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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 INFINITE DIMENSIONAL SPACES by FRANZ PETER EDWARD HARPAIN B . S c , U n i v e r s i t y of B r i t i s h Columbia, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department MATHEMATICS We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1968 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 that 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 fo r re ference and Study. I f u r t h e r agree that permiss ion fo r e x t e n s i v e copying of t h i s t h e s i s fo r s c h o l a r l y purposes may be granted 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 that copying or p u b l i c a t i o n of t h i s t h e s i s fo r f i n a n c i a l gain s h a l l not be a l lowed wi thout 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 obtain a generalization of the well known Riesz Representation Theorem to the case where the underlying space X Is an i n f i n i t e dimensional pro-duct of l o c a l l y compact, regular and a-compact topological spaces. In the process we prove that our measures on X correspond to projective l i m i t measures of projective sys-tems of regular Borel measures on the coordinate spaces. An example i s given to show that cr-compactness of the coordinate spaces i s necessary. i i TABLE OP CONTENTS •> Page 0. I n t r o d u c t i o n •. • 1 1. General N o t a t i o n iJ-2. The f a m i l y of c y l i n d e r s 7 3. The f a m i l y M of measures 10 4. The R e p r e s e n t a t i o n Theorem 18 5. Example 25 6 . B i b l i o g r a p h y 27 A CKNOWLEDGEMENTS I am deeply indebted to Dr. Maurice Sion, who suggested the topic and except f o r whose u n f a i l i n g help th i s thesis would never have come to be. I am also most gratef u l f o r the f i n a n c i a l assistance given by the De-partment of Mathematics throughout the writing of t h i s t h e s i s . 1 1. Introduction If X i s a l o c a l l y compact, regular topological space, then the well known Riesz Representation Theorem sets up an isomorphism between the family of a l l bounded Radon outer measures on X and the set of continuous p o s i -tiv e l i n e a r functionals on the family of continuous func-tions with compact support i n X . In t h i s isomorphism corresponding elements, I a l i n e a r functional and u a measure, s a t i s f y the rela t i o n s h i p i(f) = J*f d u f o r a l l continuous functions f with compact support i n X . If we now consider an i n f i n i t e product of l o c a l l y compact, regular spaces, then t h i s i s i n general no longer l o c a l l y compact with respect to the product topology, and the Riesz Representation Theorem f a i l s to hold. Iri t h i s paper we obtain a representation theorem 2 f o r t h i s case by replacing the various f a m i l i e s mentioned above by the following: ( i ) A family "6a of cylinders whose elements act l i k e compact sets f o r a "pseudo-topology" ^ , where i s closed under f i n i t e intersections and countable unions and i s a subset of the product topology. ( i i ) A family M of bounded outer measures, related to -(£ and i n much the same way as bounded Radc'n outer measures are related to compact and open sets. ( i i i ) A family JF of functions depending only on a f i n i t e number of coordinates, with respect to which they are continuous and have compact support. (iv) A family L of p o s i t i v e l i n e a r functions//on the l i n e a r span of . Under the added hypothesis of a-compactness of the coordina-te spaces we show that I J and JM are isomorphic i n such a way that corresponding elements, l i n JL and u i n JM , s a t i s f y the r e l a t i o n s h i p l(f) = Jfdu f o r a l l f i n F . Moreover we show that the elements of M can be viewed as the projective l i m i t measures of projective systems of bounded regular Borel measures. Prom the i n t e g r a b i l i t y of the members of , i t follows that a l l bounded Borel functions which depend only on a f i n i t e number of coordinates are also integrable. Thus the simple functions used by S i l o v [7] and the tame functions 3 used by Segal [6] and Gross [2] i n the development of an i n -tegration theory on Hilbert space are included among the i n -tegrable functions of the measures considered here. (Por a good guide to the l i t e r a t u r e i n th i s area see the b i b l i o -graphy i n Gross [3].). Our re s u l t s therefore not only characterize an important class of l i n e a r functionals i n terms of projective l i m i t s of regular Borel measures but also enable us to extend these functionals to a much wider class of functions through a standard i n t e g r a l of a measure. Thus the standard theory of integration becomes applicable i n t h i s s i t u a t i o n . 4 1. General Notation. .1 0 i s the empty set. .2 ID i s the set of natural numbers. .3 1 Is the set of r e a l numbers. .4 t* i s a compact family i f f f o r every subfamily of ^ , i f the in t e r s e c t i o n of any f i n i t e number of members of Jr i s non-void, then the int e r s e c t i o n of a l l members of . A i s non-void. • 5 For f a function on X to R and A c X .1 f|A i s the r e s t r i c t i o n of f to A, .2 1^ i s the c h a r a c t e r i s t i c function of A , .3 Hffoo = sup [|f(x)| : x e X} , A f + ( x ) = max ( 0 , f ( x ) ) f o r x e X , •5 support f = closure (x : f(x) > 0} i f X i s a topological space. .6 I f f o r n e u) , a n i s a set, a n e 1 , f i s a function on X to R , then .1 a . f a i f f a c a and U a = a , new .2 a f a i f f a <. a , and lim a = a , new .3 f n f f i f f f o r a l l x e X , f n ( x ) < f R + 1 ( x ) and l i m f (x) = f(x) . neu) . 7 For I an index set and X^ a set f o r each i e I , "ff X. = [x : x i s a function on I with i e l 5 x i 6 ^ i f o r e a c h i e 1} . .8 u i s a Caratheodory measure on X i f f u i s a f u n c t i o n on the f a m i l y of a l l subsets of X such t h a t u(0) = 0 and 0 < u(A) < E |i(B^.) < oo - new whenever A c |J c X . new .9 F o r |i a Caratheodory measure on X , A i s u-measurable i f f A c X and f o r every B cr X , u(B) = |i(B A A) + n(B - A) . yfl = {A : A i s u-measurable} . .10 |i i s a -outer measure on X i f f u i s a Caratheodory measure on X , ^  c f f l ^ , and f o r every A c X , u(A) = i n f {u(B) : B e ^ and A c B} . .11 u i s the Caratheodory measure on X generated by T ' and -^J i f f ^  i s a f a m i l y of subsets of X , T(A ) > 0 f o r every A e ^ , and f o r B e x u(B) = i n f { £ T ( A ) : X c *i , M i s countable Aejj Q and B e U A} . AeH .12 Por X a t o p o l o g i c a l space, M s a Radon out e r measure on X i f f u i s a Caratheodory measure on X such t h a t .1 open s e t s are u-measurable, .2 i f C i s compact then u(C) < oo , 6 .3 i f A i s open then u(A) = sup (u(C) : C i s compact, C c A} , A i f B e x then u ( B ) = i n f [u(A) : A i s open, B c A} . .13 F o r X a t o p o l o g i c a l space, n i s the t o p o l o -g i c a l measure cranked by T i f f T i s a f u n c t i o n on the f a m i l y of c l o s e d compact subsets of X , T ^ . ( A ) = sup {T ( C ) : C i s c l o s e d compact and C c A } f o r A an open subset of X , and u i s the Caratheodory measure on X generated by T # and the f a m i l y of open subsets of X . .14 Remarks. We mention here two w e l l known f a c t s about Caratheodory measures: . 1 The Caratheodory measure on X generated by T and i s i n f a c t a Caratheodory measure on X , .2 I f X i s l o c a l l y compact and r e g u l a r and T i s a f u n c t i o n on the f a m i l y of c l o s e d compact subsets of X such t h a t f o r A , B c l o s e d and compact we have 0 < T ( A ) < T ( A U B ) < T ( A ) + T ( B ) < 0 0 and T ( A U B ) = T ( A ) + T ( B ) i f A f) B = 0 , then the t o p o l o g i c a l measure cranked by T i s a Radon outer measure on X . (See f o r example S i o n [8]) 7 2. The family "6 of cylinders. Throughout th i s paper we suppose that T i s any index set and that f o r each t e T , Y^ . i s a l o c a l l y com-pact, a-compact and regular topological space. 2.1 D e f i n i t i o n s . .1 X = T f Y . t e T z .2 I i s the set of non-void f i n i t e subsets of T , ordered by i n c l u s i o n . Por i , j e I with i c j .3 X. = TT Y+- i s equipped with the product topology 1 t e i z (which i s l o c a l l y compact, a-compact and regular), A K i i s the family of closed compact subsets of X i , .5 TT^  (respectively ^j_j) i s t n e canonical projection of X (respectively Xj) onto X^ , .6 For A c X i , c y l A = TT"1 [ A ] . If no confusion i s possible we w i l l f o r t e T i d e n t i f y t and (t} , Y t and X ^ j . Thus Yt=Xj-tj=Xfc and ^(t} = * 2.2 D e f i n i t i o n s . .1 ^ = (a : there exists i e I and 0 e with a = c y l /3). Thus ^ i s the family of cylinder sets which f o r some i e I have a compact base i n X . . 8 .2 i s the closure under f i n i t e intersections of the family of complements of sets i n • .3 "^f i s the closure of under countable unions. The e s s e n t i a l properties of are the following: 2 . 3 Theorem: ^ i s a compact family. 2 . 4 Corollary: The closure of under f i n i t e unions i s a compact family. 2 . 5 Corollary: I f a e "(£ and f o r each n e w , B n e with a c U B then there exists N e w such that a c n y 0 B n ' Proofs Proof of 2 . 3 Let X be any subfamily of such that f o r any non-void f i n i t e B c we have C\ a ji 0 . . aeB Por each a E J l l e t i e I be such that a = c y l j3 f o r some j8 e U. a Let S = U i and f o r each t e S choose a, e with ae t e i Q and l e t C^ = T^ta^.] . Then C^ . i s compact i n Xfe and C^ ^ 0 . Let z be a f i x e d point i n X with z^ e C^ f o r each t e S . 9 Then C = (x e X : xfe e C t f o r t e S and x^ = z f c f o r t e T-S} i s a compact subset of X with respect to the product topology. Now l e t B be the family of non-void f i n i t e subsets of -4r • Then B i s directed by i n c l u s i o n . If f o r each B e B we l e t T„ = U i„ , then T- i s f i n i t e , 3 3 aeB a B For each B e B choose y e C\ £ f l O a. . Then y. e C. jSeB t e T B Z Z Z f o r each t e T B and i f x i s defined by xfc = y^ f o r t e T^ and x? = z. f o r t e T - T„ , then x B e r \ £ B t t B B B and x e C . Hence (x jB e IB) i s a net i n C , and since C i s compact, this net has a cluster point x . I f a e X •p then x e a f o r any B e IB with {a} c B . Therefore the net (x B;B e IB} i s eventually i n a f o r each a e Jr. Hence x e a f o r each a e ~A- and so f\. a $ . Proof of 2 . 4 See Meyer [ 5 ] , page 3 3 . Proof of 2.5 Immediate from the d e f i n i t i o n of ^ Q and 2 . 4 . The following well known elementary lemma w i l l be needed l a t e r : 2 . 6 Lemma: If i e I , A, B are open i n X i and y c A y B , then there e x i s t a, j8 e K i with a c A , £ <=• B and a U j8 = y . 10 3. The family JM of measures. 3.1 D e f i n i t i o n . M = {u : u i s a bounded outer measure on X such that .2 u(A) = sup (u(a) : a e ts and a c A] f o r A .3 |i(B) = i n f (u(A) : A e ^ and B c A) f o r B c X} . 3.2 D e f i n i t i o n s : Por any set function T on .1 T s a t i s f i e s condition (a) i f f T i s bounded, r(0) - 0 and f o r every i e I and a,/3 e 0 <. T ( c y l a) < r ( c y l a U c y l j3) < T(cy l a) + T ( c y l fl) and T ( c y l a U c y l fl) = T ( c y l a) + T(cy l 0) i f a A fl = 0 . .2 T s a t i s f i e s condition (b) i f f f o r every i e I , a e K ^ j t e T - i and sequence C i n with C n c i n t e r i o r C n + 1 f o r n e w and C n f Xfc , i f j = i U [t] and fl = {x e X. : x i i e a and x, e C } n j v n then T(c y l fln) | T ( c y l a) . (Note that we c e r t a i n l y have c y l fln f c y l a .) •3 T * ( A ) = S U P (T(°0 : a e "vS and a c A} f o r A € . 3.3 The key results of th i s section are summed up i n Theorem: Let T s a t i s f y conditions (a) and 11 (b) and u be the Caratheodory measure on X generated by and ^ . Then .1 u e M and u agrees with T # on , .2 i f i e I , \ i ± ( A ) = u(cyl A) f o r A c X ± , T^(a) = T (.cyl a) f o r a e K i and v.^  i s the topolo-g i c a l outer measure on X^ cranked by , then i s a bounded Radon outer measure and agrees with v, on Tf\ . l v ± Por the proof of thi s theorem two preliminary lemmas are needed. Lemma A: Let T s a t i s f y condition (b). Then f o r i , j e l with i c j and a e we have T ( c y l a) = sup ( T ( c y l /3) : jS € K^ and c y l j3 c c y l a} . Proof: Follows e a s i l y from condition (b) and induc-t i o n . Lemma B; If T s a t i s f i e s conditions (a) and (b) then i s countably subadditive on . Proof: Let A n e *f f o r n e tu , & > 0 and a e $ with a c u A . For each n e w A = U B where new n n mew ™ B e . So a c u u B and hence by Corollary 2.5 1 1 A new mew r u r i 12 N M there e x i s t N, M e ou such t h a t a c u u B . Let f o r n=0 m=0 1 1 1 1 1 M 0 < n < N , E ^ = U B m and i be such t h a t E = c y l A - - ' n „ n 1 1 1 1 1 n n n f o r some A n c X i • Let ^ e I be such t h a t a = c y l y n N f o r some y e M, . L e t i = i U i , then i i s f i n i t e xa a n=0 n and ±a c i . By lemma A of t h i s s e c t i o n choose fl e K.^  wi t h c y l fl c a and T ( O ) <. T ( c y l fl) + £ . Now f o r 0 < n < N , ff, [E ] i s open i n X. and N 1 n x fi c TT [a] c U TT, [ E ] . 1 n=0 1 n By lemma 2.6 and i n d u c t i o n we can f o r 0 <_ n < N f i n d N 6 e X, w i t h 6 c TT. [ E ] and 6 = U fl_ . Then n=0 N N c y l fl = U c y l |3 c (j E . By c o n d i t i o n (a) we have n=0 n n=0 n N T ( c y l fl) < S T ( C y l fl ) . Hence T ( Q ) ^  T ( c y l fl) + t • - n=0 n -N N < E x ( c y l A ) + e < £ T # ( E ) + e < I ,T*(A ) + C . I t n=0 n n=0 n new • f o l l o w s t h a t T * ( U A ) = sup ( T ( a ) : a e t$ and a c U A } < E T # ( A ) , new new n new We now proceed t o prove theorem 3«3« Proof of 3»3.1 S i n c e , by lemma B, i s countably s u b a d d i t i v e on and s i n c e i s c l o s e d under countable 13 unions we have f o r A e , u ( A ) = T # ( A ) and therefore f o r B c X , u(B) = i n f ( u ( A ) : A e and B c A ) . Furthermore, since c l e a r l y u(a) >_ T(a) f o r each a e ^ , i t follows that f o r A e ^ , u ( A ) = sup (u(a) : a e -g and a c A } . Now, since a c e r t a i n l y i s a bounded Caratheodory measure on X , a l l that remains i s to show c Ift^ . So l e t A e ^ , B c X and t > 0 . Choose B' e ^ with B c B 7 and u(B') < u(B) + £ . Let a e -g with a c B' A A and u(B'A A ) <_ \x(a) + £ . Let 0 e *g with j3 c B' - a and |i(B / - a) <_ u(j3) + t . By lemma A we can suppose that a u j8 e "S also. Then u(B 0 A ) + u(B - A ) < u(BTU)+u(B'-a) < |i(a) + |Ji(i3) + 28 < u(a U j3) + 2£ < H(B') + 26 < n(B) + 3£ • Hence |i(B'fl A ) + u(B - A ) = u(B) f o r a l l B c X . It follows that "^J c Tfl^ . Proof of 3.3.2 Let v 1 be the topological outer measure on X i cranked by . By condition (a) and remark 1.14.2 we have that i s a bounded Radon outer measure on X^ . Let A be open i n X i . Since X^ - A i s closed and X^ i s a-compact, we can f o r n e uo choose C n e K± such that C n f ( X 1 - A ) . Then A = fi ( X ± - C ) . new Since c y l ( X 1 - C N ) e **| we have 14 u.(X. - C n) = n ( c y l ( X i - C n)) = T . C c y K ^ - C n)) = sup [T(6) : fi € ta and 6 c c y l ( X 1 - C n)} = sup [ T ( c y l a) : a e K i and a c (X^ - C n)} = sup CT^(a) : a e and a c (X^ - G n)} = v ± ( x ± - o n) . Furthermore since the X^ - C n are u^-measurable as well as -measurable we have u.(A) = 11m u (X - C ) = lim v.(X - C ) = v.(A) . new new Hence and agree on open sets. I f D c X ± then v ±(D) = i n f {v ±(A) : A i s open i n X± and D c A) = i n f {u^(A) : A i s open i n X± and D c A} > \i±{D) . Hence <_ always. Now l e t B e YfL, • Given Z > 0 choose A open i n X. 15 with B c A and v±(A) < v±(B) + 6 . Since v i ( A ) = v i(B) + v i(A - B) we have v i(A - B) < Z and con-sequently |i ±(A - B) < e . But H^(A) < u (A - B) + ^ ( B ) < M i(B) + C . Hence M 1(B) = i n f {^(A) : A i s open i n X i and B c A} = i n f (v i(A) : A i s open i n X ± and B c A} = V l ( B ) . 3.4 as. related to projective l i m i t measures. Suppose that f o r each i e I , i s the a-ring generated by and i s a measure on . We c a l l [ v i : i e 1} a projective system of measures i f whenever i , j e l with i c j we have f o r A e JJ^ v i(A) = v^TT.-^A]) . We say that the projective system (v^ : i e I) admits a projective l i m i t measure v i f v i s a measure on the er-ri n g 8 of subsets of X generated by {cyl B : B e B i f o r some i e 1} such that f o r each i e I and A e , v(cyl A) = v i(A) . Such a measure v , i f i t e x i s t s , i s unique and can thus be c a l l e d the projective l i m i t measure of the system (v^ : i e 1} . 16 For more general d e f i n i t i o n s of projective or i n -verse systems of measures see Choksi [1], Mallory [4] or Meyer [5]. Now i f f o r i e I we c a l l a bounded regular Borel measure whenever i s a bounded measure on such that f o r every A e 8^  v±(A) = i n f {v i(B) : B i s open and A c B} = sup ( v ^ C ) : C e X 4 and C c A) we then have 3.4.1 Theorem: u e M i f f u i s a *f -outer measure on X and u|fo i s the projective l i m i t measure of a projective system (u^ .: i e 1} of bounded regular Borel measures on iB^ . Proof: Suppose u e JM . Then M s a -outer measure on X . If u^(A) = u ( c y l A) f o r A e H± then c l e a r l y (u^ : i e 1} forms a projective system of measures and u|ift i s c l e a r l y the projective 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 that each i s i n f a c t a bounded regular Borel measure on 0^ . Conversely l e t u be a ^ -outer measure on X and u|iB be the projective l i m i t measure of a projective 17 system {\x^ : i e I) of bounded regular Borel measures on . Let f o r each l e i and a e , i"(cyl a) = u ^ a ) . Then T i s a set function on 7J> s a t i s f y i n g conditions (a) and (b). Let v be the CarathSodory measure on X generat-ed by T # and ^ . Then by 3 -3.1 v e JM . Clearly v|ft i s the projective l i m i t measure of the system (v^ : i e 1} where v j _ ( A ) = v ( c y l A) f o r A e ft^ '. From 3 * 3 - 2 we see that = f o r each i e I . Hence ujiB = v | (B . Since 'fy c B 9 = \>\4f and therefore, since both u and v are -outer measures on X , we have u = v . Hence M M . 18 4 . The Representation Theorem. 4 . 1 D e f i n i t i o n s . . 1 Por i e I , C Q(X^) i s the set of continuous r e a l valued functions on X^ with compact support. .2 Por i e I and h e C Q(X i) , c y l h i s the function on X given by (cyl h)(x) = h(x|i) f o r every x e X . . 3 F = {f : there exists i e I and h e 0 (X^) with f = c y l h} . 4 . 2 D e f i n i t i o n s . . 1 = {l : l i s a posi t i v e l i n e a r functional on the l i n e a r span of such that . 1 there exists K > 0 with | * ( f ) | < K||f|| f o r a l l f e P , . 2 i f i , j e I with i c j , f e ^ ( X ^ ) and f o r n e uu f n e C 0(Xj) with c y l f n f c y l f then * ( c y l f n ) f * ( c y l f).} (Note that i n the d e f i n i t i o n of L above, condition . 1 does not necessarily imply condition . 2 . ) .2 Por I e , i s the set function on Tj given by T*(a) = i n f U ( f ) : 1 < f e F} f o r a e -g . Our basic theorem now i s 4 . 3 Theorem: For each I e L, there exists a unique |i e M such that the rel a t i o n s h i p 19 l ( f ) = Jfd|i* holds f o r a l l f e . Moreover the mapping I -* i s an isomorphism between and M . Por the proof of t h i s theorem we w i l l need three preliminary lemmas. Lemma C: Por I e JD , i e l and a e , T ^ ( c y l a) = i n f U ( c y l f ) : 1 Q < f e 0Q(X±)} . Proof: Suppose h e 1? and 1 c y l a <_ h . We want to f i n d f e C (X,) with 1 < f and c y l f < h . o x i ' a — — By d e f i n i t i o n there exists j e I and ge C (X.) such that h = c y l g . Let k = i U j . For z e l e t h k ( z ) = h(y) f o r some y e X with y|k = z . (Note that h k ( z ) i s i n -dependent of y provided y|k = z , and that c y l h^ = h .) Since g e C Q(Xj) and h k(z) = g(z|j) we have that h^ i s uniformly continuous on X^ . Hence i f f o r x e X^ f*(x) = i n f (h k(z) : z e X k and z | i = x) ' = i n f {h(y) : y e X and y | i = x) then- f * i s continuous on X^ . Moreover i t i s clear that c y l f * <_ h and 1 Q <. f * . Since X i i s l o c a l l y compact and regular there exists f e C(X.) with 1 < f <_ f * . Hence 1 < f and c y l f < h . I t follows that 20 T°(cyl a) = i n f U(h) : 1 c y l a < h e^F} = i n f U ( c y l f ) : 1 Q < f € 0Q(X±)} . Lemma D: Let £ e L , i € l , a e K ^ . Then f o r every £ > 0 there exists A open i n X^ with a c A such that f o r any j e I with i c j and f e c 0 ( x j ) with ||f|| < 1 and [x : f (x) > 0} c TT7"^  [A - a] we have i(cji f ) < e . Proof. By lemma C choose h e C (X.) with o v i ' 1„ < h and *,(cyl h) < T ^ c y l a) + S/2 . Let A = fx : (1 + e/1+2 t ( c y l h))h(x) > 1} . Then A i s open and a c A . Now l e t j e I with i c j . Suppose f i r s t that g € C 0 ( x j ) with 0 <^  g <_ 1 and support g c 7T~1 [A - a] . Let /3 = TT^j [support g] . 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 a and 0 respectively with V U W c A . Let v , w e C Q(X i) with 1 Q <_ v <_ l y and 1^ <_ w <_ 1^ . Then v + w < (1 + £/l+2-t(cyl h))h and therefore 21 * ( c y l v) + * ( c y l w) < * ( c y l h) + £/2 < rl(cyl a) + £ < *,(cyl v) + £ Hence <t(cyl w) <_ £ and s i n c e c y l g <. c y l w we have by c o n d i t i o n .2 of 4.2.1 th a t *,(cyl g) < £(cyl w) . Thus <t(cyl g) < e . Now l e t f e C Q(Xj) wi t h |jr|} ^  < 1 and (x : f (x) > 0} c TT~^[A - a] . Por n e u> l e t j8 n = [x : f (x) > 1/n} . Then fin e K^ and &n c i n t e r i o r 0 n + 1< Let g n e C 0(Xj) w i t h 1^ < g n < l g and l e t f n = f - g n . Then'support f c|3 . c TTT'i [A - a] , 0 < f < 1 , and hence n n+1 i j — n — by the above argument, -f,(cyl f ) <. £ . Since f n f f + we have by c o n d i t i o n .2 of 4.2.1 that <t(cyl f ) t * ( c y l f + ) • I t f o l l o w s t h a t l{cyl f + ) < £ and t h e r e f o r e -t(cyl f ) < £. i Lemma E: Por l e L , T s a t i s f i e s c o n d i t i o n s (a) and ( b ) . Pro o f . C o n d i t i o n (a) f o l l o w s e a s i l y from lemma C u s i n g w e l l known standard arguments. To prove con-d i t i o n ( b ) , l e t i e I , a e K i , t e T - i and C be a sequence i n K t with C n c i n t e r i o r C n + 1 f o r n e ou , and C n t X t . Let j = i U ( t ) and ^ = (x e X j : x | i e a and x, e C } . Then c y l fl t c y l a . Given £ > 0 , by 22 lemma D , there exists A open i n X^ with a c A such that f o r any g e C Q ( X j ) with ||g|[^ < 1 and (x : g(x) > 0} c TT~j[A-a] we have 4 ( c y l g) < ft . Choose f e 0Q(X±) with 1 Q < f < 1 A and l e t k n e c Q ( x t ) with 1„ < k < l p C n n n+l Por x 6 Xj l e t f n ( x ) = f ( x | i ) • k n ( x t ) . Then f n e C Q ( X j ) , l f i <. f and c y l f f c y l f . By lemma C choose p n h n € C Q(X.) with l f i < h n < f n and t ( c y l h n) < T ^ ( c y l flj+e, n We note that f n - h n + 1 € C 0 ( X j ) , ||f n - h ^ J ^ < 1 and (x : ( f n - h n + 1 ) ( x ) > 0) c TT~j[A-a] . Hence by lemma D , -t(cyl (f - h n ) ) < ft . Since f = f - h . + h n we v J x n n + l 1 1 ~ n n n+l n+l have * ( c y l f j = * ( c y l ( f R - h n + 1 ) ) + * ( c y l h n + 1 ) < * ( c y l h n + 1 ) + e < T*(cyl fln+1) + 2ft . Hence T ^ c y l a) < * (cyl f ) = lim 4 (cy l f ) new < l i m *,(cyl h ,. ) + ft < l i m T ^ ( C y l fl . ) + 2ft . new n + 1 . - new n + 1 Thus T < / ( c y l a) < lim T < 0(cyl fl ) and since c e r t a i n l y the - new reverse inequality holds, we have T^(cyl a) = lim T^(cyl fl ). new 23 Proof of 4 . 3 Let I e L . By lemma E rl sa-t i s f i e s conditions (a) and (b) and hence by 3 ' 3«1 the Cara-theodory outer measure on X generated by T£ and i s i n M . Now suppose f e F . By d e f i n i t i o n there exists i e I and h e C Q(X i) such that f = c y l h . If f o r every A c x we l e t uJ(A) = | / ( c y l A) then J c y l h d = j'h d ^ . If f o r a e X 4 we l e t T^(a) = rl(oyl a) and l e t v£ be t the topological outer measure on X^ cranked by T , then by 3«3«2 v£ i s a Radon outer measure on X i and u£ i i agrees with v£ on a l l -measurable sets. Hence since h e C Q(X i) we have J h d v£ = J h d u£ . Furthermore i f ^ ( g ) = -t(cyl g) f o r g e C Q(X i) then l± i s a p o s i t i v e continuous l i n e a r functional on ^ 0(X^) and by lemma C T£(O) = i n f U ± ( g ) : l a < g e 0Q(X±)} f o r a e X ± . Hence by the Riesz Representation Theorem and sa-t i s f y the relat i o n s h i p ^ ( g ) = J g d v£ f o r a l l g € C Q(X i) . 24 Hence * ( f ) = I (cyl h) = ^ ( h ) = Jh d v£ = Jh d = J c y l h d |/= Jfdn* . To show uniqueness,, suppose u e M, and <t(f) = Jfdu f o r a l l f e F . Por each i e I l e t \i±(A) = |i(cyl A) f o r A c X i , T i ( a ) = u(cyl a) f o r a e and v^ ^ be the to-p o l o g i c a l outer measure on cranked by T j _ • By 3'3.2 i s a Radon outer measure on X i and agrees with v. on TTVv. , hence also on IB. . Furthermore f o r a l l J f d v ± = J f d u ± = <t(cyl f ) = J f d u£ = J f d and therefore by the Riesz Representation Theorem = . It follows that and u;r agree on . Hence the pro-jective systems Cn i|iB i : i e 1} and (|i^|lB^ : i e 1} are equal and so t h e i r respective projective l i m i t measures, which by 3«4.1 are u|iB and |i |B , are also equal. Since ^ c is we have that u and agree on ^ and so u = n*. The mapping t -» u i s now c l e a r l y an isomorphism between L and M . 25 Example to show that a-compactness of the coor d i n a t e spaces i s needed. L e t R have the d i s c r e t e topology (which i s not a-compact) and c o n s i d e r 1 wit h the product topology. Por h e C ( R ) and x e R 2 , l e t ( c y l 1 h ) ( x ) = h ( x 1 ) and ( c y l 2 h ) ( x ) = h ( x 2 ) . L e t E 0 = G O ( R 2 ) ^ = Cf : f = c y l 1 h f o r some h e C o(R)} F 0 = {f : f = c y l 0 h f o r some h € C (R)} Using the n o t a t i o n s of t h i s paper we l e t T = {1,2} , = Y 2 = R wit h the d i s c r e t e topology and d e f i n e X , > •fy, M . P and L as b e f o r e . F i r s t we note t h a t F = P U F, U F . and t h a t s i n c e p a i r -i» ^O "•'i "'c wise I n t e r s e c t i o n s o f F„ , F-, and F 0 c o n s i s t of the zero ~o ^1 element o n l y , every f i n the l i n e a r span of F has a u n i -que r e p r e s e n t a t i o n as f = f + f , + f „ where f e F f o r o 1 2 n ~n n = 0, 1/2 . F o r f i x e d z e R 2 (which equals X) d e f i n e I by l(f) = f Q ( z ) + 2 f 1 ( z ) + 2 f 2 ( z ) f o r f i n the l i n e a r span of F . Then I e L but we s h a l l show t h a t there i s no u e M such t h a t i(f) = J f d u f o r a l l f e F . Suppose we d i d f i n d such a M e M . Then i f A = (x e R 2 : x n = z n ) ~ 1 L we have 1^ e F^ c F and hence [1(A) = J l A d u = l ( l A ) = 2 - l A ( z ) = 2 26 We next note that l r •> € F„ c F and so Izj — o ^ z > ) = J ^ z } ^ = * ( 1 { Z ) ) = 1 [ Z } ( Z ) = 1 • Furthermore since A , {z} and A -(z) are a l l u-measurable we have u(A-{z)) = u(A) - u({z}) = 2 - 1 = 1 On the other hand A-[z} c R 2-{z} which i s i n ^ . Hence u(A-(z)) < u(R 2-{z}) = sup (u(a) : a e and a c 1 - (z}} = sup U ( l a ) : a e -g and a c R 2 - [z}} = 0 since £(1^,) = 0 f o r any a e ^  with z / a . Hence A-(z} would have to have measure zero and one simultaneously, which i s 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 functions on Hilbert space, Trans. Am. Math. S o c , 105 (1962, 372-390). 3. L. Gross, C l a s s i c a l Analysis on H i l b e r t space. i n "Analysis i n Function Spaces", Chapt 4, M.I.T. Press, Cambridge, Massachusetts, 1964. 4. D. I. Mallory, Limits of inverse systems of measures, thesis, Univ. of B r i t i s h Columbia, 1968. 5. 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. Segal, Tensor algebras over H i l b e r t spaces. Trans. Am. Math. S o c , 8l (1956, 106-134). 7. G. E. S i l o v , On some questions of analysis i n H i l b e r t  space. Part I. Functional analysis a p p l i c 1 (1967) no. 2. (English translation) 8. M. Sion, Lecture notes on measure theory;, Biennial Semi-nar of the Canadian Math. Congress, 1965. 

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