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A Representation theorem for measures on infinite dimensional spaces Harpain, Franz Peter Edward 1968

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A REPRESENTATION THEOREM FOR  MEASURES  ON I N F I N I T E DIMENSIONAL SPACES  by  FRANZ PETER EDWARD HARPAIN B.Sc,  University  of B r i t i s h  Columbia,  1965  A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in  t h e Department MATHEMATICS  We a c c e p t required  THE  this  thesis  as conforming t o the  standard  UNIVERSITY OF B R I T I S H September,  1968  COLUMBIA  In p r e s e n t i n g t h i s t h e s i s  i n . p a r t i a l f u l f i l m e n t of the requirements  f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree  t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and  Study.  I further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of  this  t h e s i s 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 or by h.ils r e p r e s e n t a t i v e s .  It  i s understood t h a t  copying  or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d  w i t h o u t my w r i t t e n p e r m i s s i o n .  Department of  i ABSTRACT  In  t h i s paper we o b t a i n a g e n e r a l i z a t i o n of the  w e l l known R i e s z R e p r e s e n t a t i o n Theorem t o the case where the u n d e r l y i n g space duct of l o c a l l y spaces.  X  Is an i n f i n i t e dimensional p r o -  compact, r e g u l a r and  a-compact  topological  In the process we prove t h a t our measures on  X  correspond t o p r o j e c t i v e l i m i t measures o f p r o j e c t i v e s y s tems of r e g u l a r  B o r e l measures on the c o o r d i n a t e spaces.  An example i s g i v e n t o show that of  the c o o r d i n a t e spaces i s necessary.  cr-compactness  ii  TABLE OP  CONTENTS  • >  Page 0.  Introduction  •. •  1.  General  2.  The f a m i l y  3.  The f a m i l y  4.  The R e p r e s e n t a t i o n  5.  Example  25  6.  Bibliography  27  Notation  J-  i  of c y l i n d e r s M  1  o f measures Theorem  7 10 18  A CKNOWLEDGEMENTS  I am deeply indebted to Dr. Maurice S i o n ,  who  suggested the t o p i c and except f o r whose u n f a i l i n g help t h i s t h e s i s would never have come t o be.  I am a l s o most  g r a t e f u l f o r the f i n a n c i a l a s s i s t a n c e g i v e n by the partment thesis.  De-  of Mathematics throughout the w r i t i n g of t h i s  1  1.  Introduction  If  X  i s a l o c a l l y compact, r e g u l a r t o p o l o g i c a l  space, then the w e l l known R i e s z R e p r e s e n t a t i o n Theorem s e t s up an isomorphism  between the f a m i l y of a l l bounded  Radon outer measures on  X  and the s e t of continuous  posi-  t i v e l i n e a r f u n c t i o n a l s on the f a m i l y of continuous f u n c t i o n s w i t h compact support i n  X  c o r r e s p o n d i n g elements,  a l i n e a r f u n c t i o n a l and  I  .  a measure, s a t i s f y the r e l a t i o n s h i p all  continuous f u n c t i o n s  f  In t h i s  isomorphism  i(f) = J*f d u  u  for  w i t h compact support i n  X .  I f we now c o n s i d e r an i n f i n i t e product of l o c a l l y compact, r e g u l a r spaces, then t h i s i s i n g e n e r a l no longer l o c a l l y compact with r e s p e c t to the product topology, and the R i e s z R e p r e s e n t a t i o n Theorem f a i l s to h o l d .  Iri t h i s paper we o b t a i n a r e p r e s e n t a t i o n theorem  2  f o r t h i s case by r e p l a c i n g the v a r i o u s f a m i l i e s mentioned above by the f o l l o w i n g : (i)  A family  "6a  of c y l i n d e r s whose elements act  compact s e t s f o r a "pseudo-topology" ^ c l o s e d under f i n i t e i n t e r s e c t i o n s and i s a subset (ii)  of the product  A family  -(£ and  M  , where  like  is  countable unions  and  topology.  of bounded outer measures, r e l a t e d to  i n much the same way  as bounded  Radc'n outer  measures are r e l a t e d to compact and open s e t s . (iii)  A family  JF  of f u n c t i o n s depending o n l y on a  f i n i t e number of c o o r d i n a t e s , w i t h r e s p e c t to which they are continuous and have compact (iv)  L  A family  l i n e a r span of  of p o s i t i v e l i n e a r functions//on  way  the  .  Under the added hypothesis te spaces we  support.  show that  t h a t corresponding  IJ  and  a-compactness of the JM  l(f)  coordina-  are isomorphic i n such a  elements,  s a t i s f y the r e l a t i o n s h i p Moreover we  of  l  in  = Jfdu  JL  u  and  for a l l  show t h a t the elements of  f  i n JM , in  M  F .  can  be  viewed as the p r o j e c t i v e l i m i t measures of p r o j e c t i v e systems of bounded r e g u l a r B o r e l measures. Prom the i n t e g r a b i l i t y of the members of  , it  f o l l o w s t h a t a l l bounded B o r e l f u n c t i o n s which depend o n l y on a f i n i t e number of coordinates the simple  f u n c t i o n s used by S i l o v  are a l s o i n t e g r a b l e .  Thus  [7] and the tame f u n c t i o n s  3  used by Segal [6] and Gross  [2] i n the development of an i n -  t e g r a t i o n t h e o r y on H i l b e r t space are i n c l u d e d tegrable  functions  among the i n -  of the measures c o n s i d e r e d here.  (Por a  good guide t o the l i t e r a t u r e i n t h i s area see the b i b l i o graphy i n Gross characterize  [3].).  Our r e s u l t s t h e r e f o r e  not o n l y  an important c l a s s o f l i n e a r f u n c t i o n a l s i n  terms of p r o j e c t i v e l i m i t s of r e g u l a r B o r e l measures but a l s o enable us t o extend these f u n c t i o n a l s t o a much wider class of functions  through a standard i n t e g r a l o f a measure.  Thus the standard theory o f i n t e g r a t i o n becomes a p p l i c a b l e in this situation.  4  1.  General N o t a t i o n . .1  0  .2  ID i s the s e t of n a t u r a l numbers.  .3  1  Is the s e t of r e a l numbers.  .4  t*  i s a compact f a m i l y i f f f o r every s u b f a m i l y  i s the empty s e t .  of  ^  , i f the i n t e r s e c t i o n of any f i n i t e  number of members o f Jr  i s non-void, then the  i n t e r s e c t i o n o f a l l members of . A i s non-void. •5  For  f  a f u n c t i o n on  X  to  R  .1  f|A  i s the r e s t r i c t i o n of  .2  1^  .3  Hffoo = sup [ | f ( x ) | : x e X} ,  A  f ( x ) = max ( 0 , f ( x ) )  •5  support f = c l o s u r e  A c X  and  f  to  A,  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 of  +  A ,  for x e X ,  (x : f ( x ) > 0}  i f X is  a t o p o l o g i c a l space. .6  I f f o r n e u) , a f u n c t i o n on .1  a . f  .2  a  X  n  to  i s a set, a  n  e 1 , f  isa  R , then  a  i f fa  c  f a  i f fa  <. a  a  and ,  and  U a new lim a  = a , = a ,  new  .3  f  n  f f  and .7  For i  I  e I ,  i f ff o ra l l x e X , f ( x )< f n  R + 1  (x)  l i m f (x) = f ( x ) . neu) an index s e t and "ff X. = [x : x i e l  X^  a s e t f o r each  i s a f u n c t i o n on  I  with  5  x  .8  i  ^ i  6  u  f  o  r  a  that  i e 1}  h  on t h e f a m i l y  u(0) = 0  whenever  For  A c  yfl  and |J new  = {A  of a l l subsets of  A c X  A c B}  .  X ,  and f o r e v e r y  is  ,  X  -^J  X , ^ c ffl^  u ( A ) = i n f {u(B) : B  i f f^  i s a family  T(A)  > 0  f o r every  u(B)  = i n f { £ T(A) : X Aejj  and  B e  Por  X  U A} AeH  is  u-  B cr X ,  A e  ^ c  X  .1  open s e t s  .2  i f  X  is a  , and f o r e  ^  and  g e n e r a t e d by  , and f o r *i , Q  X  such are  i f f u  M i s countable  M s  is a  a  Radon  Caratheodory  that u-measurable,  i s compact t h e n  X ,  B e x  .  o u t e r measure on  C  such  of subsets of  a t o p o l o g i c a l space,  measure on  A  i f f u  i s t h e C a r a t h e o d o r y measure on  T ' and  X  u-measurable} .  - o u t e r measure on  every  is a  c X .  C a r a t h e o d o r y measure on  .12  i f f u  A) + n ( B - A) .  : A  |i i s a  u  X  0 < u(A) < E |i(B^.) < oo - new  i f f A c X  u ( B ) = |i(B A  .11  .  | i a C a r a t h e o d o r y measure on  measurable  .10  c  i s a C a r a t h e o d o r y measure on  function  .9  e  u ( C ) < oo  ,  6  .3  i f  A  i s open t h e n  compact, A  i f B  .13  B e x  c  A}  X  For  u ( B ) = i n f [u(A) : A  then  a t o p o l o g i c a l space,  on t h e f a m i l y  of closed  = sup { T ( C ) for  A  : C  the f a m i l y  .1  The T  X  on  mention here  and u  is T  g e n e r a t e d by  #  X .  two w e l l known f a c t s  measures:  C a r a t h e o d o r y measure and  X ,  X , and  o f open s u b s e t s o f  Caratheodory  topolo-  i s a function  compact  a n open s u b s e t o f  and  about  i s the  i f f T  i s closed  C a r a t h e o d o r y measure  We  T  n  compact s u b s e t s o f  the  Remarks.  i s open,  .  c r a n k e d by  CcA}  is  ,  g i c a l measure  T^.(A)  .14  C c A}  u ( A ) = sup ( u ( C ) : C  i s i n fact  X  on  g e n e r a t e d by  a C a r a t h e o d o r y measure  on  X ,  .2  X  If is  i s locally  a f u n c t i o n on t h e f a m i l y  subsets of and <  if  compact and r e g u l a r  X  compact we  T(A)  +  T(B)  A f) B = 0  c r a n k e d by  T  of closed  such that f o r  A , B  have  <  <  00  0 and  <  T(A)  T(AUB)  T  and  compact closed  T(AUB) =  T(A)  +  T(B)  , t h e n t h e t o p o l o g i c a l measure is a  (See f o r example  Radon o u t e r measure  Sion [8])  on  X .  7  2.  The f a m i l y  "6  of c y l i n d e r s .  Throughout t h i s paper we suppose t h a t index s e t and t h a t f o r each pact,  a-compact and r e g u l a r t o p o l o g i c a l  2.1  Definitions.  .1  X = Tf t e T  .2  t e T , Y^.  I  Y  T  i s any  i s a l o c a l l y comspace.  . z  i s the s e t of non-void f i n i t e subsets of  T , ordered  by i n c l u s i o n . Por .3  i , j e I with i c j X. = TT +equipped w i t h the product topology te i Y  i  1  s  z  (which i s l o c a l l y compact, a-compact and r e g u l a r ) , A  K  .5  TT^ ( r e s p e c t i v e l y  i  i s the f a m i l y of c l o s e d ^j_j)  X (respectively .6  For  A c X  i  s  X j ) onto  t  compact subsets of n  e  canonical  t  ,  X^ ,  , c y l A = TT" [ A ] . 1  i  and  and  ^(t}  2.2  Definitions.  .1  ^  =  i  p r o j e c t i o n of  I f no c o n f u s i o n i s p o s s i b l e we w i l l f o r identify  X  (t} ,  Y  and  t  X^j .  Thus  t e T  Y =Xj- j=X t  t  *  = ( a : there e x i s t s  a = c y l /3).  Thus  which f o r some  ^  i e I  i e I  and  0 e  with  i s the f a m i l y o f c y l i n d e r  sets  have a compact base i n X . .  fc  8 .2  i s the c l o s u r e under f i n i t e  i n t e r s e c t i o n s of the  f a m i l y of complements of s e t s i n .3  "^f  i s the c l o s u r e of  •  under countable u n i o n s .  The e s s e n t i a l p r o p e r t i e s of 2.3  Theorem:  ^  2.4  Corollary:  The c l o s u r e of  are the f o l l o w i n g :  i s a compact f a m i l y .  under f i n i t e unions i s a  compact f a m i l y .  2.5  Corollary:  with a  c n  a c  y  B 0  n  If  U B  a e "(£ and f o r each  then there e x i s t s  new,  New  B  e n  such t h a t  '  Proofs Proof of 2 . 3  Let X  be any s u b f a m i l y of  t h a t f o r any non-void f i n i t e B c we have . Por each a E J l l e t i e I be such t h a t for  some  such  C\ a ji 0 . aeB a = c y l j3  j8 e U. a  Let  S =  t e i and for  Q  U ae  i  and l e t  C^ ^ 0 . each  and f o r each  Let  t e S .  t e S  C^ = T^ta^.] . z  Then  choose C^.  be a f i x e d p o i n t i n  a, e  with  i s compact i n X  with  X  fe  z^ e C^  9  Then t  C = (x e X : x  e T-S}  e C  t  for t e S  i s a compact subset of  product t o p o l o g y . finite  fe  Now l e t B  subsets of -4r •  I f f o r each  B e B  Then  for t  each  e T^ B  t e T and  B  we l e t T„ =  choose  B  y  and i f x  x? = z. t t  B x e C .  and C  B e B  with r e s p e c t t o the  U i„ , then aeB  net Hence  x  e a  f o r any  ( x ; B e IB} B  x e a  e C\ £ f l O jSeB teT  a. .  i s eventually i n  x  Proof o f 2 . 4  See Meyer [ 5 ] ,  Proof of 2.5  Z  = y^ f o r  fc  x  e r\ £  B  C , and s i n c e  x .  {a} c B . f o r each  a e ~A- and so  y. e C. Z  i s a net i n  a  Then  Z  i s d e f i n e d by  B e IB with  i s finite,  B  B  f o r t e T - T„ , then B  f o r each  T-  a  i s compact, t h i s net has a c l u s t e r p o i n t •p  then  for  fc  i s d i r e c t e d by i n c l u s i o n .  B (x jB e IB)  Hence  x^ = z  be the f a m i l y o f non-void  33  For each  X  and  f\. a  a e X  If  Therefore the a e  Jr.  $ .  page 3 3 .  Immediate from the d e f i n i t i o n of ^  and 2 . 4 . The f o l l o w i n g w e l l known elementary  lemma w i l l be  needed l a t e r :  2 . 6 Lemma: and £ <=• B  If  i e I , A, B  y c A y B , then there e x i s t and  a U j8 = y .  are open i n  a, j8 e K  i  with  X  i  a c A ,  Q  10  3.  The f a m i l y 3.1  JM  of measures.  Definition.  M = {u : u  i s a bounded outer measure on  X  such that  u(A) = sup (u(a) : a e ts and  .2  a c A] f o r  A .3  |i(B) = i n f (u(A) : A e ^  B c A) f o r  and  B c X} .  3.2  Definitions:  .1  T  Por any s e t f u n c t i o n  s a t i s f i e s condition  r(0)  - 0  (a) i f f T  and f o r every  i e I  T  on  i s bounded,  and  a,/3 e  0 <. T ( c y l a) < r ( c y l a U c y l j3) < T ( c y l a) + T ( c y l fl) and  T ( c y l a U c y l fl) = T ( c y l a) + T ( c y l 0) i f  a A fl = 0 . .2  T  s a t i s f i e s c o n d i t i o n (b) i f f f o r every  a e K ^ j t e T - i  and sequence  C  for  n  c interior C  n + 1  new  C  i e I ,  in  and  C  n  with f X  j = i U [ t ] and fl = {x e X. : x i i e a then  T  S U  P  ( (°0 : a e "vS and T  The key r e s u l t s 3.3  x, e C } v n  n  *( ) = A  and  n j T ( c y l fl ) | T ( c y l a) .  (Note t h a t we c e r t a i n l y have •3  , if  fc  Theorem:  c y l fl f c y l a .) n  a c A}  of t h i s s e c t i o n  Let  T  for  A €  .  are summed up i n  s a t i s f y conditions  (a) and  11  (b) and and  u  be the Caratheodory  ^  .  measure on  u e M  .2  i f i e I , \ i ( A ) = u ( c y l A)  and  u  agrees  with  T  #  = T (.cyl a)  for a e K  g i c a l outer measure on i s a bounded  ,  i  and  ±  ,  v.^ i s the t o p o l o -  cranked by  Radon outer measure and  , then agrees  with  Tf\v .  on  l  X^  on  for A c X  ±  v,  generated by  Then  .1  T^(a)  X  ±  Por the proof o f t h i s theorem two p r e l i m i n a r y lemmas are needed. Lemma A:  Let T  i,j  i c j  e l with  s a t i s f y c o n d i t i o n ( b ) . Then f o r and  a e  we have  T ( c y l a) = sup ( T ( c y l /3) : jS € K^  Proof:  c y l j3 c c y l a} .  and  Follows e a s i l y from c o n d i t i o n (b) and i n d u c -  tion.  Lemma B; is  If  s a t i s f i e s c o n d i t i o n s (a) and (b) then  countably s u b a d d i t i v e on  Proof: with  T  Let A a c  u A new  .  n  e *f f o r n e tu , & > 0  F o r each  new  n  B  e 1 1  A  .  So  .  A n  a c  u u B new mew  ruri  and  = U B mew ™  a e $ where  and hence by C o r o l l a r y 2.5  12  there  exist  N, M e ou  such  that  N M u u B n=0 m=0  a c  .  Let f o r  11111  M 0 < n < N , E ^ = U B ' n „  and  1 1 1 1 m1  i n  n  for  some  A  c  X  n  i  •  Let ^  e I  be s u c h  Let  i = i  N U i n=0  n  for  some  y e M, a  .  x  and  ±  with Now  a  c i .  a  c y lflc a  and  1  1  , then  ]  i s open i n  n  a = cyl y i  i s finite  X.  and  x  n  lemma 2.6 and i n d u c t i o n we c a n f o r  6  e X,  c y l fl =  = cyl A n  s e c t i o n c h o o s e fl e K.^  By  with  that  E n  T ( O ) <. T ( c y l fl) + £ .  0 < n < N , ff, [E N TT [ a ] c U TT, [ E ] . n=0 1  that  n  B y lemma A o f t h i s  for  fi c  be s u c h  6  c TT. [ E ]  N U c y l |3 n=0 n  c  N (j E n=0  and  .  6 =  0 <_ n < N find N U fl_ . T h e n  n=0  By c o n d i t i o n  ( a ) we  have  n  N T ( c y l fl) < S T ( y l fl ) . • - n=0 -  Hence  C  T(Q) ^  T ( c y l fl) + t  n  <  N E x(cyl n=0  follows T*(  U A new  A ) + e < n  C  #  .  It  n  that ) = sup ( T ( a )  o f 3»3.1  subadditive  : a e t$ a n d  a c  U A new  } <  n  We now p r o c e e d Proof  N £ T (E ) + e < I ,T*(A ) + n=0 new •  on  t o prove  E  T  #  ( A  ) ,  new  t h e o r e m 3«3« Since,  and s i n c e  b y lemma B, i s c l o s e d under  i s countably countable  13  unions we have f o r A e for  , u ( A ) = T ( A ) and t h e r e f o r e #  u(B) = i n f  B c X ,  (u(A)  :  e  A  and  B  c  .  A)  Furthermore, s i n c e c l e a r l y  u(a) >_ T ( a )  i t follows that f o r A e ^  , u ( A ) = sup (u(a) : a e -g and  a c A} .  Now, s i n c e  measure on  , B c X  and  a e ^  ,  c e r t a i n l y i s a bounded Caratheodory  X , a l l that remains i s to show  A e ^  let  a  f o r each  and  t > 0 .  u(B') < u(B) + £ .  Choose  B' e ^  So  with a c B' A A  B c B  7  and  u ( B ' A A ) <_ \x(a) + £ .  Let  and  |i(B - a) <_ u(j3) + t .  By lemma A we can suppose t h a t  /  a u j8 e "S a l s o .  Let  c Ift^ .  a e -g with  0 e *g with  j3 c B' - a  u(B 0 A ) + u(B - A ) < u(BTU)+u(B'-a)  Then  < |i(a) + |Ji(i3) + 28 < u ( a U j3) + 2£ < H(B') + 26  < n(B) + 3£ • |i(B'fl A ) + u(B - A ) = u(B)  Hence  fora l l B c X .  I t f o l l o w s t h a t "^J c Tfl^ .  Proof of 3.3.2 measure on  X  i  Let  v  cranked by  be the t o p o l o g i c a l outer  1  .  remark 1.14.2 we have t h a t measure on  X^ .  i s c l o s e d and C  n  e K  ±  A  Let  X^  is  such t h a t  C  n  By c o n d i t i o n (a) and  i s a bounded be open i n X  i  .  Radon outer  a-compact, we can f o r n e uo choose f  (X -A) 1  .  Then  A  = fi new  Since  cyl(X  1  - C ) e **| we have N  X^ - A  Since  (X  ±  -  C  )  .  14  u.(X. - C ) = n ( c y l ( X n  - C )) = T . C c y K ^ - C ))  i  n  n  = sup [T(6) : fi € ta and  6 c cyl(X  = sup [ T ( c y l a) : a e K  i  and  = sup CT^(a) : a e  and  = v (x ±  1  - C )} n  a c (X^ - C )} n  a c (X^ - G )} n  - o ) .  ±  n  Furthermore s i n c e the  X^ - C  are  n  u^-measurable as w e l l as  -measurable we have  u.(A) = 11m u (X new Hence If  D c X  - C ) = l i m v.(X new  and then  ±  ±  ±  = i n f {u^(A) : A  Now  .  agree on open s e t s .  v (D) = i n f {v (A) : A  Hence  - C ) = v.(A)  <_  l e t B e YfL,  i s open i n  i s open i n  X  ±  X  ±  and  and  D c A)  D c A} > \i {D) ±  .  always. •  Given  Z > 0  choose  A  open i n  X.  15  with  B c A  v (A)  = v ( B ) + v ( A - B)  i  and  i  sequently  v (A) ±  < v (B) ±  +6  .  we have  i  | i ( A - B) < e . ±  But  Since  v ( A - B) < Z  and con-  i  H^(A) < u (A - B) + ^ ( B )  < M (B) + C . i  Hence M (B) = i n f { ^ ( A )  : A  i s open i n  X  i  and  B c A}  = i n f (v (A) : A  i s open i n  X  ±  and  B c A}  1  i  =  V l  3.4  .  (B)  as. r e l a t e d t o p r o j e c t i v e l i m i t measures. Suppose t h a t f o r each  generated by [v  i  i,j  and  : i e 1} e l  i e I ,  i s a measure on  with  i c j  we have f o r  We say t h a t the p r o j e c t i v e system p r o j e c t i v e l i m i t measure  for  some  We  call  of subsets of i e 1}  v X  if  (v^ : i e I)  v  Such a measure  admits a  i s a measure on the  generated by  such t h a t f o r each  v ( c y l A) = v ( A ) . i  A e JJ^  = v^TT.-^A]) .  i  8  .  a p r o j e c t i v e system of measures i f whenever  v (A)  ring  i s the a - r i n g  {cyl B : B e B  i e I  and  (v^ : i e 1} .  i  A e  ,  v , i f i t exists, i s  unique and can thus be c a l l e d the p r o j e c t i v e l i m i t of the system  er-  measure  16  For more g e n e r a l d e f i n i t i o n s of p r o j e c t i v e o r i n verse systems o f measures see Choksi [ 1 ] , M a l l o r y  [4]  or  Meyer [ 5 ] . Now i f f o r i e I B o r e l measure whenever  regular  i s a bounded measure on  = i n f {v (B) : B  ±  a bounded  A e 8^  such t h a t f o r every  v (A)  we c a l l  i  i s open and  = sup ( v ^ C ) : C e X  4  and  A c B}  C c A)  we then have  3.4.1  u e M  Theorem:  measure on  X  and  Proof:  on  Suppose  on  X .  If  (u^  : i e 1}  is a  *f -outer  u|fo i s the p r o j e c t i v e l i m i t measure o f  a p r o j e c t i v e system measures  i f f u  ( u ^ .: i e 1}  of bounded r e g u l a r B o r e l  iB^ .  u e JM .  Then  u ^ ( A ) = u ( c y l A)  M s  a  for A e H  -outer measure ±  then  clearly  forms a p r o j e c t i v e system of measures and  u|ift  i s c l e a r l y the p r o j e c t i v e l i m i t measure of t h i s system. Using 3 « 3 « 2 one can e a s i l y check t h a t each bounded r e g u l a r B o r e l measure on Conversely l e t u and  be a  i s i n fact a  0^ . ^  -outer measure on  u|iB be the p r o j e c t i v e l i m i t measure o f a p r o j e c t i v e  X  17  system  {\x^ : i e I)  on  .  Then and  L e t f o r each  T  l  e  i and  i s a s e t f u n c t i o n on 7J>  (b). Let  ed by  of bounded r e g u l a r B o r e l measures  T  v  , i"(cyl a) = u ^ a ) .  satisfying  conditions  be the CarathSodory measure on  and ^  #  a e  .  Then by 3 - 3 . 1  v e JM .  v  j_( ) = ( c y l A  that 'fy c B are M M .  = 9  v  A)  for  f o r each  A e ft^ '.  i e I .  ( v ^ : i e 1}  From 3 * 3 - 2 we see  Hence  ujiB = v | (B .  = \>\4f and t h e r e f o r e , s i n c e both  -outer measures on  X , we have  generat-  C l e a r l y v|ft  i s the p r o j e c t i v e l i m i t measure of the system where  X  (a)  u = v .  u  and  Hence  Since v  18  4.  The R e p r e s e n t a t i o n Theorem. 4.1  Definitions.  .1  Por  i e I , C (X^)  i s the s e t of continuous  Q  valued f u n c t i o n s on .2  Por X  .3  i e I  and  g i v e n by  X^  f = c y l h}  Definitions.  .1  = {l : l  and  x e X .  h e 0 (X^)  with  i s a p o s i t i v e l i n e a r f u n c t i o n a l on the  l i n e a r span of  .2  i e I  f o r every  .  4.2  such that K > 0  there e x i s t s f  i s the f u n c t i o n on  i  ( c y l h)(x) = h ( x | i )  F = {f : there e x i s t s  .1  with compact support.  h e C (X ) , cyl h Q  real  with  | * ( f ) | < K||f||  fora l l  e P ,  if  i , j e I  n e uu  f  n  with  e C (Xj) 0  i c j , f e ^(X^) with  cyl f  n  and f o r  f cyl f  then  * ( c y l f ) f * ( c y l f).} n  (Note that i n the d e f i n i t i o n of  L  not n e c e s s a r i l y imply c o n d i t i o n  .2.)  .2  Por  I e  ,  T*(a) = i n f U ( f )  above, c o n d i t i o n  .1  i s the s e t f u n c t i o n on Tj g i v e n by : 1  < f e F}  for  a e -g .  Our b a s i c theorem now i s 4.3  Theorem:  unique  |i e M  F o r each  I e L,  there e x i s t s a  such t h a t the r e l a t i o n s h i p  does  19  l ( f ) = J f d | i * holds f o r a l l I -*  f e  preliminary  the proof  and  I e JD , i e l  T ^ ( c y l a) = i n f U ( c y l f ) : 1  Proof: want to f i n d  Q  1  x  h = cyl g .  Let  dependent of  with  y  and  Q  k  = i n f {h(y)  f*  and  1  Q  r e g u l a r there e x i s t s < f  : z e X : y e X  i s continuous on  c y l f * <_ h  1  X^  and  cyl f < h . —  C (X.)  z e  let  (Note t h a t that we  ,  1 c y l a <_ h . and  ge  = g(z|j)  k  a e  h k  such t h a t h (z)  =  k  (z)  h(y)  is in-  c y l h^ = h  have t h a t  h^  .) is  .  x e X^  f*(x) = i n f (h (z)  then-  and  y|k = z , and  h (z)  continuous on  < f a —  For  y|k = z .  provided  g e C (Xj)  '  j e I  k = i U j .  y e X  Hence i f f o r  three  .  ±  h e 1? and  f e C (X,) with o i '  By d e f i n i t i o n there e x i s t s  and  < f e 0 (X )}  Q  Suppose  uniformly  w i l l need  lemmas.  Por  f o r some  M .  of t h i s theorem we  Lemma C:  Since  Moreover the mapping  i s an isomorphism between Por  We  .  <. f * .  and  k  and  X^  Since  f e C(X.)  cyl f < h .  .  z|i = y|i =  x)  x)  Moreover i t i s c l e a r t h a t X  i s l o c a l l y compact  i  with  I t follows  1  that  < f <_ f * .  and  Hence  20  T°(cyl a) = i n f U ( h )  : 1 c y l a < h e^F}  = inf U(cyl f) : 1  Lemma D:  Let  Then f o r e v e r y a c A  £ > 0  with  ||f||  < 1  i(cji  f) < e .  and  Proof.  j e I  Now  A let  with  i s open and  a c  j e I  i c j .  with  0 <^ g <_ 1  Let  .  ±  open i n i c j  choose  *,(cyl h) < T ^ c y l a) + S/2  and  Q  .  .  X^  and  with f e  [x : f (x) > 0} c TT7"^ [A - a]  A = fx : (1 + e/1+2  Then  A  with  By lemma C  1„ < h  < f € 0 (X )}  £ e L , i € l , a e K ^  there e x i s t s  such t h a t f o r any  Q  c  ( j) x  0  we have  h e C (X.) o i ' v  with  Let  t ( c y l h ) ) h ( x ) > 1} .  A .  and support  /3 = TT^j [support g] .  Suppose f i r s t  that  g € C ( j) x  0  g c 7T~ [A - a] . 1  Then  a, 6 are d i s j o i n t  compact  subsets of  A  and so l e t V , W  be d i s j o i n t neighborhoods  of  0  r e s p e c t i v e l y with  V U W c A .  Let Then  a  and  v , w e C (X ) Q  i  with  1  Q  <_ v <_ l y  v + w < (1 + £/l+2-t(cyl h ) ) h  and  1^ <_ w <_ 1^ .  and t h e r e f o r e  21  *(cyl  v) + * ( c y l  < r (cyl  <t(cyl w) <_ £  condition  let  (x  : f (x)  j8  = [ x : f (x)  Let  g  f e C (Xj)  with  Q  > 1/n}  0  Then'support  f  .  with c|3  n by  we have b y  * , ( c y l g) < £ ( c y l w)  > 0} c TT~^[A - a ] .  e C (Xj)  n  c y l g <. c y l w  .  Thus  .  Now  n  and s i n c e  .2 o f 4.2.1 t h a t  <t(cyl g) < e  h) + £ / 2  a) + £ < * , ( c y l v ) + £  l  Hence  w) < * ( c y l  Then 1^  Por fi  n  < g  . c TTT'i [A i j  n+1  t h e above a r g u m e n t ,  |jr|} ^  < 1  n e u> l e t e K^  and  < lg  n  and  &  n  c interior  and l e t  f  0  n + 1  = f-g .  n  n  - a] , 0 < f < 1 , a n d hence — n —  -f,(cyl f ) <. £ .  f ff  Since  +  we  n  have b y c o n d i t i o n .2 o f 4.2.1 t h a t <t(cyl f ) t It follows that l{cyl f ) < £ and t h e r e f o r e +  *(cyl ) • -t(cyl f ) < £. f +  i Lemma E :  Por  l e L , T  satisfies  conditions  (a) a n d ( b ) . Proof.  Condition  lemma C u s i n g dition  (b), l e t i e I , a e K  sequence i n C  n  and  t X  w e l l known s t a n d a r d  t  .  K  t  Let  x, e C } .  with  C  n  i  easily  arguments.  To p r o v e  , t e T - i  c interior  j = i U ( t ) and Then  (a) f o l l o w s  ^  C  n + 1  and for  C  from con-  be a  n e ou , and  = (x e X j : x | i e a  c y l fl t c y l a .  Given  £ > 0 , by  <  22  lemma D , there e x i s t s t h a t f o r any (x  with x  Q  Xj  6  n  such  and  4 ( c y l g ) < ft .  < f < 1  and l e t k  A  e  n  c Q  (  x t  )  p  let f (x) = f(x|i)•k (x ) n  n  cyl f  with  Q  Q  a c A  < l n+l  and  C (X.)  €  1  with  ||g|[^ < 1  we have  with  ±  n  <. f  f i  n  with  Q  1„ < k n  Por  h  g e C (Xj)  f e 0 (X ) C  p  open i n X^  : g ( x ) > 0} c TT~j[A-a]  Choose  l  A  l  f  f cyl f .  i  < h  < f  n  .  t  Then  f  n  e C (X ), Q  j  By lemma C choose t(cyl h ) < T^(cyl  and  n  n  flj+e,  n We note t h a t (x  : (f - h n  -t(cyl ( f n v  J  have  x  f n + 1  n  - h  )(x)  n  +  1  €  1 1  *(cyl f j = *(cyl  < *(cyl h  n + 1  0  X j  )  , ||f  > 0) c TT~j[A-a]  - h ) ) < ft . n+l ~ n  C (  Since  (f - h R  f n  n + 1  ) + e < T*(cyl  n  . = f n  - h ^ J ^< 1  Hence by lemma D , - h . + h n+l n+l n  ) ) + *(cyl  fl )  and  h  n + 1  we  )  + 2ft .  n+1  Hence T ^ c y l a) < * ( c y l f ) = l i m 4 ( c y l f ) new < l i m * , ( c y l h ,. ) + ft < l i m T ^ ( y l fl . ) + 2ft . new . - new C  n  Thus  +  1  n  T ( c y l a) < l i m T ( c y l fl ) - new < /  <0  r e v e r s e i n e q u a l i t y h o l d s , we have  and  +  1  since  c e r t a i n l y the  T ^ ( c y l a) = l i m T ^ ( c y l fl ). new  23  Proof of 4 . 3  Let  theodory o u t e r measure  r  sa-  l  on  X  generated by  T£  and  M .  Now suppose h e C (X ) Q  let  By lemma E  c o n d i t i o n s (a) and (b) and hence by 3 ' 3 « 1 the Cara-  tisfies  is i n  I e L .  f e F .  By d e f i n i t i o n there e x i s t s  such t h a t  i  f = cyl h .  u J ( A ) = | / ( c y l A)  If f o r  a e X  then  I f f o r every  Jcyl h d  we l e t T^(a) = r (oyl l  4  i e I  and  A c x  we  = j'h d ^ a)  .  and l e t v£  be  t  the t o p o l o g i c a l outer measure on by 3 « 3 « 2  v£  is a i  v£  Hence s i n c e  h e C (X )  on a l l Q  Furthermore i f  cranked by  Radon outer measure on  agrees w i t h  Jh  X^  i  -measurable  X  T and  i  ,  then  u£  sets.  we have  i  d v£ = J h d u£  .  ^ ( g ) = -t(cyl g)  for  g e C (X ) Q  then  i  i s a p o s i t i v e continuous l i n e a r f u n c t i o n a l on  ^ (X^) 0  l  ±  and  by lemma C T£(O) = i n f U ( g ) ±  : l  a  < g e 0 (X )} Q  ±  for  a e X  Hence by the R i e s z R e p r e s e n t a t i o n Theorem  ±  .  and  t i s f y the r e l a t i o n s h i p ^ ( g ) = J g d v£  for a l l g € C (X ) . Q  i  sa-  24  Hence  * ( f ) = I (cyl h) = ^ ( h ) = J h d v£ = J h d  = J c y l h d |/= J f d n * .  To show uniqueness,, suppose all  f e F .  A c X  i  Por each  u e M,  i e I  , T ( a ) = u ( c y l a)  let  for  i  p o l o g i c a l o u t e r measure on is a v.  Jf  ±  for  = | i ( c y l A)  for  \i (A) ±  a e  Radon o u t e r measure on  = Jf d u  <t(f) = J f d u  and  v^^  cranked by  on TTVv. , hence a l s o on  d v  and  X  IB. .  i  T  be the to-  j_•  and  By  3'3.2  agrees w i t h  Furthermore f o r a l l  = <t(cyl f ) = J f d u£ = J f d  ±  and t h e r e f o r e by the R i e s z R e p r e s e n t a t i o n Theorem It follows that j e c t i v e systems  and Cn |iB i  i  u;r  agree on  : i e 1}  and  .  u|iB  ^ c is we have t h a t  u  The mapping  i s now  L  and  M .  t -» u  and and  ^  are  measures,  |i |B , are a l s o e q u a l . agree on  .  Hence the p r o -  (|i^|lB^ : i e 1}  e q u a l and so t h e i r r e s p e c t i v e p r o j e c t i v e l i m i t which by 3«4.1 are  =  and so  Since u = n*.  c l e a r l y an isomorphism between  25  Example is  t o show t h a t  a-compactness o f the c o o r d i n a t e  needed. Let  R  have t h e d i s c r e t e t o p o l o g y  a-compact) and c o n s i d e r h  e C(R)  (cyl  L  e  spaces  E  =  0  F  Using = Y M  First  , let  2  G  O  (  R  2  1  h)(x) = h(x )  Por  and  1  = Cf : f = c y l h  f o r some  h e C (R)}  = {f : f = c y l  f o r some  h € C (R)}  1  0  the notations = R  2  . P  with and  h  0  of this  L  element o n l y , every  F = P  f  F  u e M  1  Then  such that  and  >  and t h a t s i n c e  pair-  F  c o n s i s t o f the zero  0  span of  F  has a u n i -  + f, + f „ where f e F for o 1 2 n ~n z e R ( w h i c h e q u a l s X) d e f i n e I 2  + 2f (z) 2  I e L  for f  b u t we s h a l l  i(f) = J f d u  A  f e F  A = (x e R  [1(A) = J l d u = l ( l ) = 2 - l ( z ) = 2 A  i n the l i n e a r  show t h a t t h e r e i s  forall  such a  A  X ,  "'c  M e M . Then i f ~ 1^ e F ^ c F and hence  we d i d f i n d we have  .  and d e f i n e  ,  f = f  For fixed  Q  "•'i  i n the l i n e a r  l(f) = f ( z ) + 2 f ( z )  span o f  U F, U F .  ^O  F „ , F-, ~o ^1  que r e p r e s e n t a t i o n a s .  p a p e r we l e t T = {1,2}  as b e f o r e .  we n o t e t h a t  n = 0, 1 / 2  o  the d i s c r e t e topology  wise I n t e r s e c t i o n s o f  no  (cyl  topology.  )  i»  by  the product  2  ^  •fy,  x e R  with  h)(x) = h ( x ) .  2  t  and  1  (which i s not  . 2  Suppose : x  n  1  = z ) L n  26  We next note t h a t  ^  z  > )  l  • > € F„ c F Izj — o  and so  r  = J ^ z } ^ = *(1{ )) = [ } ( ) = 1  Z  Furthermore s i n c e  A , {z}  Z  Z  and  1  •  A - ( z ) are a l l u-measurable  we have  u(A-{z)) = u(A) - u({z})  On the o t h e r hand Hence  =2-1=1  A-[z} c R -{z} 2  which i s i n ^  u ( A - ( z ) ) < u(R -{z}) 2  = sup (u(a) : a e  and  a c 1  = sup U ( l ) : a e -g and  a c R  a  since  £(1^,) = 0  .  f o r any  a e ^  with  z / a .  - (z}} 2  - [z}} = 0  Hence  A-(z}  would have t o have measure zero and one s i m u l t a n e o u s l y , which is  impossible.  27 Bibliography  1.  I . R. Choksi, Inverse l i m i t s of measure spaces. Proc. London Math. Soc.. 8 (1958, 321-342).  2.  L. Gross, Measurable Am.  3.  Math. S o c ,  f u n c t i o n s on H i l b e r t space, T r a n s .  105 (1962, 372-390).  L. Gross, C l a s s i c a l A n a l y s i s on H i l b e r t space. i n " A n a l y s i s i n F u n c t i o n Spaces", Chapt 4, M.I.T. P r e s s , Cambridge, Massachusetts,  4.  1964.  D. I . M a l l o r y , L i m i t s of i n v e r s e systems of measures, t h e s i s , Univ. of B r i t i s h Columbia,  5.  1968.  P. A. Meyer, P r o b a b i l i t i e s and p o t e n t i a l s . B l a i s d e l l ,  1966. 6.  I . E. S e g a l , Tensor a l g e b r a s over H i l b e r t spaces. T r a n s . Am.  7.  Math. S o c ,  8l (1956, 106-134).  G. E. S i l o v , On some q u e s t i o n s of a n a l y s i s i n H i l b e r t space. Part I . F u n c t i o n a l a n a l y s i s a p p l i c 2.  8.  1 (1967) no.  (English translation)  M. S i o n , L e c t u r e notes on measure theory;, B i e n n i a l Seminar of the Canadian Math. Congress,  1965.  

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