i l y of b a l a n c e d ,  tU e (J ' f o r every U e U and u is  absorbent  subsets o f  X  ,  t > 0 , then,  (J-continuous i f f  u : x e X -> u e M(C) x  is  H  (J-continuous.  Proof of P r o p o s i t i o n 2.6.  We  show t h a t (3) =>  (3) =>  (2)  take  (2) =>  (1)  let %  (2) =>  n = 1  e > 0 x  e  .  V =>  Since  y^  mass,  then  be bounded and i f  jil  0  i f  |*| <  V e nbnd 0 - / 5Cdu~| < e  |  in  X  s.t.  .  i s c o n c e n t r a t e d a t the o r i g i n  J  continuous,  > 1  = 1  Choose 1/ %d~^  (3)  .  : £ -> C  %(z)  Let  (1) =>  x £ V  (1) =>  (3)  v  v  A.  A.  For any  F  e  F  I(u) = {(w,f) Let We  n e OJ  , X • ^  n  E  ->• £  s h a l l show t h a t f o r any  s.t.  finite  ,  y (D ) = y ( { z e £' : |z| L 1 » 1 / * d y A.  , and has  = o • .  Hence, f o r any  A.  t  in  , and  u > 0  ,  < e • let  F x F* : |f (w) | >_ u} be hounded and c o n t i n u o u s , and £ > 0  there e x i s t s  x e X  V e nbnd 0  in  n  X  62,  y e x + V (Note:  V  n  =>  n  e nbnd 0  We assume t h a t For  u  z E £ .  any  |/ % d y in  - / %dy^| '< e  x  x" ).  i s normalized. , l e t [z| = sup {|z  n  .  | : k = 0,...,n - 1}  Let (i)  M = sup ( | % ( z ) |  (ii)  W  be an open  : z E £ }  ,  N  nbnd  of the o r i g i n i n  X  s.t.  w e W => u (D ) < e/16nM • , w w (iii)  t > 0  (iv)  and l e t z  e £  j  => Since  be s . t .  6 > 0  N  < e/4  i s an open  V e nbnd 0  (v)  -| V  (vi)  x + V ctW  n - 1  1  . '  nbnd  in  |z° - z ] < 6  of  X  s.t.  F e F  with  0  in  X  N  ,  ,  U  .  n  y e x + V y  k = 0  , and  j  1  e W  f o r each  be s . t .  \%(z°) - X i z ) | £ -W  any  e tW  | z | <_ t , j = 0,1  U  there e x i s t s  For  x  'and  n  >y  F  A (5) = { f  E  x  y  1  •  F* :  \v  y  y  x  F  F^ u F^ c  F  , let  x  ( f ) - V ( f ) I > 5} x 1  and k=n-l B (t) = U I (t) v v * k=0 "k . Then, y (A  (6)) = y ( U k=0 k  F  1  F  J  I  k=n-l.  "  k=o  < e/16M  y  (6)) k  k  k=n-l  F V6)(x -y ) (  k  ( 1 ) )  k  by ( i i ) and ( v ) .  -  k  ^  ^i/^ -y ) li/6Xx -y )> ( 1  0  k  k  k  k  ,  From ( v i ) , we have t h a t y e tW Hence,  — v, e W t k  by r e a s o n i n g y  b  f o r each  k e n  , and  therefore,  as above,  (B ( t ) ) < e/I6M y  .  In p a r t i c u l a r , y (B (t)) F  < e/16M  x  .  Consequently, i f  B = B ( t ) U B ( t ) V A (5) x  ,  y  y (B) < 3 e/16M  then, and,  y  < e/4M  ,  by ( i v ) ,  f e F* ~ B =>  \X(V ( f ) ) y  ( f ) ) | < e/4 x  1  Hence, 1/  Xdy  y  -  /  < /| X , V ~~ B  X d y J  -  %  0  y  =  1/  ¥  %  dy  X  F  < 2M.e/4M + -f . U(F" — 4 F  0  y y  F  + / F*~B  -  / %  |X o ¥  ^d,^}  0  -  X  ~ B) < e  X  .  i.e. y e  x  + V  n  =>  |J  Xdy^ -  /  % d y j  ¥ |d,i  0  Y  < e  .  F  3.  L i m i t s o f Continuous C y l i n d e r Measures  Let finite  X  be a H a u s d o r f f , l o c a l l y  c y l i n d e r measure .over  X  seen t h a t c o n d i t i o n s  under which  d e t e r m i n i n g when  has a  y  .  I n the p r e v i o u s  y  c -  convex space, and  is  or an  in  X  , which i s d e f i n e d  X  some k i n d  of converse to t h a t  the concept o f a weighted  notation.  Notation  For any v e c t o r V  c X  , and  F  E  space  F  X  , absorbent a b s o l u t e l y  F  v  = F cv k e r V  f o r every  C  For  any  E  t > 0  F" : f ( x ) = 0  So  f o r any I  I  f  •v  for a l l  fora l l x  c 7^}  ,  I ( t ) = {(x,f) I = K D  t > 0}  E  ,  ,  (V c\ F)° = {f e F' : |f (x) | <_ 1 = {f  convex  ,  ker V = { x e X : x e t V  F x F * : | f ( x ) | i.t} ,  • x e F , f e F  ,  = {x e F : |f (x) | >_ 1}  = {f E F* : I f ( x ) | > 1} 1  —  ,  . •  X  P r o p o s i t i o n 2.4 g i v e s  below.  We s h a l l use the f o l l o w i n g  3.1  s e c t i o n we have  s -Radon l i m i t measure" on  In s e e k i n g  p r o p o s i t i o n , we a r e l e d to i n t r o d u c e  be a  E - t i g h t a r e important f o r  In terms o f the o n e - d i m e n s i o n a l subspaces of a n e c e s s a r y such c o n d i t i o n .  y  x e V  f\ F}  system  Remarks  We note t h a t (1)  F  and s i n c e  (x,f)  F ;  e  v  F x F  e  the p r o d u c t t o p o l o g y on (2)  f o r any  t > 0  I(t)  3.2  f(x) e C F x F  i s continuous with respect  ,  ,  i s closed.  Definitions  Let (1)  c  X  be a l o c a l l y convex space.  (v,F»f)  i s a system of  6 > 0 [/  to  S-weights i n  X  if  ;  i s a f a m i l y of a b s o l u t e l y convex neighbourhoods of the o r i g i n i n F c  F  X  ,  i s d i r e c t e d , by  C  Li F  and  i s dense i n  X  and v : V e l / , F e F - > V y p measure on f When  of  e  f  F  f o r which  F* ~ (V n F)° => i s a singleton  (v,F,l/)  , a p r o b a b i l i t y Radon measure  v  (I ) f  V j F  {V}  >• 6  , we  s h a l l write  f o r each  (v,F,V)  instead  •  W i s weighted by such a system  (2)  .  (v,F,f) i f f  W i s a f a m i l y o f neighbourhoods of the o r i g i n i n  X  ker V c W  ,  W e W  there e x i s t s  V c V  such t h a t  t -> »  , uniformly  and v„ „(F ~ tW C\ F) -* 0  as  for  F e F,  ,  66.  (3)  (!) i s a weighted system i n system of  6-weights  in  X  i f fW  i s weighted by some  X  We s h a l l now s t a t e and prove the fundamental r e s u l t s of t h i s section.  3.3  Theorem  Let  X  be a H a u s d o r f f , l o c a l l y convex space, and  f i n i t e c y l i n d e r measure over If then  y  is  y  is  fx  be a  X  W-continuous  f o r some weighted system i n  X  ,  E-tight.  Corollary  Let  X  be a H a u s d o r f f , l o c a l l y  a r b i t r a r y c y l i n d e r measure over If  y  is y is  and  W-continuous  convex space, and  y  be an  X f o r some weighted system i n  X  , then  E-tight,  therefore,  y has a  c -Radon l i m i t measure on  X'  P r o o f 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 f o l l o w i n g  lemmas i n the p r o o f of Theorem 3.3.  They a r e proved a t the end of the s e c t i o n .  Lemmas  (1)  Let If  F  be a f i n i t e  £  d i m e n s i o n a l space, and  i s a f i n i t e Radon measure x e F =>  on  F  z• > 0  such t h a t  ^(I ) < e x  then  .  £(F* ~' {0}) ±e (2)  Let finite  X  be a l o c a l l y  c y l i n d e r measure  For  convex space and  over  p  a continuous  X  any dense subspace  Y  of  X  , and  e > 0  an a b s o l u t e l y convex neighbourhood of. the o r i g i n y (p"  ~ (V A F)°) <_ e  p  f o r every  F e F  , if  in  X  V  is  such t h a t  ,  r  then p„(F~ ~ ( V f \ F)°) < r  for a l l  e  F e  F  —  Proof of Theorem  v  A  3.3  By the Hahn-Banach e x t e n s i o n theorem, we F e F . , and a b s o l u t e l y convex (1)  r [ V ] = (V A F)°  V e nbnd 0  see t h a t f o r any  in  X  ,  .  0  p  Hence, by Lemma 1.5.1.2, we need o n l y prove t h a t , f o r any e > 0 (2)  , there e x i s t s  V e nbnd 0  y _ ( F * ~ ( ¥ A F)°) < e r  We  in  for a l l  X  F e  s.t.  F .  —  assume t h a t  Let to which system of  p  i s normalized.  W  be a weighted system i n  p  is  X  W-continuous, and l e t  6-weights i n  X  by which  W  with  respect  (v,F,l/)  be a  i s weighted.  For any  e > 0  , let  0 < e' < min(<5e,e)  v  v  F  W e W  s.t.  V e V  and  ,  x e w =>  t > 0  ( F ~ tW r\ F) < e'/4  y (D ) < e'/4 x x —  s.t.  ker V C  W  ,  and  F e F  f o r every  Let U = V/t  .  Suppose t h a t * 0 y (F ~ (U c\ F) ) <_ E Since  F  of. X  , and  UF  i s d i r e c t e d by f o r any  L.2.3  y (E* Since  W  of P r o o f s  - (U r% E)°)  E  , then  F  with  U  E c F  i s a subspace  subspace .  X  remarks we  .  <£  Hence, by  (1) ,  .  conclude  F e  F  and by h y p o t h e s i s ,  Hence, from Lemma (2) and  to e s t a b l i s h  (3).  ,  y ( F * ~ ( U A F ) ° ) = y ( F * . ~ t ( V n F)°) F  p  • = y (F  a  ~ t(V n F)°) + y ( F * ~ p  since F) ° c  F?  T  .  F^)  U  F  the f o r e g o i n g  t h a t (2) h o l d s .  I t remains f o r us  (V A  of  i s a f a m i l y of neighbourhoods of the o r i g i n i n  dense i n  p  E  1.4,  y i s n e c e s s a r i l y continuous,  For any  F e F  f i n i t e dimensional  F e F  there e x i s t s  ( 3 ) , and  c  f o r every  is  We  show t h a t each of the l a s t  (4) i s l e s s  than  e/2  We e s t i m a t e the f i r s t Since  (v,F,l/) f £  => v  term.  i s a system of  6-weights i n  ~ t . (Vft.F)° =>  (I (t)) = v f  V j F  two terms g i v e n i n  y j F  (I  f / t  )  f  > 6  X  , the  F* ~ ( V f t F)°  e  .  Consequently, 6 . u ( F ^ ~ t . ( V f\ F)°) F  f{v  (I (t)):  f  f  v > F  <. Vy  * v (Kt))  = J / F  1  F ; V  ^dUpdVy  I  = /p (I (t))dv F  =  X  ^F  tW  x  (  x) \,F t  I  d  F- _x x t t y  ( D  ) d v  - f ~  " V,F  < f  . l  v  +  (  V,F  ( t W A  =  )  ( x )  F )  +  [44] Ch. I l l , Thm.  by F u b i n i ' s  F  (x)  V j F  x  0  F  by S i o n  F  F  t.(V A F)°}.p (F^ ~ t . ( V f t F ) )  F J ~  £  =  /S (I (t))dv F  /y°x t t +  1 , V  l . f < 6.f  theorem,  x  ) d V  V,F  f VtW  V  F  ( x )  F' x t  ( D  t  F )  y  ~  ( F  W  V j F  A  x t  ) d V  V,F  ( x )  .  Hence, y (F* p  ~ (Uft.F)°) < e/2  We now We have x e F  V T  estimate  y„(F F  . " F ) V  that => x E k e r V => x E W =>  => y_ (I ) < e'/4 Fy x  since  F  x  Hence, by Lemma ( 1 ) , y  (F* ~ {0}) < e'/4 V  .  c  y (D ) < e'/4 x x F e F V 1T  .  1.2.6,  70.  Since r  p  F ,F V  it  [F* ~ F J ] C F* ~ { 0 } ' , V  therefore  U (F"' f  = U  v  ?  follows  - ?p  <  that  u^r'^fF*  ( F " ~ {0}) < '/4 < e/4 e  ~'{0}]j  .  Then, c e r t a i n l y , (6)  u ( F * ~ F*) < /2 F  From  Remark.  .  £  ( 4 ) , ( 5 ) , and (6) we see t h a t  (3) h o l d s .  We p o i n t out t h a t the theorem s t i l l  somewhat weaker n o t i o n  o f system o f  F : V e l / - > F c F  6-weights, i n which C  d i r e c t e d by  dense i n X  The o t h e r  Proofs  d e f i n i t i o n s remain  unchanged.  3.  P r o o f o f Lemma ( 1 ) .  For  any  x e F  and  n e w ,  and  Consequently, f o r any 5({f  x e F , . x =)= 0  e F* : f (x) f 0}) =  = l i m 5 ( 1 (-)) < e x n — new  h o l d s when we use a  U I .(-)) new  ,  and  [J F  v  is  71.  Hence i f (1)  there e x i s t s •£({f  e  y e F  F*  s.t.  0} ~ {0})- = 0 ,  : f (y) =  then  5(F~ ~'  {0}) =  We For  C ( { f e F*  shall establish  any subspace E  Let  E  = { f e F"  a  dim F = n  of  If  assume that  n >^ 2  of  2 <_ k <_ n  (2)  F  with {H ^G 3  A  : H  (1) by F  .  t h e r e e x i s t s an 5(G For  A  -'  any  .  n = 1  then (1) h o l d s .  For a n y  G^.  5(G  -  A  Then, by  of  (k - 1 ) - d i m e n s i o n a l subspace s u b f a m i l y of  Then, by  5(G  F  (2) and  ~ G )  A  A  =  G  +  i  of  G}  M  the f i n i t e n e s s of G^  of  F, , F  s.t.  , i f t h e r e e x i s t s an  (n - k ) - d i m e n s i o n a l  s.t.  {0}) - 0 ,  (2) and  k  G  0 .  the f i n i t e n e s s of  (n - k - 1 ) - d i m e n s i o n a l subspace ^  therefore  k - d i m e n s i o n a l subspace  (n - 1 ) - d i m e n s i o n a l subspace  {0}') =  We  ,  0 <_ k <_ n - 2  subspace  induction.  f o r a l l x e E}  is a  GQ = F  0} ~ {0})  : f (y) =  , let  i s an u n c o u n t a b l e , d i s j o i n t Let  + ?({-f E F*  0  : f(x) =  .  0})  : f (y) ={=  * V G  =  E,  , there e x i s t s  G  of  G  1  an  s.t.  •  0  Consequently,  aG  a + 1  - {0}) =  Hence, t h e r e e x i s t s 5(G  A  . ~  n-1  i.e.  5  (G  A  +  C  (G  A  ~  {0}) = 0- .  a o n e - d i m e n s i o n a l subspace  {0}) = 0 .  (1) h o l d s .  - G ) A  + 1  G  .,  n-1  of  F  s.t.  72.  Proof of Lemma (2)  For any  F  F e  , n e w, X e F  v  A  , and  n  t > 0  , let  k=n-l  A  A* (t) = •  { f e F" : | f (x ) | < t}  k=0  X  We s h a l l assume t h a t  u  (V A F ) ^ = (VQ r\ F ) ^  , where  i s normalized.  s h a l l f u r t h e r assume t h a t Let  E  F  e  Since subset (V  .  k  V  Since  i s the i n t e r i o r  of  V  , we  i s open.  .  v  A  E  i s separable  {x } n new  of  r\ E)° = A  V A  there e x i s t s a countable,  E  .  dense  Then,  {£ e E* : | f (x^) | £ 1}  .  kew Now,  f o r any  new  tv ^ * 1  1  :  |£  ,  °vi  ^  k=0  A  cx .....x  mew  Consequently, f o r any  6 > 0  •  n  0  n-1  , there e x i s t s  new  and  m e w  s.t.  (1+1))  P ((V^E)V A* E  V  0 n-1 Since y i s continuous, there e x i s t s (1) u e U => y ( D ) < 5/2n , u  and { y  since  Y  0'"'"' n-l y  i  n  V  E  ((VAE)°)  U e nbnd 0  +  in  {  .  X  s.t.  u  i s dense i n }  <  V  (2)  x, - y, e - U k k m  Let  F e F,, X  S - t  X  and  V  i s open t h e r e  exists  "  f o r a l l k = 0, . . . ,n - 1  be such t h a t  .  E U {y„,...,y ,} C F 0 n-1  let x = (XQ,...^^) , y =  (yQ.-.-.y^)  and  Then,  V * " x-y ™ = F  -  A  k=n-l E  )}  (D  6F  k=0  }< 6 / 2  y  *  U  x,-y m(x -y )V *k k k  y  k=0  (  : |f(X  b y (1) and (2)  n H x  k"V  1  *  Further, •  A  Y  (  1  )  C  - X A  (  1  +  7>  •  Hence,  W  J,,,,  .  ,1s . , F „ , , ,  = W y-y t t  1 } )  < u ((V  ,,F  r\ E)°) + 6  E  *  A  Y  ( 1 ) )  +  ,.F,„  W '  N  1  „ ,F  ,1  ~Vy?5  .  i.e. (3)  y ( A | ( l ) ) < y ( ( V r N E)°) + <5  .  E  F  However, i f  F  denotes the l i n e a r soan o f  y  {y„,...,y J J  0  n-1  , }  we observe t h a t A (1) 3 r " [ ( V A F )°] y „> y F  1  .  y  Since  F  e F y (V (\ F ) £ Kp  , and  Y  and  (V r\ F ) y  i s c l o s e d , we have  that  y  V y A  > y (r  ( 1 ) )  F  X F  F  [ ( V A F )°]) = y  ((V A F )°) > 1 - e  p  y'  y  Hence, by ( 3 ) , u ( ( V A E)°) > 1 - £ - 6 E  Since  6  was a r b i t r a r y , i t f o l l o w s  u ( ( V A E)°) E  >_  Consequently, s i n c e u (E' E  .  f  ~ (VA  1 - £ (V A  that  . E ) ^ £ M^,  E)°) < e  .  ,  74.  4.  Induced C y l i n d e r Measures.  I t can happen that a f i n i t e locally  convex space  been induced and we  by  will  i s given i n d i r e c t l y .  a finite  a l i n e a r map s h a l l be  X  T  on  X  to  [H]  Ch.  Y  ([11] p.  311).  X  1  , Radon w i t h  respect  space  Y  In such a s i t u a t i o n ]-~ and  c y l i n d e r measure over  IV.  I t has  X  T  which  w i l l have a  to some g i v e n  been s t u d i e d e x t e n s i v e l y , by L.  others,  i n a s e r i e s of papers  In view of the p r e v i o u s determining  we  over a v e c t o r  have  topology  on  T h i s k i n d of problem seems to have been f i r s t mentioned i n  5. Kwapien, and  be  For example, i t may  i n t e r e s t e d i n o b t a i n i n g c o n d i t i o n s on  l i m i t measure on .  u  c y l i n d e r measure  ensure t h a t the induced  X*  c y l i n d e r measure over a H a u s d o r f f ,  theory,  our  c o n d i t i o n s under which the induced  E-tight,  Using  the n o t i o n s  Schwartz,  ( [ 1 9 ] , [20],  [39]  -  emphasis w i l l  be  on  [42]).  c y l i n d e r measure w i l l  of c o n t i n u i t y and weighted system  r e a d i l y o b t a i n such c o n d i t i o n s .  4.1  Definition  For any v e c t o r and  finite  c y l i n d e r measure  the c y l i n d e r measure defined  spaces y £  as f o l l o w s : f o r each  X  and  Y  Y  ,  over over F e F  X  , l i n e a r map  induced  by  y  >  v  A  where T  p  i s the a d j o i n t of  T[F  ,  i.e. T*  : f e CT[FJ)* -> f  0  (T | F)  e  F*  .  and  T  : X -> Y  T  is  ,  We s h a l l denote t h i s induced c y l i n d e r measure  uD  T  by  . 5  We prove below that  Proof  K  i s indeed a c y l i n d e r measure over  For each  E e F„ , T X E  X  i s continuous.  Hence, by §0.4, £  is a finite  E  Radon measure on  E  S i n c e a l l . the maps c o n s i d e r e d a r e c o n t i n u o u s , then, by §0.4 and R.emark 1.2.1, f o r any  E  and  F  F  in  "  r  E,F 'F  T  E  C r  0  r  ]  r  E,F  t T  T[E],T[F]  V T[E] y  and  =  ]  =  ?  E  F  [ y  [ u  T[F]  T[F]  ] ]  =  r  E,F  V T  ]=  r  E c F  ,  V T[F]  0  y  ]  [E] ,T[F] T [ F ] [ y  ] ]  '  t h e r e f o r e , a g a i n by Remark 1.2.1,  5 i s a c y l i n d e r measure over  We now prove  4.2  with X  X  .  B  the f o l l o w i n g important  lemma.  Lemma For any v e c t o r space with  0 e U if  T  X  , family  , t o p o l o g i c a l v e c t o r space is  every continuous  U-continuous, finite  then  U Y  u a T  of subsets  U  , and l i n e a r is  c y l i n d e r measure over  of  X  T : X  U-continuous f o r Y  76.  Proof.  Let  y  be a continuous f i n i t e  For  any  x e X  <U a T ) ( D ) X  = T  X  ](D ) = . X  x For  any  and  there e x i s t s  T[U]  C  x It  follows  U =>  e  that  Our key  r,  y (D ) < e y y U e U s.t.  = y:  x  (D  )  .  in  Y  s.t.  K  V e nbnd 0  ,  . assertion, TJC e V =>  y -,3 T  (y Q T ) ( D ) X  is  X  = P  T X  (D  T X  (i-continuous.  )  < e  •  ©•  theorem on induced c y l i n d e r measures i s now  consequence of Theorems 3.3,  4.3  x  , there e x i s t s  V  Then, by the f i r s t  (T;- [D ]) 1  T x  :-.  e > 0  y e V =>  Y  , by Lemma 0.4.2,  [y  F  c y l i n d e r measure over  2.1,  and  an  immediate  the above lemma.  Theorem.  Let  X  be a H a u s d o r f f , l o c a l l y  t o p o l o g i c a l v e c t o r space, and If  T  is  (^-continuous  T  are  is  be a l i n e a r map  f o r some weighted  then f o r every c o n t i n u o u s f i n i t e y cs T  convex space,  Y on  system  be a X  to W  c y l i n d e r measure over  Y  in Y  X  ,  E-tight  therefore •k  y C\ T  has  a, c -Radon l i m i t  measure over  X  Remark.  It Corollary  i s clear  3.3 when  that  X = Y  t h i s theorem reduces and  T  to the f i n i t e  i s the i d e n t i t y  map.  case of  ,  77.  CHAPTER I I I  APPLICATIONS  We s h a l l a p p l y the theory of the p r e v i o u s c h a p t e r t o 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 H a u s d o r f f , H i l b e r t i a n g e n e r a l i z e and c l a r i f y many known theorems ( M i n l o s Badrikian  [ 1 ] , Fernique  significant  [9]).  I n the case of  spaces  [ 2 5 ] , Sazonov [ 3 5 ] ,  £^-spaces we o b t a i n  e x t e n s i o n s o f f o r m e r l y known r e s u l t s  (L. Schwartz [ 3 9 ] ,  Kwapien [ 1 9 ] ) . Our main t o o l i s C o r o l l a r y I I . 3 . 3 , which r e q u i r e s us to c o n s t r u c t weighted  systems i n the above spaces.  I n view of P r o p o s i t i o n  I I . 2 . 4 , i t i s the s e a r c h f o r such systems which l e a d s us to c o n s i d e r the f a m i l i e s  1.  S  , for r > 0  , d e f i n e d below.  Preliminaries  For any v e c t o r spaces L[X,Y]  X  and  Y  i s the s e t o f l i n e a r maps on  For any t o p o l o g i c a l v e c t o r space CM(X)  ,  X  X  to  Y  ,  i s . t h e f a m i l y o f continuous  measures over  X  finite  cylinder  78.  Remarks.  From Appendix C of f i n i t e  T h i s topology i s c a l l e d For any the  3.2.1,  we have t h a t f o r any  X  under which  u  i s continuous.  X  family  , there e x i s t s  i t i s a t o p o l o g i c a l v e c t o r space,  that p e C =>  is  and  c y l i n d e r measures over a v e c t o r space  a c o a r s e s t t o p o l o g y on and such  3.1.1  C~topology.  the  t o p o l o g i c a l v e c t o r space  CM(X)-topology, then we  call  X  X  CM-space (Appendix  a  X , 0 < r <  For any t o p o l o g i c a l v e c t o r space n  w -Radon measure S  X  with  supp .n e E  = {x e X : / | f ( x ) | d ( f ) n  r > 0  3.1.2).  , and  ra  .  ,  i s the f a m i l y of a l l s e t s  r  X  , <1}  r  For each S  on  , i f the t o p o l o g y of  S  c X  .  r, n  1.1  Remarks  Let (1)  X  be a t o p o l o g i c a l v e c t o r space.  For each which  X  r > 0  , t h e r e i s a unique t o p o l o g y on  i s a t o p o l o g i c a l v e c t o r space h a v i n g  f o r i t s neighbourhoods topology i s l o c a l l y We  shall call  of the o r i g i n .  When  If  0 < r  a e S" 3  (3)  If  < t  t h i s t o p o l o g y the  , then  there e x i s t s X  S  as a base , this  S -topology. r  r i s finer  3 e S*"  i s l o c a l l y convex, S*"  r >_1  under  convex.  t (2)  S  X  with  S  than 3 c  , i . e . f o r every  a  then, f o r each  i s a f a m i l y of neighbourhoods  r > 0  of the o r i g i n i n  X  79.  We prove only  1.1.2.  '  P r o o f of 1.1.2.  For -any f i n i t e measure space f  : 0 -> £  ,  (£2,n)  and i n t e g r a b l e  i f  1 1  • 1- — = 1 ,' P q then, by >.L;>;H«AJ&#'s i n e q u a l i t y , p = t / r and  /|f| dn 1 (/;|f| dn) r  r)?  1/p  nCn> 1/q  .  Hence,  (J|f| dn)  (l) For  r  any  S  n(x ) x  since  r  ,n  K/lfl'dT,) ' n ( a ) 1 7  1 / r  e S  r  ( t  "  r ) / r t  •  ,  < <» ,  supp n e E C K(w ')  Consequently, by ( 1 ) ,  K - n(x ) A  ( t  -  r ) / r t  and  n  i s w -Radon.  if  .  n  then S The  ,- C s t,5 r,n  assertion  To p o i n t Proposition  follows.  the s i g n i f i c a n c e of the f a m i l i e s  II.2.4 can be r e s t a t e d  as f o l l o w s .  5  we note  that  80.  1.2  Proposition  Let measure  u  X  be a t o p o l o g i c a l v e c t o r space.  over  X  E - t i g h t => u  Mien  X  i s S -continuous  t i o n and Theorem II.3.3 y i e l d is  f o r every  1  i s H a u s d o r f f and l o c a l l y  u  cylinder  ,  u is  if  F o r any f i n i t e  the f o l l o w i n g  r > 0  .  @  convex, the above p r o p o s i assertion:  W-continuous f o r some weighted  system  W  in  X  ,  r then  u  i s S - c o n t i n u o u s f o r each  r > 0  In view of t h i s , when s e a r c h i n g f o r weighted we s h a l l l o o k f o r s u i t a b l e s u b f a m i l i e s of In  general,  E-tightness.  X  S  S - c o n t i n u i t y f o r some r  systems i n  r > 0  does n o t imply  (Example 1, Appendix 4 ) .  We s h a l l need the f o l l o w i n g r e s u l t on induced c y l i n d e r measures.  1.3.  Proposition.  Let and over  X  T e L[X,YJ Y  be a t o p o l o g i c a l v e c t o r space, .  F o r any f a m i l y  C  of f i n i t e  Y  be a v e c t o r space,  c y l i n d e r measures  , i f u a. T  then, f o r each T  r > 0  is  E - t i g h t f o r every  u e C  ,  ,  i s S - c o n t i n u o u s w i t h r e s p e c t t o 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 X  S -topology i s f i n e r than the  (C o T)-topology on  .  By Appendix 3.2.2, t h i s says exactly that T  i s S -continuous with respect to the C-topology on r  Y  . Q  2.  H i l b e r t i a n Spaces.  Throughout t h i s X i.e. X  section,  i s a H a u s d o r f f , H i l b e r t i a n space ( [ 1 ] ) . i s a Hausdorff, l o c a l l y  exists a family such t h a t sets  F  nbnd 0  convex space, f o r which  of p s e u d o - i n n e r p r o d u c t s on in  X  {x e X : [x,x] <_ 1}  e r  of a l l  following.  Theorem  For  each  0 < r < S  00  ,  i s a weighted system i n  X  The p r o o f s of t h i s and o t h e r a s s e r t i o n s of  ,  .  The fundamental theorem of t h i s s e c t i o n i s the  2.1  X  has as a base the f a m i l y , [.,.]  there  the s e c t i o n .  Now,  we  w i l l be g i v e n a t the end  c o n c e n t r a t e on the consequences of the above  theorem.  2.2  Theorems.  Let  y  be a c y l i n d e r measure over  X  and  .0 < r <  00  Then, (1)  y i s E-tight  (2)  y is on  XV ' '  <=> y  S - c o n t i n u o u s =>  is y  S -continuous. lias a  c -Radon l i m i t  measure  (3)  If  K(c  ) = E  , i n particular, i f X  u  is  5 -continuous  u  is  E - t i g h t <=>  \x has a  i s barrelled,  <=>  c -Radon l i m i t measure on  X'  Using Theorems 2.2, we can now c h a r a c t e r i z e c e r t a i n d e f i n i t e f u n c t i o n s on  2.3  X  then  positive-  (Appendix 2 ) .  Theorem.  Let 0  i>  be a p o s i t i v e - d e f i n i t e f u n c t i o n on  X  and  < r < «>  Then, i> i s measure  ^ - c o n t i n u o u s =>  E, on  X'  such  there e x i s t s  K(c ^  ) = E is  f o r every  , i n particular, i f X  S -continuous  c -Radon measure  E, on  c -Radon  that  ij;(x) = J exp i Re f ( x ) d g ( f ) If  some f i n i t e  i s barrelled,  <=> t h e r e e x i s t s X'  such  i> (x) = /exp i Re f(x)d£(f)  x e X then  some f i n i t e  that f o r every  x e X  Remarks  We note t h a t Theorem 2.2.2 g e n e r a l i z e s a r e s u l t of M i n l o s ([25] p. 303 Thm. 1 ) .  Theorem 2.3 g e n e r a l i z e s r e s u l t s due to M i n l o s  ([25] P. 310), and Badrilcian when  X  i s a Hilbert  ([1] p. 16 Cor. 1 ) .  The s p e c i a l  space w i l l be d i s c u s s e d below (§2.7).  case  84.  We  p o i n t out t h a t , w i t h the v i e w p o i n t of §1.4,- the  of Theorems 2.2.2 using  and  2.3  f o r the case  r = 2  assertions  can be e s t a b l i s h e d  the technique of c h a r a c t e r i s t i c f u n c t i o n a l s  ([1], p.  by  9,  2 Lemma 1, Prohorov  [33]).  i s n o t h i n g e l s e but  Also,  i t can be shown t h a t the  the Gross-Sazonov  t o p o l o g y on  X  S -topology  ([35],  [1],  [13]  p. 65). By means of P r o p o s i t i o n 1.2 the a s s e r t i o n s above f o r  0 < r < 2  and Remark 1.1.2 from  the case  we  r = 2  been unable to g i v e a s i m i l a r d e d u c t i o n f o r the case in this  c o n t e x t , we  draw a t t e n t i o n to  As consequences  §2.6  of Theorems 2.1,  can deduce .  r > 2  We .  have However,  below. I I 4.3,  and P r o p o s i t i o n  1.3,  we have the f o l l o w i n g a s s e r t i o n c o n c e r n i n g induced c y l i n d e r measures over  2.4  X  Theorem  Let  Y  For any f a m i l y  be a v e c t o r C  of f i n i t e  space,  T e L[X Y]  c y l i n d e r measures over  y n T  is  T  S -continuous with respect  is  E - t i g h t f o r every  u e C  (p. 349)..  (2) s i g n i f i c a n t l y  generalizes  Y  0 < r <  00  ,  <=>  to the  The above theorem y i e l d s immediately below,. C o r o l l a r y  , and  3  C-topology  the c o r o l l a r i e s a result in  [11]  on  Y  given  85.  Corollaries  Let (1)  If  Y Y  be a t o p o l o g i c a l v e c t o r space, is a  y D T T (2)  If  T  is  2.5  , and  r > 0  then  E - t i g h t f o r every  u e CM(Y)  <=>  i s S -continuous. i s S"-continuous,  y a T and  CM-space,  T e L[X,Y]  is  then, f o r every  y e CM(Y)  ,  E-tight,  t h e r e f o r e has a  c -Radon l i m i t measure on . X  Remarks  Under c e r t a i n c i r c u m s t a n c e s one can r e a d i l y s t r e n g t h e n the a s s e r t i o n s of Theorems 2.2 - 2.4. Let (1)  y  be a c y l i n d e r measure over  (Theorem I I . 2.3)  X  I f t h e r e e x i s t s a base  such t h a t f o r each  U e U  the Banach space  X^  (J  for  nbnd 0  in  X  , i s separable,  then, y is  E - t i g h t => p.-has aiv ' 5 -Radon l i m i t measure on  Hence, i n those theorems i n v o l v i n g  the e x i s t e n c e of a /*c  l i m i t measure on (2)  Let  G  X'  , we may r e p l a c e _c  be a r e g u l a r t o p o l o g y on  X'  X  *  y  c -Radon  k  by with  S w |x'c  G  If  E c K(G)  , or  E = K(G) ,  then the f o r e g o i n g theorems may be m o d i f i e d as i n d i c a t e d by Theorem 1.4.3.  86.  In p a r i t u c l a r , we note that when E = K(s*)  The assertions  2.6  theorems  about the  .  ( c f . Thm.  X  i s a Montel space,  II.2.2.1)  above a l l o w us t o make some i n t e r e s t i n g S  -topologies.  Theorems.  F o r a l l 0 < r < oo  (1)  measures  the f a m i l i e s  }  5 -continuous  cylinder  coincide.  (2)  For a l l 0 < r <_ 2  (3)  Let  Y  , the S - t o p o l o g i e s  be a t o p o l o g i c a l = {T z L[X,Y]  If  of  Y  is a the  coincide.  v e c t o r space, and f o r each  : T  i s ^-continuous}  T  r  ,  .  CM-space, then, f o r a l l . 0 < r < oo  families  r > 0  ,  coincide,  Remark.  In g e n e r a l , the S " - t o p o l o g i e s do not c o i n c i d e (Example  3, Appendix  4).  S  C l e a r l y , we may i n t e r p r e t case when theorems.  X  i s a Hilbert  for r > 2  space.  a l l of our r e s u l t s f o r the s p e c i a l I n p a r t i c u l a r , we have the f o l l o w i n g  Theorems.  Let  X  Let  0 < r < oo . For any c y l i n d e r measure  be a H i l b e r t space,  n  y  X  over  r  u i s o -continuous <=> y i s E-tight <=> y has a  0 < r < oo  Let on  c -Radon l i m i t measure on X ' and \p be a p o s i t i v e - d e f i n i t e function  y  X  I/J i s S -continuous <=> * for some f i n i t e  c -Radon measure  iKx) = / exp i Re f(x)d£(f) Let  Y  £  on  X'  ,  for a l l x e X  .  be a H i l b e 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  y e CM(Y) <=>  every T  i s a Hilbert-Schmidt map ([36] p. 177).  ([42] V I I I , P i e t s c h [31], Petcynski [28]).  Let Y  be a  H i l b e r t space. For a l l 0 < r <  03  {T'.E L [ X , Y ] : T  = {T e L [ X , Y ] : T  , i s r-summable}  i s Hilbert-Schmidt}  .  (For the d e f i n i t i o n of r-summability, see [31], and [42] p. V I I . 3).  ,  88.  Remark  By Theorem every we  II.2.3.2, when  c -Radon measure  can r e p l a c e  c  by  on  X'  s  x<rhen  X  is  i s a s e p a r a b l e H i l b e r t space,  s -Radon.  X  1.4.3, Thm. We  X  and 2.7.2  are equivalent  1.5.6.2, Appendix 2.5 and 2.6).  observe t h a t even when  extends p r e v i o u s l y knoxm r e s u l t s . xtfhen  2.7,  i s separable.  We p o i n t out t h a t Theorems 2.7.1 (Cor.  Hence, i n Theorems  X  i s a Hilbert  space our work  Sazonov i n [35] d i s c u s s e s the case  i s s e p a r a b l e , o b t a i n i n g Theorem  2.7.2  f o r the case  r = 2  W a l d e n f e l s i n [48] extends Sazonov's theorem to the n o n - s e p a r a b l e case. X  Theorem  i s assumed  2.7.3  extends a r e s u l t  to be s e p a r a b l e and  g i v e n i n [11] (p. 349), where  r = 2  From Appendix 3.5 and P r o o f 2,7.4 significantly  we  see t h a t theorem  g e n e r a l i z e s the P i e t s c h - P e i e y n s k i theorem g i v e n  2.6.3  above  (Theorem 2.7.4).  Proofs  2.  We  s h a l l need the f o l l o w i n g  lemma.  Lemma  Let  X  be a l o c a l l y convex space,  r > 0  and  Let P = P(M and f o r each  jsupp n)  d i r e c t e d by r e f i n e m e n t ,  P £ P S'  P  = {x £ X :  I i n f | f ( x ) | . n ( B ) > 1} BeP f B r  e  S = S  89. Then, f o r any  F e  , Radon measure  £(F ~ tS) = lira PsP  E, on  F  and  t > 0  ,  £(F A t S ' j .  Proof of Lemma  We f i r s t make the f o l l o w i n g observations. I f P E P, Q e P, w i t h  (1)  P  (2)  For any  Q  Q  f i n e r than -P  , then  *  u > 0  ,  X ~ uS =  U  S'  u  PEP  (3)  For every Sp  P  P  E  ,  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 ) | . ( B ) > 1} , r  n  BeB feB  B  where the union i s taken over a l l f i n i t e  B c P  since  supp n e E i s equicontinuous, f o r every x £ X •> i n f | f (x) | £ R i s continuous. feB Hence, f o r any f i n i t e B Q P , (5)  x £ X -> E ( i n f | f (x) | ) . n(B) B 8 feB r  .  Hence,  B e P  ,  i s continuous,  e  and therefore, by ( 4 ) , (3) holds. From (3) we deduce that tF A Consequently, as  i s open f o r every P  that f o r any compact P  E  P  s.t.  P E P .  i s d i r e c t e d by refinement, from (2) i t follows C  in F  with  C c F ~ tS  , there e x i s t s  90.  C c t Fft.S .  Hence,  since  ' .  p  £  i s Radon and  £(F ~ tS) = sup {5(G)  F ~ tS  : C C F ~ tS  = sup U ( t Fft.S = lim PeP  2.1  Let V e 1/  f  i s open i n  : P  p  F  ,  i s compact}  P }  e  5(tF A S ' ) .  be a base f o r  nbnd  0  in  X  s . t . f o r each  there e x i s t s a pseudo-inner-product  [.,.]  on  X  f o r which V = {x ^2 r  Let For  For  each  k  V'e  each  X : [x,x]  e  £  a  §i  s  v e n  v  £ 1}  .  i n Appendix  i.l.  V ,let  F = F  .  V. e f  and  p  F e F ,let  be a p r o b a b i l i t y Radon measure on to  [.,.]  |F x F  Then, from Appendix 1.3 we see (v,F,lO  as i n Appendix  F  related  1.3.  that  i s a system of  62~weights i n  X  By Remark 1.1.3, (1)  S c  Let  S = S  Since  r  \l  nbnd 0 i n e S  r  X  .  .  i s a base f o r  there e x i s t s  V e 1/  with  supp ri C "  nbnd 0  in  X  and  supp n e E  ,  91.  Then,  •  x e k e r V => s u p  |f(x)|  = 0 = > J | f > ( x ) | d r i = 0 => x e S r  feV° i.e. (2)  ker V c S Let  t > 0 f  Using  B  .  .  e B  F o r any  and  the notation  F o t S ' C t x e F :  §  B c X'  = p r, ( B )  B  , l e t  1 / r  .f  .  g  o f t h e a b o v e Lemma, f o r a n y E BeP  P e P ,  | f (x) | . (B) > t } r  r  n  Hence, v  (F o V  tSl) < v  '  V  <_ C  _({x e F : '  '  E BeP  sup |g ( x ) | xeV F  '  E BeP  sup xeV  <C  < — C„ n(X') — r 2,r  r  E BeP  |g(x)| >l}) r  a  by Appendix  |g (x)| i^-C ° t  Z BeP  r  R  since  f e B  '  Since (3)  V j p  (F  C.  ~ tS) l ^ C n(X') <  0 0  v„ _ ( F ~ t S ) -> 0 V ,h  2  >  r  3  r,(X')  t -* °°  |f (x)| .n(B) r  B e P  that  .  , we c o n c l u d e as  sup xeV  f o r every  F r o m t h e a b o v e lemma i t now f o l l o w s v  1.3.2,  that  uniformly  f o r F e F.  F r o m ( 1 ) , ( 2 ) , a n d ( 3 ) we s e e t h a t S  2.2.1  i s w e i g h t e d by  By C o r . I I . 3 . 3 , Thm.  (v,F,l/)  1.5.6.2 a n d P r o p .  1.2  92.  2.2,2 and 2.2.3  2.3  By Thms. II.2.1 and 2.2.1.  By Remark 1.1.3 and Appendix 2.2.5, r i/j i s 5 - c o n t i n u o u s =>  i s continuous a t  0  = > ijj i s continuous =>(jjJF  i s c o n t i n u o u s f o r a l l F E p.  Hence, by 2.4 and 2.5 o f the Appendix, there e x i s t s a f i n i t e y over  S - c o n t i n u o u s c y l i n d e r measure  X s.t.  i|>(x) = J exp i Re f(x)dy (f) f o r a l l  xeX .  By Thm. 2.2.2, y has a  c-.Radon l i m i t measure  Then, f o r every  <Kx) = /  x e X  Rad<5n measure  ,  E, on  E = K(c ) X'  , and f o r some f i n i t e  ,  i>(x) = / exp i Re f(x)d£(f) We note t h a t f o r every  If,  X F  tH]  E M  F  =  F e F  fora l l x e X  and B o r e l subset  .  .F £ F , '  f o r each  y  X'  exp i Re f(x)dy (f) = J , exp i Re f(x)d£(f) .  F  Suppose now t h a t  r  E, on  r m p  >  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 F u r t h e r , by Lemma  0.4.2,  g i s a l i m i t measure of and  X  therefore i t follows  y  ,  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 o f y  . H  of  F  ,  93  y  7S  Since  E, i s  c -Radon and  7\  K(c  Lemma 1.5.1.2 and the d e f i n i t i o n of y is  ) = E y  , then,  from  we see t h a t  E-tight.  Hence, by Prop. 1.2, y is and  therefore,  S -continuous, by Appendix 2.5, r  \p i s S - c o n t i n u o u s .  1.  By Thm. 2.2.1.  2.  By Thm. 2.6.1, Cor. 2 o f Appendix 3.5, and Appendix 3.2.1.  3.  By Cor. 1 o f Thm. 2.4.  1 and 2.7.2. and 3.  a r e consequences r e s p e c t i v e l y of Thms. 2.2.3  2.3, s i n c e H i l b e r t spaces a r e b a r r e l l e d .  Since  X  and  Y  a r e Banach spaces, by [31] p. 339, Thm. 1,  2 T The  <=> T  i s Hilbert-Schmidt.  i s now a consequence of Cor. 1 of 2.4, and  3 of Appendix 3.5.  Since  X  f o r any T The  S -continuous  assertion  Cor.  4.  is  and r > 0 is  Y  a r e Banach spaces, by [42] p. V I I . 3, § 2 , ,  S -continuous  a s s e r t i o n now f o l l o w s  d i x 3.5.  <=> T  is  r-absolutely  summable.  from Thm. 2.6.3 and C o r . 3 of Appen  94.  3.  Nuclear  Spaces.  N u c l e a r spaces  comprise  H a u s d o r f f , H i l b e r t i a n spaces [47]). for  We s h a l l  the case when  one p a r t i c u l a r l y important  (Grothendieck  therefore interpret X  [14], see a l s o  family of  [36]  and  the r e s u l t s o f the p r e v i o u s s e c t i o n  i s a n u c l e a r space.  As a consequence o f the  s p e c i a l s t r u c t u r e of n u c l e a r spaces, we s h a l l be a b l e to s t r e n g t h e n c o n s i d e r a b l y the theorems c o n c e r n i n g c y l i n d e r measures over H a u s d o r f f , H i l b e r t i a n spaces. spaces  arbitrary  We p o i n t out t h a t many of the common  of d i s t r i b u t i o n s a r e i n f a c t n u c l e a r (Treves  [47] Ch. 5 1 ) .  For our d e f i n i t i o n o f a n u c l e a r space we s h a l l use a c h a r a c t e r i z a t i o n due to P i e t s c h  3.1  ([29],  [36]  p. 178).  Definition.  X  i s a n u c l e a r space i f f X  i s a Hausdorff, l o c a l l y  convex  space w i t h the f o l l o w i n g p r o p e r t y : f o r any neighbourhood neighbourhood  V  w i t h supp n C {x  e  of  0  , such  in  X  U  of  0  , and a  in  U  , there e x i s t s  w -Radon measure  that  X : J|f (x) |dn(f) <_ 1} C  X  .  n  on  another X  95.  Remarks  If  X  i s a H a u s d o r f f , 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 t h a t (1)  X  i s nuclear  iff  For any n u c l e a r 1.1.3, i t f o l l o w s (2)  the  i s a base f o r  space  X  , from  nbnd 0  in  X  (1) above, Remarks 1.1.2  and  that  r S -topologies  on  X  coincide for  r >_ 1  2 In p a r t i c u l a r ,  taking  5  as a base f o r  i s a H i l b e r t i a n space  ( [ 3 6 ] p. 1 0 2 ) .  nbnd 0  in  X  , we  deduce t h a t (3)  X  As i n Treves  (4)  E c K(s*)  (5)  Hence, i f X E = K(s*) •  [ 4 7 ] , p. 519, we can prove  . i s barrelled,  then  We p o i n t out t h a t c o i n c i d e n c e r > 0  that  of a l l the  S -topologies f o r  i s a consequence of ( 2 ) , ( 3 ) , and Theorem  2.6.2.  The theorems g i v e n below i n 3.2 a r e d i r e c t the above remarks, and a s s e r t i o n s from the p r e v i o u s  consequences of section  }  specifically,  Theorems 2.2, Theorem 2.3, and Remark 2.5.2.  3.2  Theorems.  Let  (1)  X  be a n u c l e a r  u i s c o n t i n u o u s <=> p i s  space, and  E-tight.  u  be a c y l i n d e r measure over  96.  (2)  y i s continuous => y  has an  s -Radon l i m i t  measure on  X'  k (3)  K(s  If  ) = E  , i n particular, i f X  y i s continuous y is  E-tight  y has an (4)  Let  i s barrelled,  then,  <=>  <=>  s -Radon l i m i t  measure on  X'  -ty be a p o s i t i v e - d e f i n i t e f u n c t i o n on  X  i> i s continuous => t h e r e e x i s t s an  s -Radon measure  E, on  X'  such  i>(x) = / exp i Re f(x)d£(f) . f o r a l l x e X If  E = /((s )  , i n particular, i f  >jj i s continuous there e x i s t s a f i n i t e  X  i s barrelled,  s -Radon measure  5  on  X  for a l l x e X  Theorem 3.2.2 extends a r e s u l t of M i n l o s  then  such  that  .  ( [ 2 5 ] , p. 303, Thm. 1 ) ,  c o n s i d e r e d f i n i t e c y l i n d e r measures over c o u n t a b l y normed n u c l e a r  spaces  ([11] p. 56).  V i l e h k i n extended  that r e s u l t  c o u n t a b l e s t r i c t i n d u c t i v e l i m i t s of such spaces Theorem 3.2.4 extends ([1]  .  <=>  ifj(x) = / exp 1 Re f (x)d£(f)  who  that  p. 17).  r e s u l t s due to M i n l o s  to the case of  ([11] Ch. IV  2.4).  ([25] p. 310) and B a d r i k i a n  We note t h a t the theorems of 3.2 c o m p l e t e l y r e s o l v e  a c o n j e c t u r e of I. G e l f a n d ([25] p. 310, [ 1 8 ] , p. 222), t h a t every continuous c y l i n d e r measure over a n u c l e a r space on the continuous d u a l  X  has a l i m i t  measure  X'  Theorem 3.2.1 has a p a r t i a l converse which extends of M i n l o s  finite  ( [ 2 5 ] . Thm. 4 ) .  a result  97. 3.3  Theorem  Let If  X  X  is a  be a H a u s d o r f f , l o c a l l y CM-space and  y E CM(X) then  Proof  X  =>  y  =>  y  is  E-tight, .  i s nuclear.  By Prop.  1.2,  y e CM(X) Hence, by Appendix the On  convex space.  is  S^-continuous.  3.2.1,  S^-topology i s f i n e r  than the  CM(X)-topology.  the o t h e r hand, by Cor. 2 of Appendix the  CM(X)-topology  i s finer  3.5,  than the  and Remark 1.1.3,  S^-topology.  Consequently, the Since  X X  CM(X)-topology is a  = the  S^-topology.  CM-space, i t f o l l o w s from Remark 3.1.1  i s nuclear.  that  ©  Remark.  . We sarily we  note that a H a u s d o r f f , l o c a l l y  a CM-space (Example 4.3,  Appendix  see from the above p r o o f t h a t  convex space i s not n e c e s -  4).  When  X  i s not a  the b e s t a s s e r t i o n p o s s i b l e i s the  following. If  y e CM(X)  =>  y  is  E-tight,  then, the  S ^ - t o p o l o g y and  CM-space  CM(X)-topoIogy  coincide.  98. Theorems 3.3  and 3.2.1  z a t i o n of n u c l e a r spaces  3.4  lead  to the f o l l o w i n g new  (Remark 3.1.3, Cor. 3 of Appendix  characteri3.5).  Theorem  Let X  X  be a H a u s d o r f f , l o c a l l y  i s nuclear i f f X y e CM(X)  =>  y  is a is  convex  space.  CM-space and  E-tight.  Concerning i n d u c e d c y l i n d e r measures, Remark 3.1.4 to s t r e n g t h e n C o r o l l a r y  (2) of Theorem 2.4.  the  following  is  3.5  Theorem  assertion  Let and  X  If  T  Y  be a t o p o l o g i c a l v e c t o r  T  is  and t h e r e f o r e has an  observe t h a t  be n u c l e a r ( [ 4 7 J , p. 520).  X'  y e CM(Y)  ,  E-tight, s -Radon l i m i t measure on  X  an i n f i n i t e - d i m e n s i o n a l normed space cannot As a consequence  of t h i s  t h a t c e r t a i n c y l i n d e r measures over such a space measure on  space,  .  i s c o n t i n u o u s , then, f o r every y a  We  In view of Remark 2.5.2,  immediate.  be a n u c l e a r space,  T e L[X,Y]  e n a b l e s us  X  f a c t we  can  assert  cannot have a l i m i t  3.6  Proposition  Let. X If  u  be an i n f i n i t e - d i m e n s i o n a l normed i s a f i n i t e c y l i n d e r measure over  topology of  X  i s the  l i m i t measure on  Proof.  {y}-topology,  then  space X  y  such that the  does not have a  X'  By Cor. 1.4.3, Prop. 1.2, and Appendix 3.2.1, y has a l i m i t measure on =>  y is  =>  y is  X*  E-tight 1 5 -continuous 1  =>  {y}-topology  =>  X  Since  i s c o a r s e r than the  S -topology  i s n u c l e a r , by Remarks 1.1.3 and 3.1.1. 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 h o l d , and t h e r e f o r e have a l i m i t measure on  y  cannot  X'  Corollary  Let  A  be an index s e t . For any  f i n i t e c y l i n d e r measure over  (A)  1 < p < 2  P  then  y  i s the  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) x e £ ( A ) -> exp exp - ( (It l\ x \\\ ) P  , if  P e  £  does not have a l i m i t measure on  , (£ (A))' P  100.  Proof  See (1) i n P r o o f of Example 4.2, Appendix 4, and P r o o f  3.1.1  of Appendix 3.  Remark  For  p = 2  the above c o r o l l a r y i s w e l l known (Gross  We have not seen a treatment  of the case  1 <_ p < 2  [13]).  i n the l i t e r a t u r e .  101. 4.  & -spaces.  Applied chapter y i e l d s  to  j^-spaces,  results  1 <_ p <_ °°  analagous  , the theory of the p r e v i o u s  to those f o r H i l b e r t i a n  spaces.  Since  2 I - i s a Hilbert The  results  t h i s case has a l r e a d y been d i s c u s s e d i n § 2 . 5 .  space  g i v e n t h e r e a r e s t r o n g e r than those we s h a l l o b t a i n here  f o r an a r b i t r a r y  £^-space.  Notation  Let  A  be an index s e t . {x'e  0 < r <_ °° ,  F o r any  <?• :  E  |x(c0 | < °°}  when  r <_  aeA i  £ (A) r  = \ A <• {x E C  We g i v e  £ (A)  when  b when  r =  , the t o p o l o g y generated  by the quasi-norm  3.3) : x  £ ( A ) -> r  E  r >^ 1  E aeA  |x(a) |  e R  r  , the topology generated  : x e £ ( A ) + (• E | x ( a ) I ) r  r  1/r  ;  +  by the norm  eR  , '  +  aeA we  where  ( E aeA. For any U  |x(a)| ) r  1< p  <2  1 / / r  = sup |x(a) | a£A  i f r = °° .  ,  = {x E £ ( A ) : .E . aeA P  P  00  the u s u a l t o p o l o g y , i . e . ,  r < 1  (Appendix  i : sup l x ( a ) | < °°} when asA  |x(a)|  2  < 1}  .  102.  For  any o u t e r measure £v(n) =  lim PeP(M )  ulnu = 0  on a space ft  I B P  n(B)|ln (B)| n  £  n  t o t C<x.  where  n  when  u = 0  The h e a r t o f t h i s s e c t i o n i s the f o l l o w i n g 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 <_ « , a r e  weighted systems i n £ ( A ) P  4.1"  Theorem.  Let For  any —r 5 C  satisfies  1 <_ p <_ r > 0 r S  ~r S fies  the added  the added  If  2 < p <_ oo S  (2)  If  r  < •»  (3)  If  r  , let  c o n s i s t o f those s e t s  S r,n  condition | f (x) | d n ( f ) < «> . r  and  and  0 < r < q  r = q  then  then  i s a weighted system i n and  0 < r < oo  p £ (A)  , then  i s a weighted system i n £ ( A )  (We note t h a t  P  S  r  f o r which  •  P  1 < p < 2 S  ,  i s a weighted system, i n £ ( A )  2<p<_oo  —r S  r S e S r ,n  condition  1 <_ p j<_ 2 r S>  1/p + 1/q = 1  c o n s i s t o f those s e t s  / (sup xeU P (1)  and  , let  £v(n) and when  oo  = S  r  when  p = 2  .)  e S  r  f o r which  T\ satis1  103.  The proof of the above theorem w i l l be g i v e n a t the end o f the s e c t i o n .  Now,  we  point  together with C o r o l l a r y  4.2  out i t s immediate consequences  when taken  II.3.3.  Theorems  1 < p < oo  Let (1)  -  2 < p <_ oo  If measure  y  over  — + — = 1 p q  and  -  and  £ (A) P  0 < r < q  . s=>  E-tight  p has a (Here, we  . L\»vX  •k  c -Radonimeasure on  a l s o use Prop. 1.2. and Thm.  a Banach space and i s t h e r e f o r e (2)  If  2 < p <_ « and  over  £ (A)  cylinder  ,  „r p i s • i -continuous  p is  , then, f o r any  r = q  (£ (A))' P  II.2.1, n o t i n g  that  £ (A)  is  P  barrelled.) , then, f o r any  c y l i n d e r measure  p  ,  P  —x p is  S -continuous  p is  E-tight •k  p has a (3)  If over  1 <_ p <_ 2  £ (A) P  =>  => _ Liy*>'^  p  c -Radonimeasure on and  r > 0  (£ ( A ) ) '  , then, f o r any c y l i n d e r measure  ,  p is  S -continuous  p is  E-tight •k  p has a  =>  =>  _  c -Radonimeasure on  p (£ ( A ) ) '  p  104.  U s i n g Theorems 4.2 we can r e p r e s e n t c e r t a i n f u n c t i o n s on  £ (A) P  positive-definite  as F o u r i e r t r a n s f o r m s o f measures on  (£ (A))' P  The p r o o f s o f the a s s e r t i o n s g i v e n below a r e s i m i l a r t o the p r o o f o f Theorem 2.3, and a r e t h e r e f o r e  4.3  omitted.  Theorems  Let  l < p < ° ° , - - + — =1 — — p q  f u n c t i o n on (1) _ I f  , and  \j> be a n o s i t i v e - d e f i n i t e  £ (A) P  2 < p <_ «> and 0 < r < q  ', then,  r is measure  *  S -continuous E, on  <=> f o r some f i n i t e  (£ (A))' P  I f '2 < p < » —r  and  r = q  for a l l  E, on  If  1 <_ p <_ 2 ~r  (£ (A)) P  !  and  E, on  r > 0  /•  c -Radon  f o r a l l x e £ (A) P  P  .  , then, f  ^  (£ (A))'  .  ,  ^ i s 5 - c o n t i n u o u s => f o r some f i n i t e measure  P  *  iKx) = / exp i Re f (x)d£(f) (3)  x e £ (A)  , then,  ^ i s S - c o n t i n u o u s => f o r some f i n i t e measure  c -Radon  ,  U>(x) = / exp i Re f ( x ) d£(f) (2)  <  c -Radon  ,  ip(x) = / exp i Re f(x)d£(f) f o r a l l x e £ ( A ) . Concerning induced c y l i n d e r measures, Theorem 4.1 y i e l d s the P  f o l l o w i n g r e s u l t s when taken t o g e t h e r w i t h Theorem I I . 4 . 3 .  105.  4.4  Theorems.  Let  1 < p < -  , —+ — = 1 , Y P q  00  be a v e c t o r space,  a f a m i l y of f i n i t e c y l i n d e r measures over (1)  If  2 < p <_ <*> T  <=>  is  and  0 < r < q  r  f o r every  y e C  has a  2 < p <_  c -Radon l i m i t  T for  is  every  is  every  (3)  If  T for  is  every  for  every pa  E-tight  1  is  II.2.1.)  C-topology  on  Y =>  on  Y =>  (I ( A ) ) '  C-topology  , E-tight  has a  r = q  P  , then,  S - c o n t i n u o u s w i t h r e s p e c t to the  =>  , c -Radon l i m i t measure on  r e s u l t s due L. Schwartz  the case when  (£ (A))'  =>  r > 0  As consequences of  Y  , then,  c -Radon l i m i t measure on  and  y e C T  on  ,  y e C  y D T  C-topology  4.2, we a l s o use Prop. 1.3 and Thm.  r = q  has a  1 <_ p <_ 2  P  ,  y e C  y p T  measure on  ^ - c o n t i n u o u s w i t h r e s p e c t to the y e C  y • T for  and  00  .T e [£ (A),Y]  ,  (Here, as f o r Thm. If  , and  be  , then,  S - c o n t i n u o u s w i t h r e s p e c t to the  y u T  (2)  Y  C  and  (& (A))' P  of Theorems 4.4 we have the f o l l o w i n g e x t e n s i o n s 139] and Kwapien [ 1 9 ] . A  i s countable.  They c o n s i d e r o n l y  106.  Corollaries.  (1)  If  2 < p <_ «>, 0 < r < q , y e £ (A)  and  r  T : x e £ ( A ) •+ ( x ( a ) y ( a ) ) . e / ( A ) aeA  ,  P  then, f o r every u a T (2)  If  y £ CM(£ (A))  has a  2 < p < co,  = q , y e £ (A)  with  r  r  a " (A))*  c -Radon l i m i t measure on  | y(ct) | 1 l n | y ( a ) [ 1  £  ,  |  <  co  ,  aeA and T  : x e £ (A)  (x(a)y(a))  P  e £ (A)  ,  r  - x then, f o r every y n T (3)  If  y e CM(£ (A))  has a  ,  c -Radon l i m i t measure on  1 £ p <_ 2 , r > 0 , y e £ ( A )  F  , and  r  T : x e £ ( A ) -»• ( x ( a ) y ( a ) )  (£ (A))'  , aeA then, f o r every y e CM(£ (A)) , * y a T has a c -Radon l i m i t measure on P  e £ (A) r  A  We g i v e here the p r o o f of onl\  P (£ (A))  Corollary  T  (1) .  proofs are s i m i l a r .  Proof of C o r o l l a r y ( 1 ) .  For each e and  n  D  e  a e A  , let  e (£ (A)) P  a  ?  : x e £ (A) ->• x ( a ) e C P  the d i s c r e t e measure on n({e  a  )) =  i  v(a)  i  r  (£ (A))  f o r each  p  r  a e A  ,  with  The o t h e r  107.  Then, supp n e E and  f o r any  ,  x e £ (A)  ,  P  £ I ( .*)J = J|f 00 | dn(f) T  r  I t follows that an immediate  4.5  T  •  r  is  S - c o n t i n u o u s , and  consequence  of Thm.  4.4.1.  the c o r o l l a r y i s now #  Remarks.  (1) "  If  A  i s c o u n t a b l e , then  by Theorem II.2.3.2, every fact (2)  s -Radon. We  when \  and  The  P  ye  £ (A) q  and with  f o r e g o i n g theorems may  r = q T, aeA  .  (£ ( A ) )  Consequently, !  is in  then be s u i t a b l y m o d i f i e d .  i s the b e s t r e s u l t  possible  If  | y (a) | | In | y (a) | | = » q  i s as g i v e n i n the c o r o l l a r y ,  Appendix  i s separable.  c -Radon measure on  p o i n t out that C o r o l l a r y 4.4.2  2 < p <_ <»  T  (£ (A))'  then by Example 4.2  of  4, p e CM(i>5(A))  there e x i s t s  such t h a t  p Q T  fails  to be  E-tight. (3)  With  the n o t a t i o n of (2)  4.4.1, we T • and  see  is  above, as i n the p r o o f of C o r o l l a r y  that  S^-continuous,  t h e r e f o r e , by Lemma II.4.2, pp  T  is  S "-continuous. q  From Remark (2) above, and Theorem I I . 3 . 3 , i t now f o r any S  q  1 1 2 < p < ° ° , i f 1 =1 , then p q i s not a weighted system i n £ (A) P  follows  that  108.  However, P r o p o s i t i o n 1.2 system i n  £ (A)  we  P  suggests t h a t when s e a r c h i n g  ought to l o o k  f o r a subfamily  Remark (2) then i n d i c a t e s t h a t subfamily  of  When  any  f o r us  q  2 <_ p < °=  a system of for  S  to  set  x z (E -> exp K  i s i n f a c t an  S  q  appropriate  , the c o n s t r u c t i o n which we  K  £ (A)  depends on  P  use  f o r producing  the f a c t  that  ,  Z  -  of  consider.  5-weights i n  finite  f o r a weighted  |x(a) |  e ®  q  aeK K <p  i s a D o s i t i v e - d e f i n i t e f u n c t i o n on (Remark (1) of P r o o f s 1 <_ p < 2 longer  , then,  ( 4 ) , and  q > 2  and  Proof 2.2 the  c o n s t r u c t i o n of a system of  any  finite x  e  ?  set  K  X(  K  [38J  5-weights i n X  : R  +  .  can  construc-  show t h a t  , 1 < p < 2  P  -> £  The  We  £ (A)  If  above i s no  p. 532).  1 < p < 2  would be p o s s i b l e i f there were a  =1  of Appendix 2.)  f u n c t i o n given  p o s i t i v e - d e f i n i t e (Schoenberg  t i o n t h e r e f o r e breaks down when  1 1 — + — p q  , where  ,  such t h a t f o r  ,  |x( ) | )  I aeK  q  a  e €  K was  p o s i t i v e - d e f i n i t e on  (J  .  I f such a f u n c t i o n  X  existed,  then, by Appendix 2.2.4, (i)  X(  x e £ (A) q  Z  |x(«) | ) q  e C  aeA would be when X  : R  a p o s i t i v e - d e f i n i t e f u n c t i o n on  q > 2 +  -> <E  , one  can  such t h a t  N o n e t h e l e s s , we £ (A) P  by  , 1 <_ p < 2  show as i n [5]  that  £ (A)  .  q  there  However,  does not  exist  (i) holds. can  still  , i f we  use  the c a n o n i c a l imbedding x E i (A) £ (A) P  2  x  £  o b t a i n a system of the  .  system of  6-weights i n  5 —weights induced  (Remark (2) of P r o o f s  4.)  109.  (5)  Remarks  (4), Proposition  1.2,  ~r led  us  to b e l i e v e  family 1 < p  that  the  S  C  S  , r > 0  , w o u l d be  for a weighted  We of  q  point  out  i n the  (Remarks may  be  (4)  that  out  and  p >_ 1  Appendix 1). with  a  suitable  system i n  £ (A)  ,  P  s u c h an  , we  q  identification £ (A)  and  P  — + — = 1  and the  1 , the  appearance  finite-dimensional  Thus, a l t h o u g h  £ (A)  r e l a t i o n s h i p between  (£ (A))' P  have avoided d o i n g  this,  might have suggested  £^(A)  was  level  crucial  as  that  the  to our  argument.  Proofs  2,  4.  Together with for  for  hypotheses a r i s e s at  identified  carrying  Proofs  4.1.1,  2 1  (6)  p r o o f of Theorem  r  t o s t u d y when s e a r c h i n g <  and  any  p  >_ 1  notations  o f A p p e n d i x 1.1  and  , let  q  p X  V  the  =  P  £ (A)  =' {x  I. I  ,  P  X  e  : f  e  P  : Ixl 1  <  'p  -  If(x)}  X* -> sup  q  x V  P  1}  .  e  For  any  finite  |.|  q „,K  r  =  K c  A  : f e (f,V  P  , let  + sup { | f ( x ) |  : x e V ft <C } K  p  ,  r  c ,x  K  K  P Let ^ F =  K {€  : K c  A  is finite}  d i r e c t e d by  inclusion,  110.  For any  2 <_ p <^ »  and f i n i t e  K c A  K • K Yp be the p r o d u c t measure on £ D K K v : £ e F -> Y  and  , let generated by  y  on  ? which i s Radon.  Remarks  (1)  By Appendix  1.2, f o r each  (v ,F,V P  (2)  For each  P  )  i s a system of  1  p < 2  (V f\ C ) ° c then, by Appendix 2 (v ,F,V ) p  C )° K  6 -weights i n P  X P  f o r every f i n i t e  K  1.2, f o r e v e r y  i s a system o f  We observe that f o r any (1)  }  , since  (V A  K  2  4.1.  2 < p < «>  S c nbnd 0  in  r  1 <_ p < 2  c  A  ,  ,  6 -weights i n X 2 p  p  X P  1  and  (Remark  r > 0  ,  1.1.3).  and (2)  ker V  V  = {0} e S  Now, f o r any ' f  B  S =' S  e B , g  B  f o r every  r,n  =_  1 / r n  e S  r  S e  . B c  , and each  (B).f  X' p  , let  B  and s = sup { | f |  : f e supp n } .  Then, as i n P r o o f 2.2.1, f o r any t > 0  (5)  , we have  P e P  K  P  e  K  (6)  P  v (£ ~ ' P  K  tS) =  \~ Z M\  : Z  , ~ and, by the lemma of P r o o f s 2, K  K c A  ., and  that  v ( C ^ tS') < v ( { x C P  , finite  V  n  BeP  K  lim V «E A PeP(S) P  K  tS') . p  > D)  ,  C ,  Ill,  Case 1.  (2 < p <_ °°, 0 < r < q) . Since  supp r\ z E 0 < s <  ,  ro  Then, f o r any  t > - r/^CX') , s p B c supp n => |1 g j < 1  Hence, by Appendix  .  1.2.3, the r i g h t - h a n d - s i d e of (5) i s  m a j o r i z e d by  <  1- r C i .r p,r (Since E n BeP  < n  q / r  _  /vM j . ±- n^r ,q-r+l, q ^ q / r , , n(X') + —q 2^Cp Pq-r - ^ s V ' (x') p p e  q/r > 1  q / r  v  v  y  fc  (B) = n  , then q / r  (x') E BeP  [p(B)/ (x')]  q / r  n  p  P  ( x ' ) E n(B)/r,(x') = n BeP •  q / r  P  (x').) ' P  Whence, by (5) and ( 6 ) , (7)  Y «E K  ~ tS) < ± - C n ( X ' ) ' + i - 2TTC - r p,r p q p r  K  S  P  fc  t  Since the c o e f f i c i e n t s of independent of  K  1/t  and  (4^±I) V S  / r  1/t  .  (X')  q-r  p a r e f i n i t e and  i t follows that  K K (8)  Y By  P  (  C  ~ tS) -> 0  as  t ->• °° u n i f o r m l y f o r a l l f i n i t e  ( 1 ) , ( 2 ) , and ( 8 ) , 5  i s wei ghted by  (v ,F,V ) P  .  Kc  A  .  112.  —V  Case 2  (2 < p < », r = q, S e 5 )  .  •  If c =  sup 0<u<s  u In u  then 0 < C For  oo  1 1/r t > — n (X') ., by Appendix s p  any  of  <  1.2.3, the r i g h t - h a n d - s i d e  (5) i s m a j o r i z e d by  (9)  C P  '  E B P  q  |r, (f g ) | „ + 2TTC E ' BeP q  £  The f i r s t  | r_,£ g) | ' '  q  q  r  p  q  | In | r ( i g ) | ' q  term of (9) i s m a j o r i z e d by •—  C  s ri(X')  .  q  q p q p The second term o f (9) i s m a j o r i z e d by t  2^C  ^  p  E  lint!  s  ^ (!f | ) n(B)[|lnt| q  2TTC S ( X ' ) + — . P P  q  n  2TTC S  q  n  t  + l | l n ( B ) | + | In | f | | ]  q  B  q  P  q  q  t  + — t  2irC q  Hence, by (5), (6) and Y (C P  E BeP  c n(X')  P  B  q  n(B)|ln (B)| n  ' .  P  (9),  tS) 2TTC  < -  — q  t  [c s n(x') + — p.q P q q  +  E  q  n  Ji^l t  £ v ( n ) + 2,TC C ( X ' ) ] P P  q  [ 2 n C  q  p  By t h e h y p o t h e s e s , the c o e f f i c i e n t s o f f i n i t e and independent o f (10) By  v (C K P  K  ~ tS) -> 0  as  K  .  r  i s weighted by  l / t  q  and  |lnt|/t  q  are  Hence,  t -> oo u n i f o r m l y f o r a l l f i n i t e  (1), (2) and (10) S  P  (v ,F,V ) P  P  K C A  113.  Case 3  (1 <_ p <_ 2 , r > 0 , S e S ) r  By Appendix 1.2.3, f o r any of  (5) i s m a j o r i z e d E  °2,rB £p Hence, by v (£ 2  t > 0  by  l K K>l2, ' r  (  K  (5) and ( 6 ) , ~ tS) = l i m Yo(C ^ t S ' ) ' PeP(S)  K  K  K  2  =  C  . lim PeP  t  1 ~  c  l i m 2  r  t  PeP  5  = 7 C t  , the r i g h t - h a n d - s i d e  Z  '  r  /*  E BeP  ( s u p  1  BeP  P  sup „ xeUr\C P ( s u p  feB  |f ( x ) | ) . r , ( B ) r  I f ( x ) l ) ) n(B) • 1  xeU  p  |f(x)|) dn(f) •  ( s u p  r  xeU P  By h y p o t h e s i s , the c o e f f i c i e n t .of of  K  . 1  as  and independent  t -> °° u n i f o r m l y f o r a l l f i n i t e  ( 1 ) , (2) and (11), ~r S  i s finite  Hence  2 K (11) v„(C ~ tS) -> 0 K By  1/t  2 " i s weighted by (v ,F,V )  K C A  .'  114.  APPENDIX •  In t h i s Appendix we e s t a b l i s h a number o f r e s u l t s and c o n s t r u c t i o n s which a r e n e c e s s a r y f o r the d i s c u s s i o n s o f Chapter I I I . last  In the  s e c t i o n we g i v e some counterexamples which complement the c o n s i d e r a -  t i o n s of Chapter I I I .  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  Spaces.  In t h i s s e c t i o n we s h a l l c o n s t r u c t 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 spaces. to produce systems o f P L 1  The e x i s t e n c e o f these measures e n a b l e s us  <5- eights i n H i l b e r t i a n spaces and i n W  .  Notation  K  i s a f i n i t e set.  For any 1 5 . p l  p  ,  0 3  q (x  e c  K  (x e C  P  f e (C )"  For any  K  q  :  z n  r  v  lx( )l a  P  5 l)  : sup |x(a)| < 1} aeK  when when  ,  sup | f ( x ) xeV P  (We note t h a t  V ° = {f e ( t ) " p ' K  : If.l < 1}) 'q ~  p < °°, p = »  & -spaces, P  115.  A i s the Lebesque measure For  on  any f i n i t e d i m e n s i o n a l space I = {(x,f)  e  <C ' . F  ,  F x F* : | f ( ) | >_ 1}  .  x  The c o n s t r u c t i o n s o f t h i s  s e c t i o n w i l l be based on the a s s e r -  t i o n s g i v e n below.  1.1  Lemmas  2 <_ p <_ «> and r > 0  Let (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 , (9  : £ + R  P  such  +  that = J  exp - |w|  q  (2)  continuous  When  2 < p <_ <»•  0  (i)  < C  <  (exp i Rewz)(9  (z)dw(z)  for a l l w e £  , there e x i s t s  co  P such  that  (ii)  0 (z) < C / | z l p  p  Hence, when  r < q  for a l l z e t  2 + q  .  1  , f o r any  u > 0  ,  2TTC 2  (m)  f l+ l / u  qr  ]  i f r < q  q-r  z| z <u  r  1<  Q (z)dA(z) < ?  2TTC  lnu  i f r = q  and (iv)  (3)  f  u<|z| When  Q (z)dX(z) < 2TTC Jul  -  P  p = 2 0 (z)  2  n  ,  n  P  q  1 1  ,  1  = -— exp 4T(  L  Izi 4  T — j  2  f o r every  z e £  ,  116.  Notation  For  2 <_ p <_ m  each  , let  :B c C + f  Y  'p  1  '  dA £ R  J  B  , '  +  u  and P = Y P ( { z e € : i I zi l >-  6  For  any C  For  9  r >  C  Z  <M )  r  <*X(z)  Z  2 < p <_ ° o  each  .  , let  0  H |  = r  1})  • 0 < r < q  and any  >  be the c o n s t a n t of Lemma  P  , let  1.1.2,  and 2TTC  q-r q-r C  +  J  i  ' |z|  ^(z)dA(z)  |z|<l  if  r < q  i f  r = q  P  P>r 2TTC  +  /  |z|<l  P  |z| 0 r  '  (z)dA(z)  P  Remarks  We note t h a t i n view o f Lemmas Y  > 0  p <  C  < co,  p 0  and  1.1.2,  i s a p r o b a b i l i t y Radon measure on  5 0  1.1.1  <  C  <  CO  .  €  ,  117,  1.2  Lemmas  Let  2 <_ p K Yp  (1)  <_ oo  and  the  measure  y P  on  €  K Yp  i s a p r o b a b i l i t y Radon measure on  K « 0 K f f e («T) ~ V => Y (I ) > <5 P P - P  (3)  F o r any sequence  {f } C n new  B = {x e C  If  p = 2  , then  (<C^)  C  K  , and  Ifn (x) 1  : -E new  K  generated by  ,  (2)  (i)  K €  be the product measure on  , let  > 1}  r  1  r > 0  1  q = 2 and  new (ii)  If  p > 2 , r <_ q  C  l^ lq — ^  ^  n  o  r  e  v  e  r  y  new  E If | + 2^C ( " ) E If I n'q p q-r 'n'q new new r  p,r  C p,q  1.3.  , and  q  r + 1  i f r < q  2  1  E If l + 2TTC E 'n'q p new ^ new  If | | l n | f I I '.n q n q  q  q  1  r  1  1  n  1  1  ,  if  r = q  on  F  n  Lemma.  Let If  F  be a f i n i t e - d i m e n s i o n a l v e c t o r  [.,.] i s a pseudo-inner p r o d u c t on V = {x e F : [x,x] <_ 1}  space. F  that f e (ker V) ' ~ V ° => £ ( I ) 1 3  £  and  ,  then, t h e r e e x i s t s a p r o b a b i l i t y Radon measure  (1)  , then,  6 2  •  £  such  118.  (2)  For any sequence £({x  e F :  {f } n neoo  in  F  ,  E | f ( x ) | > 1}) < C. nea) r  E new  (sup|f(x)|) xeV  r  P r o o f s 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 f o l l o w i n g f a c t s , (See  a l s o Levy [21] Ch. V I I . ) For any  0 < q <_ 2  , there  exists a s t r i c t l y  positive  continuous  9  2 : R  P  R  +  s.t, (i)  exp - | t | = /[exp i ( t . u ) ] 0 (u)dA(u)  where,  f o r any  t.u = t u Q  t e R  2  , u z R  2  for a l l  ,  +  0  and  |t| = / ( t If  2  q < 2  + t )  .  2  , there  lim  |t|  2  +  exists 9 (t) = c P  q  i • i  0 < c  q  Hence, i f  ( i i ) C = sup |t| 0 (t) , 2+q  P  xeR  P  then, (iii)  0 < C  P  <  and  (iv) 6 ( t ) < C / | t | P  —  P  2 + q  fora l l  &  q  < <» s . t .  t e R  2  ,  119.  Consequently, f o r any  0 < r <_ q  / | t | 6ip (t)dx(t) < pc  , and  / !t|  r  u > 0  ,  . — 1 — 2d+ q x e t )  r  • ~ i< t!<u • |t!  =  j • (p^ l£p£u  2TTC P  + q  ~) ''"dp  using polar  By i n t e g r a t i n g the l a s t  term i t follox-Js  (v)  2TTC £  q-r  t  1<  For  T\Q <U —  2TTC  [l-l/u  ~ 2TTC P  q = 2  , using  | | lnu  the f a c t 2  (exp i xy)exp - —  R  computation shows  <9<t)  (vi)  that  q  r  ]  < ^[l+l/u - q-r  q  r  ]  , if r < q ,  (t)dA(t) < P  (l//2rr)|  a direct  coordinates.  r =  q  that , dx = exp - ~ -  2  ,  that  =^exp  2  Hence, f o r any  if  r > 0  ,  •r"  (vii)  / | t | 0 (t)dA(t) < » . 2  Let 2 T : z e <C -> (Re z, Im z) e R and f o r each  &  0 < q <_ 2 •, o T .  = 6 P  P  The a s s e r t i o n s of Lemmas 1.1 now above,  f o l l o w from  and the p r o p e r t i e s o f the map  for  ,  any  z  and  w  in  (C  T  (i) - ( v i i )  , namely,  ,  Re wz = (Tw).(Tz) ' , and  T  i s an i s o m e t r i c , measure p r e s e r v i n g ,  homeomorphism.  120.  1.2.  Notation  F o r any  K * (<S )  f e  - f / | f |  <?  : x  f  "K,f Y p  K  - f  f  >  q  ff  e  r  , let  -> f ( x )  e £  K [Y ] p  and l e t T  1 .'2.1  '  m e a s u r e on  1.2.2  This  ~f  follows  e  (1)  from the f a c t that  a e K = 1  a  f o r any  r  K * (£ )  f e  a  p r o b a b i l i t y Radon  > and  w  e £  ,  "K f dy ' ( z )  i Re f ( w x ) d y ( x )  II [/(exp aeK  = exp -  i Re w f ( e ) x ( a ) ) d y a  Iw  1 ' aeK  f(e )|  (x(a))] P  q  b y Lemma 1 . 1 . 1 ,  a  lw! |f|  = exp -  q  1  .  q  q  a R a d o n m e a s u r e on a f i n i t e - d i m e n s i o n a l s p a c e i s  d e t e r m i n e d by  (2)  s  , let  -^ \p (w) = J e x p i Re wz  =  2.3),  l  , e £ {a}  = /exp  Since  Yp  £  For each  Then,  — ~K f ( e x p i Re w z ) d y ' ( z ) e £  : w e £ ->  i t s Fourier  i t follows  Ifl  =  1 =>  that K Y  '  f  = Y  transform  uniquely  ( B o c h n e r ' s Theorem, A p p e n d i x  121.  However, f  =1  .  and K * 0 I f e (C ) - V => P  I  U  1  f > 1 'q -  Hence,  = 6 P  , by (2) above.  1.2.3.  (I)  E  Since  new functions  n  K C  on  ^(B)  f (x)  •  i s a series of p o s i t i v e  '  Yo~  -2  ,  = /l .d ^ B  < /( Z  Y  |f (x) )d ^(x) r  n  Y  new =  l ni  E  f  new = E new  \fj  = C„ 2, r  E  new  /|f;W| dY|(x) r  2  j\z\  z  d  X  2  If  1  I*  n Y  2  (. ) Z  by (2) o f P r o o f  n'2  1.2.2.  ( i i ) Let H  = {x e C  n  H =  C\ new  K  : | f (x)I < 1} n — 1  f o r each  new  H n  and h  K :' x e € -> 1  E new  if |f ( x ) | n 1  r  x  e  C  if x e H  K  ~ H .  ,  m e a s u r a D  le  122. We have t h a t  =  I „({x e £ : |f'(x)| > l/.|f I }) Y  new  P  n  n  q  -K,f =  (2)  Z new  < 2TTC — p and f'(x)I n  functions  £ new  1  Lemma  If | n'q  Y p  H  =  £ new  < —  I new  =  Z new  Z  new  <  Letting  I  n  a  by ( 2 ) of P r o o f 1.2.2.  q  J  igx«r^>>  new  |£ i „ / 1„ | f ' ( x ) | d ( x ) 'n'q' H n p n r  1  |f | n'q  If | 'n'q  ~K,f' / |z| d (z) ' ' ' 'p ,'z|<l/|f j ' — n'q  r  r  1  / | z | 6 (z)dA(z) ' i i p '  r  r  v  r  | z | 6 (z)dA(z) p | z| <1  Lemma 1.1.2 ( i i i ) ,  J  n  Y  If T [ / | z | 6 (z)dA(z) 'n'q-' p z <1  a = P>r  T  1  M - ^ ^ J q  Hence Y  f  1.1.2.(iv).  1  1 7  1  , we a l s o have t h a t  /i hd wi [£  / l hd ' H p —  : |z| > / I I > >  C  r  on  H  '({z e  i s a s e r i e s o f p o s i t i v e v -measurable • 'p  1  new  n  Y_  r  we have t h a t  by (2) o f P r o o f 1.2.2. +  / | z | 9 (z)dA(z)] / I I p 1< z <1/ f n'q  , from  r  X  J  |fn |q  Z neoj  (3) /  1  2TTC r  1  If i 'n'q  (a + P-[l + p,r q-r  ])  if  if  r =  qr  r <  < E  If  new Consequently, i f  | (a  r < q  2TTC | l n | f I  +  r  'n'q  p,r  p  I)  'n'q' ^  1  , then, from  (1), (2) and  q  (3),  2TTC Y  ( B ) < 2TTC Z P P new =  C  | f | + q  n  Z  p,r  q  |f  |  r  'n'q  new  Z new  +  If n  | (a ' r  q  P  r  q  2TTC ( ~ ^ )  Z  q  p  + "  q-r  P-[l+|f| r  n  |f |  r  'n'q  new  q  1  ])  q  ,  s i n c e a l l terms are p o s i t i v e . The case  r = q  i s e s t a b l i s h e d from  (1), (2) and  (3)  similarly.  1.3  We  first  choose a  suppose  that  [.,.]  i s non-degenerate.  [.,.]-orthonormal b a s i s T : z e ( C  ->  and *  f e F  K  Z z . a e F „ a aeK  •+ f = f  o  T" e  for  F  .  I f so  Let  .  , K *  (C  )  .Then, T Further, (1)  i s a homeomorphism.  f o r any  x' e C  [ T x ' j T y ' ] = <x',y'>  product i n  C  ,  and (2)  sup x V  |f(x)| = |f|  e  Whence, i f h  ~ 2 Y  °  T  K  , y' e C , where  K  , and <.,.>  f e F denotes  ^  , the  inner  124. then (I ) = 2< ) f  5 l  L F  Y  and C(B) = y^ax  C  1 E  : Z |f (x')| new  K  r  n  > 1})  The assertions now f o l l o w from (3) and Lemmas 1.2. When of  F  [.,.]  s.t. F  (Possibly,  i s degenerate, l e t F ^ be any subspace  i s the d i r e c t sum of F^ and ker V  F = {0}) x F^  [.,.]|F  since  . We have that i s non-degenerate,  F^/TN ker V = {0} Let  £  be the p r o b a b i l i t y Radon measure on F ^ determined  as above, and E : HcF+5(H/>  F ) e R  +  1  .  Then, (3)  ^ i s a p r o b a b i l i t y Radon measure on F  Since  F  i s the d i r e c t sum of F ^ and ker V  x e X  there e x i s t s a unique representation x = x^ +  , with  x^ e F^  , f o r any  and x^ E ker V  Consequently, (4) f e (ker V ) => sup | f ( x ) | = sup | f ( x ) | xeV xeVKF and therefore, from the non-degenerate case above, 3  (5)  f e (ker V ) ~ V° => r  ( f ) e F* ~ (V A F^)^  a  r  F  p  (f)  f  => e;(i ) = ^ ( I r  ) > 5  2  .  If  f  n  e F  sup xeV and  therefore If  from  f  »  (ker V)  f o r any  new,  then  If (x) | • = » (2) of the lemma h o l d s .  e (ker V)  f o r every  the non-degenerate 5(B)  a  = C  2  r  '  then  c a s e above,  = £ '({x e F. : 1 1 < C  new,  Z new  |f (x)| n 1  Z new  sup |f (*)| XEVAF^  Z new  sup xeV  r  >  1})  r  n  •|f (x)|  r  by  (4).  126.  2. • P o s i t i v e - d e f i n i t e  In  this section  positive-definite  2.1  F u n c t i o n s on V e c t o r Spaces.  we g i v e a number o f u s e f u l  f u n c t i o n s on v e c t o r  results  concerning  spaces.  Definition  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 if) : X -> C and f o r any  function  on  X i f f  , <Z C  n e to, {x^,. . . > _^} £ X , {z , . . ., x  n  n-1  ,  _  E  Z  k,£=0  k £ ^ Z  ( x  k ~ £  We s h a l l need the f o l l o w i n g  X  }  - ° •  elementary a s s e r t i o n s about  positive-  d e f i n i t e f u n c t i o n s on groups.  2.2  Propositions  Let (1)  If  X  be a commutative group.  if; i s a p o s i t i v e - d e f i n i t e  0 <_ (2)  Let  If  If  T : Y -> X  cf  and q»ip  X  , then  function  on  i s a homomorphism, then  if)  X  , and  Y  be  group.  definite function (3)  on  CO) < °° .  if) be a p o s i t i v e - d e f i n i t e  a commutative  function  on  0  T  i s a positive-  X  if) a r e p o s i t i v e - d e f i n i t e i s a positive-definite  f u n c t i o n s on  function  on  X  X  , then  (4)  If X  (ilO. , J Jed  , and  ip : X  i s a net of p o s i t i v e d e f i n i t e functions C  i s such I|J . (x)  4(x) = lira jeJ  on  that for a l l x e X  ,  3  then, i s a p o s i t i v e - d e f i n i t e f u n c t i o n on (5)  If  X  i s a t o p o l o g i c a l group, and  f u n c t i o n on  X  2.3  [4] p.  X <=>  \p i s c o n t i n u o u s a t  f i n i t e - d i m e n s i o n a l spaces we  of a w e l l known, r e p r e s e n t a t i o n Bochner  is positive-definite  , then  \p i s c o n t i n u o u s on For  X  theorem  0 e X  have the f o l l o w i n g  (Rudin [34] p. 19  version  1.4.3,  58).  Theorem  Let  F  be a f i n i t e - d i m e n s i o n a l v e c t o r  space.  i s a c o n t i n u o u s p o s i t i v e - d e f i n i t e f u n c t i o n on  F  iff t h e r e e x i s t s a unique f i n i t e Radon measure such t h a t  4<(x) = / exp  Using the above theorem, establishes  i t s following  the p r o o f .  (The theorem  vector  spaces.  See a l s o  i Re f(x)d£(f)  as i n [11]  (p. 349)  [48]).  on  F  f o r a l l x'e one r e a d i l y  i n f i n i t e - d i m e n s i o n a l analogue. g i v e n i n [11]  £  i s formulated only  We  omit  for real  F  128.  2.4  Theorem  Let  X  be a v e c t o r space.  ij) i s a p o s i t i v e - d e f i n i t e f u n c t i o n continuous  f o r every  F e F  ,  such  X  I^|F  with  i f f  t h e r e e x i s t s a unique f i n i t e X  on  c y l i n d e r measure  u  over  that I|J(X)  = / exp i Re f ( x ) d u  (f) f o r a l l x £ X  .  Remark  When we c a l l  u  and  i|; a r e r e l a t e d as i n the f o r e g o i n g theorem,  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 o f The f i n a l  continuity  theorem o f t h i s s e c t i o n  properties  Proposition  of c y l i n d e r measures.  II.2.6, i t f u r t h e r m o t i v a t e s  i s useful  [33]).  f o r determining  As an adjurxctto "continuous  II.2.5.1.  Theorem  Let over X  (Prohorov  the t e r m i n o l o g y  c y l i n d e r measure", i n t r o d u c e d i n D e f i n i t i o n s  2.5  u  X  with  X  , and  V  uV e V If  be a v e c t o r space, be a f a m i l y  f o r every  u  be a f i n i t e  of balanced,  absorbent  is  subsets  u > 0  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 u  c y l i n d e r measure  l/-continuous <=> <j J  1  S  of  u  , then  ^-continuous.  V  of  Corollary  Let  X  measure over  be a t o p o l o g i c a l space, and  p  be a f i n i t e  cylinder  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 p i s continuous <=>  of  p.  , then  iji i s c o n t i n u o u s .  Proofs 2  2.2.1  Taking follows  2.2.2  n = 1, x^ = 0  = 1  , the a s s e r t i o n  immediately from Defn. 2.1,  F o r any n e w , n-1 1  Z  From  {x^,..., _^}C X n  {z^,..., _ } ^ ® z  n  Z  0  1  T(X  1<  l<  [48] we have Cf(x) = Cp (-x)  " V  =  £  Z  E  k,£=0  k £ * x * k Z  ( T  k  X  " x £ T  Z  n - l  }  C  for a l l x e X , and  '  S  M = (cpCx, - x ) ) " }_ * k £ k, £—U n  i s a positive-definite  Hermitian matrix. Hence, t h e r e e x i s t s an M = TT* where  T = (t  ^- °  that  A  V \ '  }  X  Hence, by Defn. 2.1, f o r any ne u>, {x ,...,x .} c X 0 n-1 {  n  n-1  k £ * k  , and  x  _  k,£=0  2.2.3  , and  n x n-matrix  T  s.t.  ,  ) , T* = ( t * )  , and  t*  = I  ,  •  •  >  130.  Consequently, n-1  . I  _  V ^ k "V ^ k ~ V  n  . n-1  _  n-1  k,£=0  V **  n-1  n-1  1  E  1  s=0  k,£=0  {  H  s=0  W  and t h e r e f o r e  2.2.4  \ sK ?' k £ ' ''  ( E  K  '  S  new, _  K  (x  X  n  t  (x  l , S  K  } =  ^  y  in  X  1  ^^  1.4.1  2•3  This Analysis the  _  k£ V k "V Z  X  ( [ 3 4 ] , p. 19  -°  ( 4 ) , we have t h a t f o r any  (^(0) - if»(x - y ) )  x  .  1.4.3).  theorem i n Harmonic  However, i t i s r e a d i l y d e r i v e d ([4] p. 58).  of II.2.4 and I I . 2 . 6 , f o r each  ,let ijj (w) =  f exp i Re wzdu (z)  f o r every  We note t h a t (1)  J  theorem i s a s p e c i a l case of a g e n e r a l  Together w i t h the n o t a t i o n s x e X  rt  follows.  r e a l case t r e a t e d by Bochner  2.5  {z ..... z , } C U n-1  ,  | * ( x ) - i K y ) | 1 2<JJ(0) Re The a s s e r t i o n  Z  , and  r j e J k,£=0  From Rudin [ 3 4 ] , p. 18, and  1  {x^, . . . ,x ,}c X u n-1 n-1  . V* * k"£ k,£=0 . E  2.2.5  }  s k £ s^ k " V -° > )(z  K  (x _X  y  \p i s p o s i t i v e - d e f i n i t e .  For any n-1  b  i|> (w) =  IJJ(WX)  f o r every  z e (C  z e €  from  u  Suppose t h a t  is  (/-continuous.  Since  z e £ -> exp i Re z e (C i s bounded and c o n t i n u o u s , by (1) and Prop. II.2.6 we have (2)  <|) i s  [/-continuous I}J i s  Suppose t h a t definite'for  every  e > 0  (/-continuous. S i n c e ip/c  c > 0  *(0) = 1 Given any  assume t h a t  • , choose  ,  e  t > 2//e*  ,  V e f s.t. x  e V  =>  1  By (1) and the f a c t (3) If  i s positive-  , by Prop. 2.2.1 we may  o <« < f<^=h  and  that  -  IJJ(X) <  Re  that  V  e'  .  i s balanced,  x e V , z e <C , |z| <_ 1 => z = u^ + i u  t h e r e f o r e , by (4)  u  2  2  e ' <C  ., then  ( 3 ) , f o r any + u  2  i  12  |z|  x e V  1 - ^ (z) < e =  2  2 + u  2  T  .  , and  ,  <_ 1 => 1 - i|> (V) < e'  .  Hence, by the lemma g i v e n by Kolmogorov i n [ 1 7 ] , f o r any (5)  U ( D ) = y ({z e « : | z | > 1}) < ~ X  X  < e Since  e  was a r b i t r a r y , u is  i t follows  (/-continuous  that  ^  (e' + ~ ) l t 2  x e V  132.  Proof of C o r o l l a r y  We  note  2.5  that  nbnd 0  b a l a n c e d , absorbent  in  sets  V  X  has  , with  Hence, by the above theorem and Prop. /A i s c o n t i n u o u s  a base e.V £ V 2.2.5,  <=>  u  is  (/-continuous  <=>  iff i s  (/-continuous  <=>  ij>  continuous.  is  V  c o n s i s t i n g of  f o r every  E > 0  3.  CM-spaces.  For any f a m i l y space  X  C  of f i n i t e  , we s h a l l d e f i n e the  properties.  c y l i n d e r measures over a vector-  C-topology  on  finite We note  and  T : X -> Y  that' f o r s e t s  i s a t o p o l o g y on G  X  and  Y  , topology  G  on  Y  ,  : G e G}  X  and  .  We s h a l l  r e f e r t o i t as t h e - t o p o l o g y on  X  T  Definition  Let  X  be a v e c t o r space.  (1) . F o r any f a m i l y the  C  of f i n i t e  C-topology  base a l l s u b s e t s  V  V = x + e  for X  some is a  x e X  of  X  A  ,  having f o r a  with  y  , finite  whose topology i s the  X  X  {y e X : y (D ) < e} y  CM-space i f f X  Concerning  c y l i n d e r measures over  i s the t o p o l o g y on  yeH  (2)  by the f a m i l i e s of  , _1  3.1  spaces  c y l i n d e r measures.  '.{T [G]  induced by  and g i v e some of i t  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 v e c t o r  whose t o p o l o g i e s a r e e x a c t l y those determined continuous  X  H C C  , and  e > 0  i s a topological vector  space  CM(X)-topology.  C - t o p o l o g i e s we have the f o l l o w i n g  assertions.  134. 3•2  Propositions  Let  Y  be a vector space, and  c y l i n d e r measures over (1)  Y  C  be a family of f i n i t e  Y  i s a t o p o l o g i c a l vector space under the  C-topology,  which i s the coarsest such topology w i t h respect to which  C  is  a family of continuous c y l i n d e r measures. In p a r t i c u l a r , when and  C = CM(Y) the  (2)  Y  i s a t o p o l o g i c a l vector space,  ,  C-topology i s coarser than the o r i g i n a l topology of  For any vector space  X  and  T e L[X,Y]  Y  if  C o T = {y a T : y e C> • , then, the  C Q T-topology i s the topology on  C-topology and  induced by the  T  We s h a l l now show that the c l a s s of i n t e r e s t i n g t o p o l o g i c a l vector spaces. vector spaces are  X  CM-spaces.  Banach space which i s not a  CM-spaces contains many  However, not a l l t o p o l o g i c a l  In Appendix 4 we give an example of a CM-space (Example  4.2).  135.  3.3  Definitions  (1)  Let  X  be a v e c t o r space  b  : X -> R"*~  i s a pseudo-quasi-norm on  b(0) = 0 f o r any  x  and  y  in  X i f f  , X  ,  b(-x) = b(x)  ,  b(x + y) < b ( x ) + b ( y )  ,  and z e € -»- b ( z x ) e R~*~  (For any f a m i l y as i n Y o s i d a  {b.}.  j e J  (2)  0 E. (D  of pseudo-quasi-norm on  [49] p. 31, one can show t h a t  space under the c o a r s e s t every  T  i s continuous at  t o p o l o g y on  X  X  X  ,  i s a topological vector  making  b^  continuous f o r  ).  F o r any measure space  (A,n)  , and  r,. „A „ . L (A,n) — {f G £ '• f i s  b and when  ,  , , r i ,-1 r n -measurable, / |f| dr\ < }' , 00  : f e i (A, ) ^ / |f! dn e R r  r  r > 0  r  +  n  ,  r >_ 1 , | -I  : f e L (A,n) - (/ | f . | d ) r  r  n  1 / r  e R  +  .  Remarks  When  0 < r < 1  ,  b^  i s a pseudo-quasi-norm on  L (A,n) 1  which i s t h e r e f o r e a t o p o l o g i c a l v e c t o r space under the c o a r s e s t making  b^  continuous.  ,  topology  136.  When is |'|  r >_ 1 , |•|  therefore a l o c a l l y  i s a pseudo-norm on  L~(A,n)  > which  convex space under the c o a r s e s t topology making  continuous. We s h a l l h e r e a f t e r assume t h a t  L (A,n) , r > 0  , carries.the  a p p r o p r i a t e t o p o l o g y i n d i c a t e d by the f o r e g o i n g o b s e r v a t i o n s .  We s h a l l need the f o l l o w i n g lemmas, which are. o f independent  interest.  3.4  Lemmas  (1)  Let X on  X  .  be a v e c t o r space, and  If  X  IIJ continuous  making  X  such  that  coincides with  the c o a r s e s t topology making  b  a f i n i t e c y l i n d e r measure  y  over  X  i s the c o a r s e s t t o p o l o g y on  is  c o n t i n u o u s , then, t h e r e e x i s t s X  whose c h a r a c t e r i s t i c  function  , and, the  {y}-topology  r e s p e c t to which Let X  b  on  be a v e c t o r space.  X  X  isa  continuous Let  If  v  T7  I|J  on  X  V e I/, t h e r e e x i s t s  satisfying  the h y p o t h e s i s  then,  CM-space under the c o a r s e s t t o p o l o g y making  f o r each (A,n)  b^  V cV  be a measure space.  f e L (A,n) r  with  {b ,} .. i s a f a m i l y of V Vey  such t h a t f o r each  a positive-definite function g i v e n i n (1) above,  X  i s continuous.  pseudo-quasi-norms on  (3)  be a pseudo-quasi-norm  \p is a p o s i t i v e - d e f i n i t e f u n c t i o n on  the c o a r s e s t t o p o l o g y on  (2)  b  F o r any  exp - b ( f ) e £  i s a p o s i t i v e - d e f i n i t e f u n c t i o n on  L (A,n) r  0 < r <_ 2  ,  137.  The f o l l o w i n g  theorem and i t s c o r o l l a r i e s i n d i c a t e  t h a t many  of the t o p o l o g i c a l v e c t o r spaces c o n s i d e r e d i n t h i s paper a r e i n f a c t CM-spaces.  3.5  Theorem  X  is a  having a family following  CM-space whenever f  X  o f neighbourhoods  i s a topological vector  of  0  space  which s a t i s f i e s the  conditions:  (i)  {eV : V e V, c > 0}  (ii)  For each  i s a base f o r  v  in  X  V e C  , t h e r e e x i s t s a measure space  , and  T  r  0 < r  nbnd 0  <_ 2  y  e L[X,L  V  (^,r\ )]  ,.such  . (A^,!"^)  ,  that  r  V = {x e X : J | T ( x ) | v  d n < 1)  .  v  Corollaries.  (1)  Let  (A,n)  be a measure space.  For any  0 < r <_ 2  ,  IT  L (A,ri)  is a  CM-space.  In p a r t i c u l a r , (A) (2)  (3)  Let  is a  CM-space.  X  be a t o p o l o g i c a l v e c t o r space.  X  w i t h the  S -topology i s a  Every H i l b e r t i a n space i s ' a  For any  CM-space.  CM-space.  0 < r <_ 2  ,  138.  Proofs 3  Notation  For  any v e c t o r space  {(x,f)  X , e > 0 , y e X  x*  Ke)  =  D  = (f e F j = |f(y)| >  y > c  e X x  and f o r any f a m i l y V(C,E)  =  H  C  :  UEC  Remark  e}  ,  ,  of f i n i t e c y l i n d e r measures over e X  {X  : | f ( x ) | >_  ,  p  (D  X  ) X  ,  X  e}  <  £  Since D  = D  x, e  for a l l x e X  x  e > 0  and  ,  e it  follows  that  V(C,e) = E A yeC  3.2,1  {x e X : y (D ) < e} x  We s h a l l o n l y prove f o l l o w s immediately  x  the f i r s t  assertion.  from the d e f i n i t i o n .  V = {V(H,E) : H c  C  The second  Let  is finite,  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 that  1/ has the f o l l o w i n g  (i)  0 e V  (ii)  \J i s a For  each  f o r every  then  i f we show  properties.  V e 1/  filterbase. Vet/  ,  (iii)  there e x i s t s  U e l'  (iv)  V  i s absorbent,  (v)  V  i s balanced.  s.t.U+ U c V  .  (Treves [47] p. 21)  139.  P r o o f s o f ( i ) - (v)  (i)  and  F o r any  therefore  (ii)  e > 0  ,  0 e V  F o r any  f o r every  0 < 6  y (D J y y>o  <  E  , y e C  > y (D ) - y y,e  Hence, i f V(e_.,C'J  V e V , and  y e X  ,  , and  e = min { e ^ e . ^  .  e V , j = 0,1  '  then. V ( , C „ U C\) C e  0  (iii)  Let  1  2  V = V(e,fO  For any  H V(e.,C.) j-0,1 , and  U = V(e/2,H)  x e X , y e Y  |f(x)  + f(y)|  .  2  , and  < |f(x)|  .  f e F , , (x,y)  +  |f(y)|  (II.2.6),  ,  and t h e r e f o r e , I  x + y  ( e ) C I (e/2) U  Consequently, y  x+y  < y  I (e/2)  x  f o r any  t  x e U , y e U  . , and  y e H  ,  (D ) = y^ (I (e)) x+y,e F, . x+y (x,y) (I (e/2)) + y  F  x  (x,y)  (I (e/2))  p  (x,y)  = y (D , ) + y (D . ) < e x x,e/2 y y,e/2  .  i . e . . U + U C V (iv)  F o r any  .  x e X , e > 0 , t > 0  and 0 < u < t => D , C x,e/u  B , x,e/t  , and  y e C  ,  140.  Consequently, s i n c e lim rv-K° = y  y  y^  i s finite,  , (D . ) = lim x/n x/n,e n-x"  ( new  D  A  x,ne  Hence, f o r any  y (D ) x x,ne  ) = y (0) = 0 x  .  V e 1/ , t h e r e e x i s t s  new  s.t.  and  z e C  x/n e V i.e.  V  (v)  i s absorbent. F o r any  with  y e C , x e X , e > 0  | z | <_ 1 y zx  -  (D  ,  ) = y (D, , ) = y (D , .,) zx,e x |z|x,e x x,e/|z|  < y (D ) — x x,e  since  Hence, f o r any zV C V  3.2.2.  D  V e 1/  , > .C D x,e/|z| x,e ,  .  As i n Lemma I I . 4 . 2 , f o r any  y e C  T  _ 1  e > 0  [A  and f i n i t e  ' (y e Y  yeH =  =  C\ yeh  A  {x e X : y  (D x  {x e X :  follows that  neighbourhoods  : y (D ' y  yeH  3.2.1.  x e X , e > 0  , and .  ,  Hence, f o r any  It  .  y  - 1  of  0  in  from Prop.  C  ,  ) < e}] £  »  (y a T) (D '  ) < e}  X  .  C  : V e V} i s a base f o r the X  , where  V  of  0  in  3.2.. 1 and the l i n e a r i t y  Y  of  C a T-topology  i s as d e f i n e d i n P r o o f  However, from P r o o f 3.2.1 we see t h a t  C-topology neighbourhoods  H of  ) < e)  T  x  '{T [V]  subfamily  \J i s a base  The a s s e r t i o n now T  f o r the follows  Lemma  Let quasi-norm on  F  be a f i n i t e - d i m e n s i o n a l space.  F  , then  b  Let K  be a b a s i s o f F  P r o o f o f Lemma. Every  x e F  i s continuous  has a unique  on  If b  i s a pseudo-  F  representation  £ z (x)a , aeK a  and  the norm x e F ->  generates  x. -> 0 => 3  z (x.) -> 0 a J a  F  net (x.). _ J J £J  E aeK  in F  a  3  f o r each  a e K  f o r each  => a e K  =>  E b ( z (x.)a) -> 0 => b ( x . ) -> 0 aeK 3  J  y  b  in F  , since  3  b(x.)  Hence  ,  I z (x.) I -> 0 =>  (x.)a) -> 0 3  a  <  E b ( z (x.)a) aeK  i s continuous  .  J  at  0 e F  .  However, f o r any  x and  , |b(x) - b (y) | £ b ( x - y)  and  therefore continuity of  of  b  on  +  a•  the t o p o l o g y o f  F o r any  b(z  I z (x) I e R  E „ aeK  F  b  at  , 0 e F  implies continuity  142.  3.4.1  By the above Lemma, 4)|F and  i s continuous f o r every  F e F  ,  t h e r e f o r e , by Thm. 2.4, t h e r e i s a c y l i n d e r measure  X  whose c h a r a c t e r i s t i c  X  i s a t o p o l o g i c a l v e c t o r space under the c o a r s e s t t o p o l o g y  making  f u n c t i o n a l i s ip .  y  continuous.  the  over  From t h e h y p o t h e s i s ,  Hence, by Cor. 2.5 and Prop.  3.1.1,  {y}-topology  =  c o a r s e s t t o p o l o g y making  y  =  c o a r s e s t t o p o l o g y making  \p c o n t i n u o u s  =  c o a r s e s t t o p o l o g y making  b  3.4.2  By Prop.  3.1.1 and Lemma  3.4.3  L e t a : B E M •> a E B n •B For any  P e P(M )  d(P) by  continuous.  3.4.1.  .  let  be the f a m i l y o f f i n i t e  s u b s e t s of  P  directed  inclusion.  Then, f o r any b  ,  continuous  (f) = r  f E L (A,n)  lim PeP(M )  n  Consequently, t E R  , E |f(a )| *nCB) BeK  lim Ked(P)  r  .  since exp - t £ R  i s continuous,  we have t h a t i 1/r ,r lim II exp - |n(B) f (ct ) | Q£d(P) B E Q r 1/r f E L (A,n) -> n(B) ( jj) i s l i n e a r f o r every , we deduce from Lemma 1.1.1 and Props. 2.2.2 - 2.2.4  exp - b ( f ) = r  lim PeP(M ) n  Since B £ M that  f  f E L (A,n) 1  a  exp - b ( f ) E £  i s positive-definite.  143,  3.5  F o r each  V e f ,let =  b  v  1  when  r^ <1  1/r^  when  r^ >1  : x e X ->• (/|T (x) |  d ^  v  and \J>  : x e X  For each by  exp - b (x) e C  V e 1/  , we have t h a t  i s a pseudo-quasi-norm on  and f o r every  t > 0  X  ,  , r  v V e  tV = {x e X : b ( x ) <_ t  }  Since the topology o f  X  i s c o m p l e t e l y determined by  i t s neighbourhoods o f  0  , i t f o l l o w s t h a t the  topology o f continuous ifj  X  i s t h e c o a r s e s t t o p o l o g y making V e V  f o r every  i spositive-definite,  .  b^  However, by Lemma 3.4.3,  and s i n c e  t e R  +  •+ exp - t e (0,1]  i s a homeomorphism, i t f o l l o w s t h a t  = The  Corollaries  the c o a r s e s t topology on  X  making  ^  continuous  the c o a r s e s t topology on  X  making  b^  continuous.  theorem i s now a consequence o f Lemma 3.4.2.  (1) and (2) are. immediate consequences o f the theorem.  Proof of C o r o l l a r y space  (3)  Recalling  the d e f i n i t i o n of a H i l b e r t i a n  (§111.2), we need o n l y make the f o l l o w i n g o b s e r v a t i o n . Let  [.,.]  X  on  X  be a v e c t o r space.  , t h e r e e x i s t s a measure space  index s e t and n  i s c o u n t i n g measure on  such t h a t  A  (A,n) (A ), and  0  [x,x] = / (Treves  F o r any p s e u d o - i n n e r - p r o d u c t  |T(x)| dn i  [47], p. 115 - 116.)  for a l l x e X  .  i s an  2 T e L[X,L (A,n) ] ,  144.  4.  Examples  4.1  Example  There e x i s t s a Banach space u  over  X  such y is  Proof  Let  X  and f i n i t e  cylinder  measure  that S^-continuous  A  but i s n o t  be a s e t .  Together  w i t h N o t a t i o n 1.1, l e t  X = £^~(A)  w i t h the u s u a l topology  [•',.] : ( x , y ) e X x X +  4  K.  f  (Notation I I I . 4 ) ,  Z x(a)y(a) e £ aeA  : x e X -»- exp - [x,x] e C K C A  For any f i n i t e T  E-tight.  K ; w e £ -> f  (x) =  w  , K * e (C )  , where  K for a l l x e C  Z x(a)w(a) aeK  Since [x,x] =  Z aeA  |x(a)|  2  for a l l  x e £ (A) 1  ,  as i n the p r o o f o f Lemma 3.4.3, we deduce t h a t  \p i s a  p o s i t i v e - d e f i n i t e function  x e X ->- /[x,x]  i s a norm on continuous By Thm.  £^(A)  f o r every  2.4,  on  , we f u r t h e r  , and  .  Since  deduce t h a t  t h e r e e x i s t s a c y l i n d e r measure  K x e C  \{J|F i s  F e r  whose c h a r a c t e r i s t i c f u n c t i o n K C A  £~^(A)  is  \\>  .  Then,  , as i n P r o o f 1.2.2,  y  over f o r any  £^~(A) finite  / exp i Re f ( x ) d ( 2 Y  o T ^ ) (f) .= <Kx)  / exp i Re f ( x ) d u ( f ) = / exp i Re f ( x ) d y (Note t h a t  T  (f) .  i s a homeomorphism, and t h e r e f o r e  y  0  r  i s Radon.) Hence, by Thm. 2.3,  (1)  V-Y^,!-  Consequently,  1  •  f o r any  t > 0  , w i t h the n o t a t i o n  of P r o o f s  III.4,  y K(r [tvJ]) = y^KCtO^n c  =  Y^Qw  e  c  K  :  S U  P  aeK =  £ )°)' K  R  n aeK  !  w  ( a ) |  £  t})  / e (w(a))dX(w(a)) | / \ i |w(a)I<t 2  However, /  6 (z)dX(z) < 1  | z|<t since  ,  2  '  / 6 (z)dA(z) = 1  and  2  (Lemmas 1.1). inf finite  I t follows  that  y K(r [tvJ]) = 0 £  i s strictly positive  ,  K  K c A  and t h e r e f o r e , by Lemma 1.5.1.2,  .y  On the o t h e r hand, by P i e t s c h there e x i s t s  S_ e1, u  E-tight.  [30] p. 82, Prop. 4,  s.t.  [x,x] <_/| f ( x ) | d n ( f ) and  cannot be  for a l l x e X  ,  therefore x e X ->- [x,x] e R  Hence  <JJ i s  y  S^~-continuous.  is  +  i s .^-continuous.  S^-continuous.  Consequently, by Thm. 2.5,  4.2  Example  Let If  A  be a s e t ,  2 < p < i s such t h a t  y e £ (A) q  E aeA  , and  00  |y( )| |ln|y(a)||  1 P  q  =1  = -  q  a  and T : x e £ (A)  (x(a)y(o))  P  then t h e r e e x i s t s  e £ (A) , q  y e CM(£ (A))  such  q  that  y p T  i s not  E-tight.  N o t a t i o n . Together w i t h the n o t a t i o n s o f any  t _> 0, ,  §2.1 and P r o o f s I I I . 4 ,  for  let b(t)  = t |lnt|,  t > 0 , and  q  b(0) = 0  .  We s h a l l need the f o l l o w i n g lemma ([41] Lemma2].)  Lemma.  Let w : A -> <L w i t h  |w(a)|  1  for a l l a e A  There e x i s t s a c o n s t a n t  0 < C < <» such t h a t  for  ,  K Y  every f i n i t e  ({z  e £  P  Proof o f Lemma  K  :  K  A  E |z(a)w( )| aeK  q  a  L e t 6p  > 1}) >_ e " - exp 1  be the f u n c t i o n of Lemma 1.1.1.  From [3] p. 263, 0 <  lim |v|-*°  |v|  E b(|w(a)|) aeK  q + 2  6 (v) p  <  147.  Hence, t h e r e e x i s t s (1)  0 < C' <; «>  0 (v) > C 7 | v | ^  for a l l v E C  + 2  By T a y l o r ' s theorem,  s.t.  f o r any  0 < t < 1  1 - exp - t = t exp - t '  Ivl  with  > 1  ,  f o r some  0 <_ t ' < 1  and t h e r e f o r e (2)  1 - exp - t >_ t e  for a l l  1  0 £ t <_ 1  .  Let (3)  C = 27re C'  .  _1  Then, f o r each (4)  f(l  >_ e "  a e A  (v)dA(v)  q  a  /|vw( ) | 6  1  ,  - exp -- | v w ( ) | ) 0 (v)dA(v)  q  a  by ( 2 ) ,  0<_| w ( a ) | <1  ie'V  /|w( )|  •  q  a  l£|v|<_ l/|w(a) | = 2 e~ C K*»lfy 1  q + 2  ' ' V  1/p  ,  7r  dA(v) by (1)  1  dp  l<p<_l/|w(a) | = C b(|w( )|)  by ( 3 ) .  a  For any f i n i t e B  R  = {z  K C A C  e  :  K  ,if  Z |z( )w( )| aeK a  > 1}  q  a  ,  then, Y D V (  l  /P 1  P  = / ( l - exp -  - / l  (  z  K Z aeK  (  1  "  > / ( I - exp -  1  -  II f aeK  Z aeK  6  X  "  P  z  a  K Y  q  Y  aeK q  a  l (a)w(a)| )d ^(z)  S  |z( )w( )| )d  P  (z)  P  ( z ) ( l - exp £ ~B  = e  )  Z aeK  |z(a)w(a)| )d (z) q  K  Y  P  |z(a)w(a)| )d (.z) q  K  Y  - (1 -  P  exp - | z(a)w(a) | d  (z(a))  q  Y P  .  .  e" ) 1  ,  148.  However, f o r each  a e K  ,  / exp - | z ( a ) w ( a ) | d ( z ( a ) ) = 1 - / ( l - exp - | z ( a ) w ( a ) | ) d ( z ( a ) ) q  q  Y p  <_ 1 - C b ( w ( ) )  Y ?  by (4) above.  a  Hence, Y  K  0O  P  K  > e"  - n aeK  1  _  > e ^  [1 - Cb(w(a))]  II exp - Cb(w(a)) aeK  -1  exp - C  P r o o f o f Example  since  1 - u < e  fora l l  U  u > 0 . ,  E b(w(a)) aeK  4.2  If h  then  : x e £. (A) -> exp q  h  i s continuous.  P r o o f s 3., for  h  £ aeA  |x(a)|  q  e £  By Lemma 3.4.3 and the lemma o f  i s p o s i t i v e d e f i n i t e and  every f i n i t e  ,  d i m e n s i o n a l subspace  h|F  F  of  i s continuous £ (A) q  .  Hence, by Thms. 2.4 and 2.5, (1)  t h e r e e x i s t s a continuous  y over  £ (A)  finite  cylinder  with c h a r a c t e r i s t i c functional  q  measure h  Clearly, (2)  y e CM(£ (A))  .  q  Choose (3)  t > 0  s . t . | y ( a ) | / t <_ 1  Let  0 < 6 < e"  .  1  For any f i n i t e s u b f a m i l y K K K * h : z e € h (z) e (€ ) K  li (z)(x) K  K  =  E aeK  for a l l a e A  x(a)z(a)  of  A  ,  let  , with fora l l  x e C  K  .  149,  then, as i n e a r l i e r p r o o f s  u  (4)  - Yp  o  \  (Proof  1.2.2 ( 1 ) , Proof  4.1(1),  •  Hence, f o r any f i n i t e  Kc A  s . t . a e K => y ( a ) ^ 0  ,  (y a I) ( ( ( C ) * - t ( V rv £ )°) K  K  K  £  .  K  = y „((£ ) K  £  P  ~ t T * " " ^ . r\ C ) ° )  A  1  <E  K  that  £  = yh{z  = T[£ ]  K  ;  K  e £  :  K  P  by Lemma 0.4.2, and the f a c t  K  P  K  Z | ^ z ( a ) | aeK  q  >_ e ^ - exp - C E b ( | y ( a ) | / t ) aeK  > 1})  by the Lemma.  Now, £ b ( | y ( a ) | / t ) = °° f o r any aeA Therefore e"  1  there e x i s t s f i n i t e  t > 0  J c A s . t .  - exp - C E b ( | y ( a ) | / t ) > 6 ae J  , aeJ=>  y ( a ) =j= 0, and  .  Hence, (y G T) ( ( £ ) J  - t(V A  A  T  • £ Since  J  t > sup aeA  and w i t h  |y(a)[  it  p  .  , and  p  £ )° J  0 < <5 < e ^  were a r b i t r a r y ,  111,4,  ,  f o l l o w s from Lemma 1.5.1.2 t h a t y • T  4.3  J  the n o t a t i o n o f P r o o f s  r tV °) = t ( V A j(  C )°) > 6  P  i s not E-tight.  Example  There e x i s t s a Banach space which i s n o t a C M-space.  Proof.  Let CQ = {x e € I I  |  : x e c„ -> sup  *  £  : lira x ( n ) = 0 }  =  £  . .,  x(n)  e R  , +  ,  (CO)  and  2 T : £  CQ  be the c a n o n i c a l imbedding  As i s w e l l known, generated  c^  by the norm  i s a Banach space under the t o p o l o g y | "|  From P i e t s c h [ 3 0 ] p. 8 3 , Remark 2.2, (1)  T  i s not  S^-continuous.  From Kwapien [ 1 9 ] we have t h a t 2 u e O I ( C Q ) => y D T and  has a l i m i t measure on  Q  Hence, by Prop.  on  ,  t h e r e f o r e , by Cor. 1 . 4 . 3 , u e CM(c ) => y Q T  (2)  (£ ) '  T c  Q  is  is  E-tight.  III.1.3,  S"*"-continuous w i t h r e s p e c t to the  CM(cQ)-topology  .  From ( 1 ) and ( 2 ) i t f o l l o w s t h a t the  CM(CQ)-topology  does n o t  c o i n c i d e w i t h the norm t o p o l o g y , i . e . CQ i s not a CM-space.  4.4  Example  For any  r >_ 4  , the  2 coincide with  the  S  -topology.  r S - t o p o l o g y on  2 £ (w)  does n o t  oof  2 2 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  S -and  that the  For each n  S -topologies do not coincide.  new {n}  a = n n  , let  -2/r  ,  r  2 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 e x i s t a * 2 * w -Radon measure n on (£ (w)) w i t h supp n e E , s . t . (/|f(x)| dn(x)) 2  < 1 => |Tx|  1 / 2  < 1  r  .  Hence, l  T  x  l  1 /|f(x)| d  for a l l  2  r  Consequently, f o r any  I a = n<k  E n<k  2  n  .  2  k e w  |TeJ  <_ f ( sup |x| ll  x e £ (w)  2  </  E | f( e ^ | d n<k 2  |f(x)|) dn(f) 2  n  (f)  ,  2  since  {e } n new  £ (w)  . Since  supp n e E , and  that v L  new  i s a orthonormal basis of the H i l b e r t space  2  a < n  00  k e w  was a r b i t r a r y , i t follows  152,  However, a  this i s impossible,  2 - 4 / r ^ -1 = n > n n —  since ,  Y, 1/n  , and  new  Hence m  T  •  c  i s not  2 . o -continuous,  Remark.  In view o f Theorem I I I . see t h a t f o r every I (to)  r >_ 4 i s not a  2.6.3, from the above example we  , CM-space.  153.  BIBLIOGRAPHY  1.  A. B a d r i k i a n : Remarques sur l e s theoremes de Bochner et P. Levy. 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