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Cylinder measures over vector spaces Millington, Hugh Gladstone Roy 1971-12-31

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-CYLINDER MEASURES OVER VECTOR SPACES by HUGH GLADSTONE ROY MILLINGTON B.Sc, University of West Indies, Jamaica, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of MATHEMATICS We accept this thesis as conforming to the required standard. The University of British Columbia March 1971 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver 8, Canada ii. Supervisor: Professor M. Sion ABSTRACT In this paper we present a theory of cylinder measures from the viewpoint of inverse systems of measure spaces. Specifically, we consider the problem of finding limits for the inverse system of measure spaces determined by a cylinder measure y over a vector space X * For any subspace Q of the algebraic dual X such that (X,n) is a dual pair, we establish conditions on u which ensure the existence of a limit measure on Q . For any regular topology G on £2 , finer than the topology of pointwise convergence, we give a necessary and sufficient condition on y for it to have a limit measure on £2 Radon with respect to G We introduce the concept of a weighted system in a locally convex space. When X is a Hausdorff, locally convex space, and . £2 is the topological dual of X , we use this concept in deriving further conditions under which y will have a limit measure on £2 Radon with respect to G We apply our theory to the study of cylinder measures over Hilbertian spaces and Jl^-spaces, obtaining significant extensions and clarifications of many previously known results. iii. TABLE OF CONTENTS Pages INTRODUCTION 1 CHAPTER 0: PRELIMINARIES 3 1. Set-theoretic Notation2. Outer Measures and Integrals 4 3. Radon Measures 6 4. Induced Radon MeasuresCHAPTER I: CYLINDER MEASURES OVER VECTOR SPACES 9 1. Inverse Systems of Measure Spaces 9 2. Cylinder Measures over Vector Spaces 16 3. Non-topological Limit Measures 20 4. Radon Limit Measures 33 5. Finite Cylinder Measures 45 CHAPTER II: CYLINDER MEASURES OVER TOPOLOGICAL VECTOR SPACES . 52 1. Notation 52. E-tight Cylinder Measures 54 3. Limits of Continuous Cylinder Measures 64. Induced Cylinder Measures 7CHAPTER III: APPLICATIONS 77 1. Preliminaries2. Hilbertian Spaces 82 3. Nuclear Spaces 94 5. £P-spaces. 101 APPENDIX: 114 1. Special Measures on Finite-dimensional spaces 114 2. Positive-definite Functions on Vector Spaces 126 3. CM-spaces 133 4. Examples 14BIBLIOGRAPHY: 153 ACKNOWLEDGEMENTS I wish to thank Dr. Sion for his invaluable guidance throughout the writing of this thesis. I wish to thank also Dr. Greenwood and Dr. Scheffer for their helpful suggestions. Finally, I would like to thank Miss Barbara Kilbray for her accurate typing of this thesis. INTRODUCTION Cylinder measures were first introduced independently by I.M. Gelfand (Generalized random processes, [10]) and K. Ito (Sta tionary random distributions [15]) as a more general kind of stochastic process (J. Doob [7]), and arise naturally in probability theory when one defines a stochastic integral ([4] p. 137, [7] p. 426, [15] p. 211). On the other hand, the demands of theoretical physics (in particular, quantum field theory and statistical mechanics) have led to a considerable interest in the theory of integration over function spaces ([13], I.M. Gelfand and, A.M. Yaglom [12], I. Segal [43]), where the integrals considered are defined with respect to some cylinder measure (e.g. as in L. Gross [13] p. 53-54). In the study of cylinder measures researchers have concen trated on two main approaches: in one, a cylinder measure is viewed as a linear map on a vector space into the space of measurable functions on some probability space (Gelfand [10], Ito [15], Gelfand and Vilenkin [11] Ch. IV , Fernique [9]); in the other, it is viewed as a set function on a family of cylinder sets of a vector space ft (Minlos [25], Gelfand and Vilenkin [11] Ch. IV, Prohorov [33], Badrikian [1], L. Schwartz [39]). Inherent in both approaches is the notion of an inverse (or projective) system of measure spaces ([25] p. 293, [11] p. 309, [33] p. 409, [1] p. 2, [3.9] p. 832, [9] p. 34). In this thesis we view a cylinder measure as an inverse system of measure spaces indexed by the finite dimensional subspaces of a vector space X . Moreover, we do so without any a priori choice of the "target" space ft on which the limit measure is to live. The basic problem of finding a limit of the system on a space ft of linear functionals is then analyzed with variable ft in Chapter I. The key idea there is to examine the measure theoretic size of ft in relation to the algebraic dual X . To this end the notion of "almost" sequential maximality is introduced. Next, in Chapter II, we consider the more standard problem of finding a Radon limit measure on ft when X is a topological vector space and ft is its topological dual. When X is Hausdorff and locally convex, by introducing the concept of a weighted system in X , we establish a condition for the existence of such a Radon limit in terms of the notion of continuity with respect to a weighted system. In Chapter III we apply the theory of Chapter II to the study of cylinder measures over Hilbertian, nuclear, and £^-spaces, thereby extending and clarifying several previously known results. In the appendix we establish mainly technical results used in the proofs of Chapter III and present several counter-examples. 3. CHAPTER 0 PRELIMINARIES 1. Set-theoretic Notation. In this work we shall use the following notation. (•1) 0 is the empty set. For any sets A and B , A ~ B = {x £ A : x i B} . w is the set of finite ordinals. R is the field of real numbers. R+ = {t e R : t _> 0} . <£ is the field of complex numbers. In proofs we shall abbreviate "such that" to "s.t." (2) For any set X and family H of subsets of X , - LJH = u H , ]~\H = n H , HeH HeH P(H) = {H'C. H : H' is countable, disjoint, and = [_|H} . For any A c X , H|A = {HA A : H E ff} . (if is a compact family iff for any n" c H , if A ch" is finite => (I H , then [\H* ^ 0 . 4. For any topology G on X , K(G) = {K c X : K is closed and compact in G} (3) For any set X and A c x > 1, : x e X -> 1 e <E if xeA , A OeS if x e X ~ A , For any f : X -> Y , B c Y , f|A : x E A+ f(x) e Y , f [A] = {f (x) : x e A} f_1[B] = {x e X : f(x) e B} For any sets X and Y,IcXxY,xeX,yeY , Ix = (Y e Y : (x,y) e 1} , IY = {x £ X : (x,y) el}. 2. Outer Measures and Integrals. Our measure-theoretic approach is essentially that of Caratheodory, as given by M. Sion in [44] and [45]. (1) For any set X and Caratheodory measure n on X , M is the family of n~measurable sets. n is an A-outer measure iff A c. M , and for any A c X , n ^(A) = inf{n(A') : Ad A' e rt} n is an outer measure iff n is an M -outer measure. n Throughout this work all measures considered will be outer measures, n is the Caratheodory measure on X generated by T and A iff A is a family of subsets of X with 0 e A , + T : A -> R with T(0) = 0 , and for any BC X , n(B) = inf{ E T(H) : f/c A is countable, Bcr|jH} . HeH (X,n) is a measure space iff X is a set and n is an outer measure on X (2) Integration. We observe that for any measure space (X,n) , P(M ) is directed by refinement. In general, we shall be considering complex-valued functions on X and therefore also complex-valued integrals. However, we point out that for any n-measurable f : X -> R+ , ' / fd = lim I (inf f[B])*n(B) n PeP(M ) BeP n lim E' (sup f[B])-n(B) . PeP(M ) BeP Further, for. any f : X ->• R+ , the outer integral /*fd = lim Z (sup f[B])'n(B) n PeP(M ) BeP n is a well-defined point in R U {°°} 6. (3) Radon Measures In this paper, many of the measures we consider will in fact be Radon measures. We give the relevant definitions below. For any set X and topology G on X , ri is a G-Radon measure on X iff (i) n is a G-outer measure on X , (ii) K e K(G) => n(K) < ». , and for every G e G , (iii) n(G) = sup'{n(K) : KCG , K e K(G)\ . For any G-Radon measure n on X , supp n = support of n (4) Induced Radon Measures Let Y be an abstract space. For any finite measure space (X,n) and T : X -> Y , T[n] is the Caratheodory measure on Y generated by n 0 T-1 and {ACY : T_1[A] e M^} We shall use the following lemmas. Lemmas (1) For any A e M^-j > T 1[A] e M and therefore T[n](A) = n(T 1[A]) 7. (2) For any space Z and U : Y -> Z , U[T[n]] = (U 0 T)[n] (3) If X and Y are topological spaces, T is continuous, and r\ is Radon, then T[n] is Radon. Proof of Lemma 4.1. Let & = (Ac ! : T_1[A] E M } . n First we note that for all A e A , T[nJ(A) = n(T_1[A]) . Let A e Hpj- j • Since A is a a-field and T[nJ is an . A-outer measure, there exists A' e A s.t. A c A' and T[n](A) = T[r|](A') . If T[n](A) = 0 then (1) 0 < n(T_1[A]) <_ n(T_1[A'J) = T[n](A') = 0 . In general, since T[n](Y) < 00 , T[n](A' ~ A) = 0 and therefore by (1), n(T~1[A' ~ A]) = 0 and T_1[A' ~ A] e M n Hence, since T ^[A'] e M , T_1JA] = T_1[A'] - T_1[A' ~ A] e M n The second assertion now follows immediately from the fact that T[n]|A = n o T"1 . 1 Proof of Lemma 4.2. We need only observe that, by 4.1 above, '{BCZ : (U 0T)_1[B] e M^} = {B C Z : U_1[B] e and for any B CZ Z s.t. (U 0T)_1[B] e , n((U o T)_1[B]) = T[n](U_1[B]) . I Proof of Lemma 4.3. Let A e MTrn] anci £ > 0 . By Lemma 4.1, T_1[A] e , and therefore, since n is a finite Radon measure, there exists a compact K c T "*"[A] s.t. n(T_1[A.]) - n(K) < e . ' • Then T[K] c A is compact and T[n](A) - T[n](K) <.n(T_1[A]) - n(K) < e Hence, since e > 0 was arbitrary, (1) T[n](A) = sup{T[ri](C) : C C A is compact} . Since T[n] is finite it then also follows that T[n](A) = inf {T[n] (G) : G^A is open} , and since T[n] is an outer measure we therefore conclude that for all BC Y , (2) T[n](B) = inf {T[n](G) : G O B is open} . consequently T[n] is Radon. ® CHAPTER I CYLINDER MEASURES OVER VECTOR SPACES As indicated in the introduction, we shall treat cylinder measures as being special inverse systems of measure spaces (Choksi [6]). In the following section we introduce the basic notions and results that we shall require about such systems. 1. Inverse Systems of Measure Spaces. Throughout this section, F is an index set directed by a relation < For any E e F , (Xp,y ) is a measure space, E For any E and F in F with E < F r_, „ : X_, X„ is surjective, L, r r L with being the identity map. 10. Definitions (X , p ) E is an inverse sj'Stem of measure spaces relative r r c £ r to the maps r , E, t iff, for any E,F , and G in F with E < F < G , rE,G = rE,F ° rF,G ' and, for all A e M , r~^[A] £ MF , yF(r^F[A]) = uE(A) Let (XLpjPp)^,^^ be an inverse system of measure spaces relative to the maps r E, b If ft is a set, and for each F e F , p_ : ft -> X„ is surjective, then, we call (ft,0 a r r limit relative to the maps p of the given inverse system of F measure spaces, iff for each E and F in F with E < F PE = rE,F ° PF ' and, F is an outer measure on ft such that for all A e M , E p'V] e M , ap'V]) = UE(A) . For the rest of this section we assume that (X , p ) r r r r £ r is an inverse system of measure spaces relative to the maps r . b,.b 1.2 Definitions For any set ft , and surjective maps p : ft ->• X such F F that for any E and F in F with E < F , PE = rE,F ° PF ' '(1) Cyl(ft,P) = {P~V] : F e F, A e Mp} , (2) T(ft,p) : p'1^] e Cyi(n,p) -»• up(A) e R+ , (3) n„ is the Caratheodory measure on Q generated by ft, p T _ and Cyl(ft,p) . ii, p When there can be no ambiguity we shall omit the subscripts ft and p Remarks The following assertions are readily established. (Choksi [6], Mallory and Sion [23]). (1) Cyl(ft,p) is a field. (2) T is well-defined and is finitely additive on Cyl(ft,p) (3) Cyl(ft,P) cMr . (4) (ft,n) is a limit relative to the maps p of the given r inverse system of measure spaces iff n|Cyl(ft,p) = x . (5) There exists an outer measure E, on ft such that (ft,£) is a limit relative to the maps p of the given inverse system r of measure spaces <=> 12. T is countably additive, in which case n is such a measure. We now suppose that (ft>n— —) is a limit relative to the maps p , ft,p F ft C ft , and for each F e F , Pp = Pp|ft is surjective. We shall be interested in determining when (ft,nn ) itself " 5 P is a limit relative to the maps p F 1.3 Lemmas. With the above notation and hypotheses, (1) ) is a limit relative to the maps p of the given ft, p F inverse system of measure spaces iff rrr-(A) = n— —(A f\ ft) for all A e Cvl (ft, p) . Si,p ft,P v ' (2) For any F'c F , let A(F') be the set of all f e ft such that there does not exist g e ft with Pp(g) = PF(f) for every F e F' . If, for every {F } C F with F < F for each new n new n n+1 then n--(A({F } )) = 0 , ft,p n new n--(A) = rnr-(A O ft) for all A e Cyl(ft,p) Proofs 1. Let Cyl(fl.p) = Cyl , Cyl(n,p) = Cyl , T Tft,p ' T Tft,p ' ft, p ft,p 1.3.1 Since the maps p are surjective, for each A in Cyl , ——— Q there exists a unique A e Cyl s.t. A = A A ft • Then, A e Cyl -> A e Cyl is bijective and T(A) = 7(A) for all A e Cyl . Hence, for any B o fi , n(Bnfi) = inf{_E_ 7(H) : He Cyl is countable, HeH BnftcjjH} = inf{ E T(H) : He Cyl is countable, B A ft cjjH} HeH = n(B n n) . Consequently, if (ft,n) is a limit, then, by Remark 1.2.4., for any A e Cyl , 7(Ac\ ft) = n(A) = T(A) = 7(A) = 7(A) . On the other hand, if 7(A) = n(A f\ ft) for all A e Cyl , then, again by Remark 1.2.4., for any A e Cyl , n (A) = "7(1 A 0) = 7(A) = 7(A) = x (A) , and therefore (ft,n) is a limit. 14. 2. For any subfamily F' of F ; let ,—1. Cyl(F') = {Pp [B] : F e F' , B e Mp} We shall show that for any A e Cyl there exists A'c ft s.t. rf(A') = 0 and rf(A ~ A') = rf (A (\ ft) . In which case, rf(A) = rf(A ~ A') + rf(A a A') = Tf(A ~ A') = rf(A f\ ft) and the lemma follows. For each new , let Cyl be countable with A r\ ft C I lH and £ 7(H) < 7f(A f\ ft) + 1/n . n Heff For each new , choose countable F c F with n (1) {A} u H c Cyl(F ) . n n Since (X^ ^p)pep ^s an inverse system of measure spaces relative to the maps r , we may further assume that (2) F is a sequence {F .}. in F with F . < F . n n,j jew n,j n,j+l for each jew Let A = A(F ) n n and A' = U A n new Since Tl(A ) = 0 for every new , then (3) n(A') = 0 .' Let new . For each f e A ~ there exists g e ft s.t. 7 (g) = ?(f) for all F e F . 15, In particular, by (1),. g e A . Hence, for some H e H , with H = p [p-''"[H] ] n (J " G for some G e F , n g e H , and consequently, f e p^tpgCf)] = PG1[pG(g)] c p/tPgtH]] = H . It follows that A ~ Anc UHn , and therefore (4) 7(A ~ A ) < n(Aa ft) + 1/n . n — Since A c ft ~ ft for each new, n A' C ft ~ ft • Hence A <A ft c A ~ A' , and therefore, by (4), for each new, "n(A r\ ft) <_ n"(A ~ A') <_ n(A ~ A ) <_ n"(A r\ ft) + 1/n Consequently, ~r\(A n ft) = ri(A ~ A') . 16. 2. Cylinder Measures over Vector Spaces.-We shall view a cylinder measure over a vector space X as being.an inverse system of measure spaces whose indexing, set is the family of finite dimensional subspaces of X In this paper we shall consider only complex vector spaces, and we shall hereafter refer to them simply as vector spaces. By the term subspace we shall always mean vector subspace. We note that if F is a finite-dimensional vector space, then there is a unique Hausdorff topology on F under which it is locally convex (the Euclidean topology). Since this is the only topology on F that we shall ever consider, explicit reference to it is hereafter omitted. Throughout the remainder of this work, we shall use the following notation. For any vector space X , X is the set of linear functionals on X to § , w is the topology on X of pointwise convergence, For any AcX , A° = {f e X^ : |f(x)| <_ 1 for all x e A} . Fv is the family of finite-dimensional subspaces of X A directed by C . When there can be no ambiguity we shall omit the subscript X For any subspaces E and F of X with E C F > r : F -*• E is the restriction map, E, F i.e. for all f e F , rEjF(f) = f|E . 17 In what follows, E and F will always denote finite-dimen sional vector spaces. For any subspace ft of X , (X,ft) is a dual pair iff r Ift is surjective for every F e F Remark. With the viewpoint of inverse systems discussed in the preceeding section, taking Fv as our index set and letting A X„ = F" for each F E Fv , a A we note that, for any E , F and G in Fv with E c. F C G , A the restriction map r is surjective and continuous, E,l and rE,G rE,F ° rF,G Thus, we shall make the following definition. 2.1 Definition. (1) Let X be a vector space. u is a cylinder measure over X iff y : F e F •+ y , a Radon measure on F , F is such that (F ,y„)„ r is an inverse system of measure spaces relative F r E r to the restriction maps r . (2) p is a cylinder measure iff p • is a cylinder measure over some vector space X Remark. Let X be a vector space. If r -» * u : F e r ->• u , a finite Radon measure on' F , r then, by §0.4, p is a cylinder measure over X iff for any E and F in F with EC F , yE = rE,F[lJF] • Let ft be any subspace of X . For any E and F in with E c F , Hence, when (X,ft) is a dual pair, the viewpoint of Definition 1.1.2 applies, with PE = rE X^ f°r eacl1, E e F • We shall therefore make the following definition. 2.2 Definition. | ft = r E,F Let X be a vector space, jn, a cylinder measure over X and ft be a subspace of X such that (X,ft) is a dual pair. 19. For any outer measure £ on ft' , ^ is a limit measure of u on ft iff (ft,5) is a limit relative to the restriction maps r ft of the inverse system of measure spaces (F ~, y_)_ r r ,A F Fer Remarks. From the theory of inverse systems of measure spaces we know several conditions under which we can put a limit measure on the pro jective limit set L , where L = U e n E" : £ = rw ^(O , E c FV . EeF E E)F F Since there exists a set isomorphism r : X* -> L such that r_ v(f) = (r(f))_ for all f E x" and F E F , r , A r it follows that L is sequentially maximal (Defn. 3.4). Hence, by a theorem of Bochner ([4] p. 120), we deduce that JA/ always has a limit measure on X . However, little has been said about the properties such a limit measure can have. Therefore, in the next section, we shall construct one having special approximation properties. v'c Unfortunately, for most practical purposes X is far too unwieldy. We shall therefore be studying the problem of putting limit measures on subspaces of X 20. 3. Non-topological Limit Measures. Given any cylinder measure y over a vector space X , and subspace ft of X such that (X,ft) is a dual pair, we shall determine sufficient conditions on y for it to have a limit measure on ft . Throughout this section we shall use the following notation. X is a vector space. For any cylinder measure y over X , and subspace Q of X Cyl (n) = {fi o r^A] : F e F, A e Mp} , T : ftfNr"1 [A] £ Cyl (ft) -> y (A) e R+ , y, ft r, A y s y^ is the Caratheodory measure on ft generated by T n and Cyl (ft) . y ,ft y y y>x-and y = yx* • In what follows, ft will denote a subspace of X such that (X,ft) is a dual pair. From Definition 1.1.2 and Remarks 1.2 we get the following assertions. 21. 3.1 Propositions. Let u be a cylinder measure over X (1) For any outer measure E, on ft, E, is a limit measure of y iff Cyl (ft) c. Mr and E, I Cyl (ft) = T 0 . y C y y,ft (2) If there exists any limit measure of y on ft , then y^. is a limit measure of y In view of Proposition 3.1.2, when looking for a limit measure , of y on ft , we shall concentrate on y^ When ft = X , we have the following result. 3.2 Theorem For any cylinder measure y over X , y is a limit measure of y If C = {r'^jK] : F e F, K c is compact} F,X then C is a compact: family, and for any A e M , , y* V(A) = sup{yX(C) : C CA , C e C } . (We note that C^ is also a compact family.) 22. Proof For each F e F (1) y_ is Radon and cr-finite. r Hence, for any A e Cyl^(X ) , (2) T*(A) = SUP{T*(C) : C C A , C e C} Since MX(A) 1 T*(A) for all A e Cyl^x*) , v?e also deduce from (1) that (3) y is o-finite. Hence, by Thm II.2.5 of [23], the assertions of the theorem will follow once we show that C is a compact family. (Also see [24]). For any C. = r"1 V[K.] £ C, j = 1,2, let F be the linear soan of F U F , and K = A r ^ „[K.] 1 2 j=l,2 Fj'F 1 Then K is compact and C. <% C„ = r \jK] 12 F,X Hence, (4) C is closed under finite intersections. For any C'^ Cs.t. ] [ot =|= <f> for every finite a CZ C , let A = { fl ct : aC C* is finite) We note that A is a filterbase ([8] p. 211). In view of (4), for each finite a CC , let o P| a = r^ ^[K ] for some F& e F and compact K c F" , and Y = U(F : a C C is finite) . a 23. From the remarks preceding (4),' we see that for any finite subfamilies a and 3 of C , aC3=>FaCF3 ' and therefore Y is a subspace of X n Let U be a maximal filterbase in X ([8] p. 218) which is a subfilterbase of A ([8] p. 219, Thm. 7.3). Then, for each finite a C C , (r„ [U]) _T is a maximal filterbase in F ' and F ueJ ' a there exists u e U s.t. r [u] c K F a a * Since K is compact and F is Hausdorff, this ultrafilter a a -converges to a unique point f e K a a' We note that if F = F„ , then f = fn . Also, for any finite a 3 a 3 • . subfamilies a and 3 of C with a C 3 , , r ^ (fj = f • F ,F v 3 a ' a 3 since the restriction map is continuous and rF ,X = rF ,Ffl ° rF.,X ' a a 3 3 A Consequently, there exists a unique g e Y s.t. g i F = f for each finite a C C' 1 a a •k If f e X is any linear extension of g , then, for each finite f e r;\xtrF ,x(f)] = ^\A]C^\X[V = ^ a C C . "I r "I r r n _ "I ? ,X[fa] C rF , a a Hence, fl'c + • . It follows that C is a compact family. Next, we consider the problem of finding a limit measure of y on an arbitrary ft Since y is always a limit measure of y , application of Lemma 1.3.1 yields the following basic result. 3.3 Lemma. For any cylinder measure y over X , y has a limit measure on ft iff (1) y*(A) = y"(Ar\ft) for all A e Cyl (x") . y However, we are interested in finding intrinsic conditions on our systems which will guarantee the existence of a limit measure on ft One such condition is the following, which is of considerabl importance in the general theory of inverse systems of measure spaces (Bochner [4], p. 120, Choksi [6], Mallory and Sion [23]). 3.4 Definition ft is sequentially maximal iff for any sequence {F } in F with F c F ,- for each n neco n n+1 •k new , and f e F such that r_, _, (f ,-) = f , there ' n n F ,F ,1 n+1 n n n+1 exists g E ft such that r„ = f for each new F ,X n n 25. Remark. We note trial: X is sequentially maximal. Consequently, A the fact that p is a limit measure of p follows also from a theorem of Bochner ([4], p. 120). Since p is always a limit measure of yj, , application of Lemmas 1.3.2 and 3.3 yields the following. 3.5 Proposition If ft is sequentially maximal, then every cylinder measure over X has a limit measure' on ft However we have the following. 3.6 Observation. If X is a topological vector space containing a bounded, countable, linearly independent subset, and ft is its continuous dual, then ft is not sequentially maximal, (e.g. whenever X is an infinite-dimensional, metrizable, locally convex space). Proof Let {a : n e w} be a bounded, countable, linearly n independent subset of X , and for each new let Fn be the linear span of {a ,...,an} . Then, for any f e X with f(a ) = n for every new. 26. (1) f[{a ; n e w}] c £ is unbounded. n Hence, there cannot exist g e ft s.t. elF = f F for 01 n 1 n every new. For if so, then g U F is continuous, and therefore §[{an : 11 £ ^s bounded, which contradicts (1). "Si Since, in the theory of cylinder measure, ft is often the continuous dual of metrizable I.e. space ([11], [39]), it follows that the condition of sequential maximality does not apply in many important situations. In order that we might take fuller advantage of Lemma 3.3, we therefore weaken the notion of sequential maximality. 3.7 Definition Let y be a cylinder measure over X U is y-sequentially maximal iff for any sequence {F } in F with F rF ,-, for every n new n n+1 new, and e > 0 there exxsts A e M_ for each new, such that n F n Z yp (An) < e new n ' and for any sequence {f } with ^ n new f e F" ~ A , r (f ) = f , n n n F ,F n+1 n n there exists g e ft such that r (g) = f for each. new. n' . 27. The following key theorem of this section is now an immediate consequence of Lemma 1.5.2 and the above definition. 3.8 Theorem Let y be a cylinder measure over X If ft is y-sequentially maximal, then y has a limit measure on ft . We now establish a condition on y which ensures that ft is y-sequentially maximal. 3.9 Definition Let y be a cylinder measure over X •k For any family H of subsets of X , y is H-sequentially tight iff for any sequence {F } in F with F c. F , -, ror each n miii n n+1 n E (a , A E M with y (A) < « t and e > 0 , 0 0 • there exists H £ H such that yp (r^1 F [A]) ~ rp X[H]) < e for all n £ GO . n 0' n n' 3.10 Theorem Let y be a cylinder measure over X If y is H-sequentially tight for some family H of w -compact subsets of ft , then ft is y-sequentially maximal, and therefore y has a limit measure on ft 28. We point out that under certain conditions y-sequential maximality of ft is also a necessary condition for y to have a limit measure on ft 3.11 Proposition Suppose that the Mackey topology on X induced by ft ([47] p. 369) restricted to any subspace of countable dimension is metrizable. For any cylinder measure y over X , if y has a limit measure on ft , then ft is y-seq.uentially maximal. Proofs 3. Lemma. Let {F } be a sequence in F with F o F for 1 n new n n+1 each new, * A K be a w -compact subset of X For any sequence {f } with J n n new f e r_, [K] and r (f ,,) = f . n F ,X F ,F n+i n n n n+1 there exists g e K s.t r (g) = f for all new . r , A n n Proof For each new , (1) r^1 [fJnK-H • n Since rF [fn+1] = fn . n n-rl 29, (2) r"1 [f ] o r 1 [f ] . F , X n F ,. n+1 n n+1 •k * Also, since r is w -continuous and K is w -compact, n -1 * (3) K n r [f ] is w -compact. r , A n n Since w is a Hausdorff topology, it follows from (1), (2) and (3), that rS (K r\ r"1 x[fn])> $ , new n' and the lemma follows. Let {F } C F with F c. F for each new, n new n n+1 {B}. C M_, , with u_ (B.) < oo for each jew , je-w FQ Fq j and Fn = U B. ,. 0 • J jew Since u is H-sequentially tight for some family H of k w -compact subsets of Q , given e > 0 , for each jew choose a w -compact K c 9, s.t 8UWF (^,F [Bj] ~ rF ,X[Kj]) < . n 0', n J n J Let Cn= .U (IV [Bj] ~ RF ,x£Kj]) jew 0' n J n' J Ao = co Vi = Vi -n n-rl 30. Then, for any k e w , n=0 n k and k k 1+1 n=0 n j=l Hence E p„ (A ) < e F n new n If is a sequence s.t. for each new, f e F* ~ A , r_ _, (f ) = f n n n F ,F n+1 n n n+1 then, for some jew , and hence for every new, Consequently, by the Lemma, there exists g e ft s.t. r „(g) = f for all new, and it follows that ft is /A.-r , A n n sequentially maximal. The last assertion is immediate from Thm. 3.8. Proof of 3.11 For any {F } c F with F c F ,, for each new, let n new n n+1 Y = U F with the restricted topology, new A = {f e X : there does not exist g e ft s.t. rF = rF X^ f°r a11 n E ^ ' n' n' 31, 8 = {r^1 X[A] : n e to, A E M } n' n be the o-field generated by B , and n' the Caratheodory measure on X generated by |B and B i Since the topology of X restricted to Y is metrizable, choose a sequence {V, }, of absolutely convex neighbourhoods k keco of the origin in X s.t. {V, n Y}, is a base for the neigh-K KEU) bourhoods of the origin in Y . Using the Hahn-Banach extension theorem, one readily checks that keo) new n n It then follows that (1) . A E © , and, since Cyl^(X ) c M , (2) A £ M , y* We note that (3) A c X* ~ fi . >v k k Since y is o-finite and y (A) = y (A O ft) for all A E Cyl (X*) , from (2) and (3) it follows that (4) y*(A) = 0 * * Since B is a field and T is countably additive on Cyl (X ) (Thm. 3.2, Remark 1.2.5), we have that n|B = T*|B and y"|8 = x*|B 1 y1 1 y1 •k . However, |B has a unique countably additive extension to V5 . Hence, since CM n M n y T,|$=y*|<&, and therefore, by (1) and (4), n(A) = 0 . Consequently, given any e > 0 , there exists {B . } . C 8 -s. t. 3 JEW ' * AC U B. and Z T (B.) < e . i y i ' '• jew jew For each j e to , let B =rFX [B'] , B^ £MF , J n. J n. } 3 and B. 4= r"1 ,r[B'] for any n. < n. and B' e M J n n For each n e w , let An = UtB* : 3 £ a), nj = n} Then, (5) E y (A ) < e . r n new n Further, if {f } is any sequence s.t. for n new each new, f e F ~ A , r_ _, (f ..) = f n n n F , F n+1 n n n+1 then, there exists f e x" ~ U r [A ] s.t. r , A n new n r„ „(f) = f for each new. F ,X n n Since A C U B. = U r"1 V[A ] , from the definition of A 3 F ,X n jew new n follows that . (6) there exists g e Q s.t. r? (g) = f for all new. n' Since the sequence {F } in F with F tg. F for each n n new n n+1 new was arbitrary, we conclude that 0. is y-sequentially maximal. 33. 4. Radon Limit Measures In this section we shall consider the problem of finding Radon limit measures. The technique we use was communicated to us by C. Scheffer. In this section we shall use the following notation. X is a vector space, ft is a subspace of X such that (X,ft) is a dual pair. For any topology G on ft and cylinder measure y over X. , • g : ACX%inf W(rF,X[A]) : F z F} ' gA : G e G + sup {g(K) : K e K(G) , Kc G} , y is the Caratheodory measure on ft generated by g.,„ and G We shall hereafter assume that y is a fixed cylinder measure over X , G is a regular Hausdorff topology on ft which is finer than w restricted to ft. We have the following important assertions, 4.1 Propositions (1) y_ is a G-Radon measure on ft , and y |G = g., b G (2) Cyl (ft) a M y yG (3) If there exists any G-Radon limit measure of y on ft , then \in is a limit measure of y (7 34, In view of the above propositions, when searching for a' G-Radon limit measure of y , we shall restrict our attention to \in b Following Scheffer [37] we make the following definition. Our terminology is slightly different. 4.2 Definition For any family H of subsets of X , u is H-tight iff for any E e F, A e with ^^.(A) < <*> , and e > 0 , there exists HeH such that u^Cr"1 „[A] - r_ [H]) < e for all F e F with E CF . r Jti, r  , A We point out that the above definition is a "uniform" version of the definition of H-sequential tightness (Defn. 3.9). We now have the following key theorem concerning the existence of a G-Radon limit measure of y 4.3 Theorem u has a G-Radon limit measure on 0, <=> y is K(G)-tight. Remark The above theorem extends a result due to Mourier [26], and Prohorov [33] (§5 Lemma 3). However, our approach is somewhat different from theirs. Theorem 4.3 has a useful corollary. 35. Corollary If X is a metrizable, locally convex space, a is its continuous dual, and {V } is a base for the neighbourhoods of the n new origin in X , with vn+1 vn f°r every new , then, y has a w -Radon limit measure on ft <=> y has a limit measure on ft <=> y is {V°} -tight, n new Proofs 4. Notation Let H = (Cyl 'CO)). A (w"|ft) , y = vG , and T = T y,ft We shall need the following lemmas. L.l For any A e'Cyl^(ft) , (1) T(A) = inf {T(H) : A C H e H} (2) YCA) <I(A) . 1^2 For any K c K (G) , (1) r [KJ is compact for every E e f , VtWt -wv^oW^U*. (2) g(K) = y (K) = inf' (T(H) : K C H e H} (3) For any E and F in F with E c F , and A e , KE(RE,X[K]) > ^FCrF,X[K]) • 36, Proof of L.l.l For any E e Fv and B e M , since r ' is X E E, X k w -continuous and y is Radon, T(ft C\ r'^tB]) = yE(B) = inf {y^,(G) : B c G C E" , G is open} > inf (T(H) : ft f\ ^"^[B] c H e H} > x(fi A r'^tB]) . Ji, A Ji , A Proof of. L.1.2 We note that H C G , and for every HeH, gFT(H) <_ T(H) . Hence, y(A) = inf { I g.,.(H) : H'c'H is countable and Ac |JH'} HeH' <_ inf { E T(H) : H'C H is countable and Ac {JH1} HeH' < inf {x(H) : Ac H e H} = T(A) , by L.l.l. Proof of L.2.1 We only observe that for every E e F, y^, is Radon, * k K(G) C f((w ) , and r„ v is w -continuous. b, A Proof of L.2.2 For every E e F, y„ is Radon and r is w -continuous. ii L, X Therefore, g(K) = inf {yw(G) : E e F, r_. V[K] C G <=. E" , G is open} Ji Ji, A = inf (T(H) : K c H e H} . ' On the other hand, by L.l.l, y (K) = inf { E T(H) : H'c H is countable, Kc UH'} HeH' = inf {x(H) : K C H e H}, since, K is G-compact, H<z G,M is closed under finite unions, and x is finitely subadditive on Cyl (ft) bv 37. Proof of L.2.3 We have that [K] C r E,F E,X [K]] Hence, ^E(rE,X[K]) = ^F(rEVrE,X[K]]) l^F(rF5X[K]) 4.1.1 To show that y is a G-Radon measure, by Sion [44] Ch. V, Thm. 2.2, we need only show that (1) g(<j>) = 0 , g is positive, monotone, subadditive and additive on K(G) , (2) Y(K) < ro for a11 K e K(G) , Except for additivity, the properties of g are immediate from L.2.2. We shall now establish the additivity of g on K(G) Let K and K be in K(G) with K n K = | . Since K(G) c K(w"|ft) , and • w ft is regular and Hausdorff, there exists G. e w"|X, K.C G., i = 1,2 , with -GlfN G2 = * " However, H is a base for w ft , and is closed under finite unions. Consequently, since K ,K„ are w ft-compact, there exists H. e H K.C H. » 3 i = 1,2 with H n H2 = <f> 38, Then, by L.2.2, g(V + g(K2) = yfl(Kl) + uQ(K2) I inf {y-CA^) : K. c A. e M \ 3=1,2 ° ^ 2 I inf {y„(Aj) : A e M , K. c A3 <z H.} ' 3 = 1,2 " ^ J J TO* " = inf. {y (A (j A ) : AJ e M , K. C A2 c H. ; j = 1,2,} , yft 2 2 <_ y^ft^ u K2) = gC^U K2) . Hence, by the subadditivity of g , gd^u K2) = gO^) + g(K2) , Since and K2 were arbitrary it follows that g is additive on K(G) . It remains for us to prove (2). Let K e H(G) . For any F e F, since y is Radon and r [K] r r ,X is compact (L.2.1), •ft r\ r~*x[r [K]] e Cyl^(ft) and by L.1.2, Y(K) <_ y(0 ri.r^x[rF)X[K]]) £ T(n ri r^Er [K]]) = yF(rF)X[K]) < oo . , Hence, y(K) < oo for all K e K(G) . 4.1.2 Let F e F and A z V If y (A) = 0 , then, by L.1.2, Y(ft A r^lAj) = 0 . and therefore ft A r"1 [A] e M F,X y 39, / 5 Otherwise, since u„ is Radon, choose a Borel subset B of F r with A C B and n (B ~ A) = 0 r By the preceding observation, ft 0 r"1 [B ~ A] e M . a, A y However, r ft is (j-continuous since w ft C G , F, X and'by Prop. 4.1.1, y is G-Radon. Hence, ft f\ r""1 „[B] E M , F,X y and therefore, ft f\ rp^x[A] = ft ri r^x[B] ~ ft r» r~^[B ~ A] e M . We shall now establish another useful lemma. L.3 For every K e K(G) , g(K) <_ Y(K) . If y is a limit measure of u , then, for every K E K(G) , . g(K) = y(K) . Proof of L.3 Let K E K(G) .. By Prop. 4.1.1, y(K) = inf',{gA(G) : K C G e G} >. g(K) . If y is a limit measure of u , then, by L.2.2 and Prop. 3.1, g(K) = inf (T(H) : K c H e HI = inf {y(H) : K c H e H} > y(K) , since y is G-Rado'n and H C G 40. 4.1.3 Let £ be any G-Radon limit measure of y on ft • For any K e K(G) , ?(K) = inf U(G) : K C G e G} <_ inf U(H) : KCHEH} since H C G , = inf {T(H) : Kc II e H} by Prop. 3.1.1 = g(K) by L.2.2 <_ Y(K) by L.3. Hence, as E, and y are both G-Radon measures on ft , £ (A) <_ Y (A) for all A c ft . In particular, by Prop. 3.1.1, for any A e Cyl^(fi) , T(A) = 5(A) < Y(A) , and-therefore, by L.1.2, y(A) = T(A) • From Props. 3.1.1 and 4.1.2 it now follows that y is a limit measure of y 4.3 By Prop. 4.1.3, if y has any G-Radon limit measure on ft , then y is a G-Radon limit measure of y . Hence, for any E e F and A. e with PE(A) < <*> , -1 M . ..-1 Y and therefore ft n rE X[A] e M , y(ft r\ r^tA]) < » , y(ft O r^tA]) = sup {y(K) : K e K(G) , Kc ft Ar^tA]:? . Hence, for any e > 0 , there exists K e H(G) with Kc ft ftr [A] £j , A and y(ft A rE1x[A] ~ K) < e In which case, for any F e F with E C F , yF(r^F[A] - rp)X[K]) . = Y(n ^ r^x[r^F[A]) ~ « " r^r^K] ]) <_ Y(fi n r'^fA] ~ K) < e . It follows that y is K(G)-tight. . We now show that /<(G)-tightness of y is a sufficient condition for y to have a G-Radon limit measure on ft . IN view of Props. 4.1 and 3.1.1, we need only show that (1) y is K(G)-tight => y|Cyl (ft) = T . 'If, for every A e Cyl (ft) , (2) T(A) = sup (g(K) : K e K(G) , KC A} , then, for every A e Cyl (ft) , since y is a-finite (L.1.2), Y(A) = sup {Y(K) :K e K(G) , K C A} >_ sup {g(K) : K e K(G) , KC A.} by L.3, = T(A) by (1). Hence, by L.1.2, Y(A) = T(A) for all A e Cyl (ft) , y Consequently, (1) will have established when we show that, (3) y is K(G)-tight => (2) holds for all A e Cyl (ft) . Suppose y is K(G)-tight. Let E e F and B G Mg with yE(B) < 03 • Given e > 0 , since u,., is Radon, there exists a closed C c B s yE(B ~ C) < e/2 . Since y is «(G)-tight, there exists e K(G) s.t. for every F E F with E C F , yF(rEyB] - rFjX[K]LJ) < e/2 . Let K = Ki^rE!x[c] • Since C is closed and r [ ft is G-continuous, (4) K e K(G) . Further, (5) . K C ft r\ r'1 [B] . Now, for any F e F with E C F ,' (6) rF,X[K] = rF,X[Kl] * rE!F[C] • Hence, VrE!F[B] " rF,X[K]) = V(rE>] ~ rF,X[Kl])U <F[B] ~*1]Y™» lVF(r^p[B] - r^f^]) + vE(B - C) < e . Consequently, by L.2.3 and L.2.1, g(K) = inf {yp(rF X[K]) : E c F e F} >, inf {y^r"1^]) - e : E C F e F} = yE(B) - e . Since e was arbitrary it follows that (2) holds for all A e Cyl («) with T(A) < » . However, since y is a-finite for all F e F , T(A) = sup {T(A'): A'c A , A e Cyl (ft) , x(A') < »} , Hence, (2) holds for all A e Cyl (ft) . Proof of Cor. 4.3. Let K = {V° : new} . We note that (1) ft = \JK , and (2) K c K(w*|ft) By (2) and Thm. 4.3 we need only show that (3) y is ,.:?. /(-tight whenever y has a limit measure on ft Suppose.that y has a limit measure on ft . Let E e F and A e M with Vu(A) < °° * Since y is Radon, choose a closed C C E s.t. Ji CCA and yE(C) > y^(A) - e/2 , and for each new , let * Since C is closed and r is w -continuous, then, by (2), hi, X (4) K e K(w"|ft) for each new. Further, by (1) (5) ft rs r'^C] = U Kn . new Since y„ is an outer measure, and K c K ,, for each new ft n n+1 we deduce from (5) that (6) y^(ft t\ r^tC]) = sup y (K ) . ' new n By Prop. 3.1.2, C7) y^ is a limit measure of y , Hence, (8) y^(ftn r^x[C]) = yE(C) < yE(A) < » . Let e > 0 . By (6) and (8), there exists new s.t. 44. Then, by (7), W > VC) ' £/2 > V.A) " £ • Hence, by (4) and L.2.2, (10) g(K ) > y (A) - e . We have that, for any F e F with E e F , rF,X[Kn1Cr;>ClE"!F[Al • and therefore, = y (A) - y (r [K ]) < y (A) - g(K ) b r r , A n — & n < e , by (10). However, K c , and therefore, for every F e F with E F , n n V^F^ " rF,X^n]) < E • Since e > 0 , E e F , and A e with Vg(A) < 00 , xrere all arbitrary, it follows that y is K-tight. 45. 5. Finite Cylinder Measures We shall specialize the results of the foregoing sections to the case of finite cylinder measures. By introducing the notion of a finite section of an arbitrary cylinder measure, we shall show that, with regard to the problem of finding limits, we can concentrate on finite cylinder measures. 5.1 Definition. u is a finite cylinder measure iff u is a cylinder measure over a vector space X and for some F E Fv ,' X yp(F") < « . (We note that u (F ) is independent of F E F ) r A For the rest of this section we assume that X is a vector space, ft is a subspace of X such that (X,ft) is a dual pair, 6 is a regular, Hausdorff topology on ft which is finer than the w -topology restricted to ft , p is a cylinder measure over X results The following lemmas indicate that the hypotheses of earlier can be simplified when considering finite cylinder measures. 46. 5.2 Lemmas. If y is a finite cylinder measure over X , then (1) • yX(A) = y/V(A r\ ft) for every A e Cyl (X*) <=> y'?(ft) = y"(x") . For any family fl of subsets of X , (2) y is H-sequentially tight <=> for any sequence {F } in F with F c F ,, for each n new n n+1 new, and e > 0 , there exists HeH such that y (F ~ r v[H])-< e for all new. r n r , A n n (3) If r V[H] e M for every F e F and HeH , then, r , A c y is H-tight <=> for any e > 0 there exists HeH such that yp(F" ~ rp X[H]) < e for all F e F . Proof of 5.2.1 Certainly, if y"(ft n A) = y"(A) for all A e Cyl (x") , then y*(ft) = y"(x') . On the other hand if y (ft) = y (X ) , then, for any A e Cyyx*) , y*(XX) = yX(ft) = y" (ft A A) + y" (ft ~ A) <_ y*(A) + y* <x" ~ A) = y*(X*) -k ' * Hence y (ft r\ A) = y (A) 47. Proof of 5.2.2 We observe only that for any F e F, A e and HeH , PF(A ~ rFjX[H]) < yF* ~ r [H]) . Proof of 5.2.3 Together with the observation of Proof 5.2.2 above, we note that for any E and G in F with Ec G , and HeH, VE* ~rE,X[H]) ^G(G* ~rG,X[H]) The assertion is now immediate. The following theorems are now immediate consequences of, respectively, Lemma 3.3, Theorem 3.10, and Theorem 4.3 5.3 Theorems If y is a finite cyliner measure over X , then :L^..>\ ] - . '! . (1) y has a limit measure on ft <=> y"(ft) = y"(x") . (Silov [46]). (2) y has a limit measure on ft if, for any sequence (F ) in F with F c F ,.. for each n £ 43, J H n new n . n+1 ' and e > 0 , there exists a w -compact K c ft such that yr(F* - r [K]) < e for all new. r n r , A n  . • . 48. (3) (Mourier-Prohorov, [3 ] ^5 Lemma 3) • u has a G-Radon limit measure on ft <=> for any e > 0 there exists K e K(G) such that U_(F" ~ r V[K] < e for all F e F . r r , A Remark. We point out that Theorem 5.3.2 does not seem to have been previously stated in the literature. We shall now show that the problem of finding limits for arbitrary cylinder measures can be recuced to that for the finite case. First, we make the following definition. 5.5 Definition ^ is a finite section of u iff for some E e F and A e Mg with WgCA) < 00 , 5 is the cylinder measure over X such that for every F e F, ^F = rF,G[yGlrE,F"1[A]J ' for some G e F with E c G and F c G Remark. If £' is a finite section, of u then | is well-defined, 5 is in fact a cylinder measure over X , and = u"(rE X_1[A] A B) for all B c X^ . (This remark is proved below.) The following theorems are then readily established. 49. 5 .6 Theorems. (1) y has a limit measure on ft iff every finite section of y has a limit measure on ft . (2) u is K(G)-tight iff every finite section of y is K(G)-tight. Hence, y has a G-Radon limit measure on ft iff every finite section of y has a G-Radon limit measure on ft Proof of Remark 5.5 *• • _i For any F e F and a c F with r [a] t Mn , in view of Remark 2.2.1 and Lemma 0.4.1, ^G(rF,G1[a] " rE,G1[A]) is independent of the choice of G e F with E c G and F C G Hence, so also is £ F We note that for any F e F with E C F , (1) UFlrE p1[A] is Radon. Consequently (2) ?F = yp|rE F1[A] and is Radon. Hence, by Lemma 0.4.3, (3) E, is Radon for every F e F. For any F and F^ in F with F c F^ •, if G e F with E U F u F c, G , then by Lemma 0.4.2, h = rF,G[yGlrE,GltA]] = ^.F^F.F^G^E.G1^111 50. Hence, by (3) and Lemma 0.4.1 5 is a cylinder measure over X-We shall now prove that (4) £*(B) = u*(r_. "V] n B) for all B c x' L, A Let k k H = {H e Cyl (X ) : H is w -open} , a = r^'V] . We have that H C Cyl (X*) A Cyl (X*) and by (2), T^(H) = T (H A a) for every HeH Hence, by Thro.. 3.1, (5) ?"(H) = T"(H A a) for all HeH y a Let B c X If y (B) < TO , then, since y is an H -outer measure, for any e > 0 there exists HeH s.t. a .B c H and y*(H) < y*(B) + e . Since a e M , we have that y« A • A  A y (B n a) + y (B ~ a) = y (B) > y (H) - e k k = y (H n a) + y (H ~ a) - e and therefore •A. a. y" (HA a) < y" (B A a) + e . k Since y is a-finite it follows that for any B C X , (6) y*(B n a) = inf {y"(H A <*.} : B C H e H } . a A A Since £ and y are both H -outer measures, (5) and (6) o together imply that (4) holds. Proof of 5.6.1. By Lemma 3.3 w£ need only show that (1) yX(A) = y*(An ft) for all A e Cyl (X*) iff (2) E, (X ) = E, (ft) for every finite section E, of y From Remark 5.5 it is immediate that (1) => (2). However, (2) => y"(A) = y' (A n ft) for all A e' Cyl (x") with y (A) < oo . For any A e Cyl^(X ) , since y is a-finite, choose an increasing sequence {A } in Cyl (X ) s.t n new J y y (A ) < 00 for all new and A = U A new n Since y is an outer measure, we then have that p (AO ft) = lim y*(A A ft) = lim y (A ) = y (A) . n n new new Hence (2) =3> (1). Proof of 5.6.2. Let E, be a finite section E e F and A e M with y„(A) < E E Remark 5.5, for any F e F with V^F [A] - rF,X[K]) = ~ of y determined by some oo . By (2) in the proof of E C F , and K e K(G) , rF,X^} • The assertion now follows from Lemma 5.1.2, and Thm. 4.3. CHAPTER II CYLINDER MEASURES OVER TOPOLOGICAL VECTOR SPACES In this chapter, we are primarily interested in determining when a cylinder measure over a Hausdorff locally convex space X will have a limit measure on the topological dual X' which is Rad^n with respect to some given topology G on X' . Since (X,X') is a dual pair, the theory of the previous chapter applies with 0, = X' A Hence, if G is regular and finer than the w -topology restricted to X' , then, by Theorem 1.4.3, y will have a G-Radon limit measure on X' whenever y is H-tight for some family H c K(G) . We shall take G to be one of three standard topologies, and these suggest that we take for H the particular family E defined below. Our main concern is then directed towards finding conditions under which y is E-tight. 1. Notation , We point out that our topological vector spaces are not • assumed to be necessarily Hausdorff. In the rest of this paper we shall use the following notation. For any vector space X and V c X , V° = {f e X" : |f(x)| <_ 1 for all x e V} . 53. For any topological vector space X , nbnd 0 in X is the family of'neighbourhoods of the origin in X k k E is the family of all sets K c X such that K is w -closed and K c for some V g. nbnd 0 in X •k X' = {f e X : f is continuous} and for every F e F , A. rF=rFlX' • In addition to the w -topology restricted to X' , we shall consider the following two topologies: c is the topology on X' of uniform convergence on the compact subsets of X , s is the topology on X' of uniform convergence on the bounded subsets of X Remark We note that E c K(w*) and •k | k A W X' C C C S 54. 2. E-tight Cylinder Measures. Throughout this section X is a topological vector space and y is a cylinder measure over X When X is locally convex and Hausdorff we notice that E is nothing else but the family of w -closed equicontinuous subsets of X' Hence, fr oiu TrsvGs [47] , Props* 32.5 and 32.8, WG h.avG that E c K(c*) . Consequently, application of Theorem 1.4.3 yields the following assertion. 2.1 Theorem. Let X be Hausdorff and locally convex * • y u is E-tight => p. has a c -Radon limit measure on X' If E = K(c ) , in particular, if X is barrelled ([47] Thm. 33.1) then, y is E-tight <=> y has a c -Radon limit measure on X' Sometimes E-tightness of y can also imply the existence of a limit measure on X' which is Radon with respect to the s -topology. For example, if X is a Montel space ([47] p. 356) then E = K(s ) ([47], Prop. 34.5); or, if X is a nuclear space ([47] p. 510) then Ec K(s") ([47] Prop. 50.2). Hence, on applying Theorem 1.4.3, we obtain the following theorems. 2.2 Theorem (1) If X is a Montel space, then, y is E-tight <=> y has an s -Radon limit measure on X' (2) If X is a nuclear space, then, y is E-tight => y has an s -Radon limit measure on X' Even if E £ K(s ) , E-tightness of y can still imply that y has an s -Radon limit measure on X1 2.3 Theorem. Let X be Hausdorff and locally convex. For any V e nbnd 0 in X let X^ = nV® with the topology induced by the norm |* : f e X^ + sup|f(x)| e R+ x«=V If there is a base \l for nbnd 0 in X such that for each V £ 1/ , X^ is separable. then, (1) y is E-tight => y has an s -Radon limit measure on X' (2) If X' is a separable Banach space under s , then indee * „ k < every finite c -Radon measure on X' is s -Radon. Proof of Theorem 2.3 We first establish the following lemma. Lemma. For any V e nbnd 0 in X s.t. X^ is separable, if G is the family of all open subsets of X^ , then Gc cr-field generated by (c |x^) 56. Proof Let ff =' {f + EV° : e > 0 , f e X^} . For any e > 0 and f e X' , g e X' -y eg e X1 and g e X' -> f + g e X' are homeomorphisms with respect to c , since X! is a topological vector space under c 0 * * * . Hence, since V is w -closed and w c c , H consists of c -closed subsets and therefore f/ c cr-field generated by (c |X^) However, X^ is separable and metrizable. Consequently, G c C a-field generated by (c |X^) We now prove the theorem. (1) By Thm. 1.5.6.2, we may assume that u is normalized. In that case, by Lemma 1.5.1.2, we need to prove that for any e > 0 ., there exists K e K(s ) s.t. u„(F ~ r_[K]) < e for every F e F . r r With the notation of 1.4, since u is E-tight, then, by Thm. 2.1 and Props. 1.4.1, ]i' ^ is c -Radon limit measure on X' of , and by L.3 of Proofs 1.4, . (1) uc*(K) = g(K) for all K e K(c*) . Since u is E-tight, for any e > 0 , there exists V e nbnd 0 in X s.t. X^ is separable, and for all F e F , yp(F* ~ rF[V°]) < e/2 . » 57. Hence, by L.2.1 of Proofs 1.4, 1 - g(V°) < e/2 , and therefore, by (1), (2) yc>v(X' - X^) < 1 " VC*(V°) < e/2 . If 5 : Ac X^ + u^(A) e R+ ,. then 5 is a c |X^-outer measure on X^ * i By the lemmas, and the fact that c [X^ c 6 , it follows that £ is a G-outer measure on X^ However, X^ is complete ([47], Lemma 36.1, see also p. 477)-X^ is also separable and metrizable. Hence, by Prohorov L32] Thm. 1.4, there exists K e K(G) s.t. (3) 5(X^ ~ K) < E/2 However, s |x^ c G , hence K E K(s*) • Then, certainly, K E K(C ) , and by (1), (2) and (3), g(K) = u^CK) > 1 - e . From the definition of g , and L.2.1 of Proofs 1.4, u„(F" ~ r_,[K]) < s for all F e F . r r (2) From the lemma we have any c -Radon measure on X' is an s -outer measure (G = s ) . The assertion now follows from Prohorov [32] Thm. 1.4 We are led by the above theorems to study conditions under which will be E-tight. In view of Theorem 1.5.6.2 we shall concentrate on the case when u is finite. Conditions for u " to be E-tight will then be given in terms of the one-dimensional subspaces of X . We begin by indicating a necessary such condition. 2.4 Proposition Let X be a topological vector space, r > and p a finite cylinder measure over X If p is E-tight, then, for any e > 0 , there * , k exists a w -Radon measure n on X with, supp n e E , such that x e X , J|f(x)|rdn(f) <_ 1 => y„ '({f e F* : |f(x)| >_ 1}) < F X. x where F^ is the space spanned by x Proof. We assume that p is normalized. With the notation of I.4, by Th. I.4.3, u , is a w -Radon limit measure on X for p ' W" Since p is E-tight, by L.2.1. of Proofs 1.4, for any e > 0 there exists V £ nbnd 0 in X s.t. 1 - g(V°) < E/2 , and therefore, by L.3 of Proofs 1.4, p",(X* ~ V°) < e/2 . For any x e•X , let I = {f e XW : |f(x)j > 1} . If r) : A C x" - y" , (A A V°) e R+ , then, k £ W" k n is a w -Radon measure on X with supp n e E and for any x e X s.t. J ] f (x) {r dn <_ 1 , PF ({f e F* : |f(x)| > 1}) = V**(IX> x = V(Ix^°) + V(Ix ~ - 2 N (I -} + 2 2 $ 1I dr' + e/2 < f /|f(x)|rdn(f) +f if + f = e • m The above proposition suggests the following definitions. 2-4 Notation. For any vector space X , x e X and cylinder measure y over X , F^ = space spanned by x,. D =' {f e F* : If (x) I > 1} x x 1 1 — and \ = yF 2.5 Definitions. Let X be a vector space, and U be a family of subsets U of X with 0 e U (1) For any finite cylinder measure y over X , y is U-continuous iff for any e > 0 there exists U £ U such that x£U=>y(D)<£ x x (2) For any cylinder measure y over X , y is U-continuous iff every finite section of U is U-continuous. (3) For any topological space Y and T : X -> Y , T is U-continuous iff for every neighbourhoods V of T(0) there exists U e U such that T[U] c V . (4) When X is a topological space, for any finite cylinder measure y over X , y is continuous iff y is f-continuous for some family V of neighbourhoods of the origin in X only the standard The discussion of limits in the rest of the chapter requires above concepts. However, to explain their relation to notions of continuity we introduce the following definitions. 60. Let X be a vector space. For any new and x = -(x^,. . . ,x _^) - e Xn , F = linear span of {x„,...,x , x U n-1 • Cfx J f e F* -y (f(xQ),...,f(xn_1)) e <En , and for any finite cylinder measure y over X , X For any new, M(|]n) is the family of finite Radon measures on ^n endowed with the vague topology; i.e. for any net (n.) . , in M((j;n) and n E M(f]n) , Tij +n in M(fpn) iff J fdn.. -»- / fdn for every bounded continuous f : <£n -> We note that y e M(£n) for all n e OJ and x e Xn X We now have the following well-known proposition (Gelfand, Vilenkin [11] P. 310, Fernique [9] p. 37, which shows that one can naturally associate certain continuous maps with a continuous cylinder measure. (see also Appendix 1.7). 2.6 Proposition Let X be a topological vector space. (1) y is continuous iff (2) y : x e X -> y^ e M($) is continuous at 0 iff (3) for each new, y11 : x E Xn y e M((£n) is continuous with respect to the product topology on X 61. Remark From the proof of the above proposition one readily checks that the following assertion also holds. If U is a family of balanced, absorbent subsets of X , with tU e (J 'for every U e U and t > 0 , then, u is (J-continuous iff u : x e X -> u e M(C) is (J-continuous. Hx Proof of Proposition 2.6. We show that (3) => (2) => (1) => (3) (3) => (2) take n = 1 . (2) => (1) let % : £ -> C be bounded and continuous, %(z) = 1 if jil > 1 0 if |*| < | Let e > 0 . Choose V e nbnd 0 in X s.t. x e V => 1/ %d~^ - / 5Cdu~| < e . Since y^ is concentrated at the origin in t , and has finite mass, then J = o • . Hence, for any x £ V , y (D ) = yv({z e £' : |z| L 1» 1 / *dyv < e • A. A. A. A. (1) => (3) For any F e F , and u > 0 , let I(u) = {(w,f) E F x F* : |f (w) | >_ u} Let n e OJ , X • ^n ->• £ be hounded and continuous, and x e Xn We shall show that for any £ > 0 there exists V e nbnd 0 in X s.t. 62, y e x + Vn => |/ %dyx - / %dy^| '< e . (Note: Vn e nbnd 0 in x" ). We assume that u is normalized. For any z E £n. , let [z| = sup {|z | : k = 0,...,n - 1} Let (i) M = sup (|%(z)| : z E £N} , (ii) W be an open nbnd of the origin in X s.t. w e W => u (D ) < e/16nM • , w w (iii) t > 0 be s.t. x e tW for each k = 0 n - 1 , (iv) and let 6 > 0 be s.t. zj e £U |zj | <_ t , j = 0,1 , and |z° - z1] < 6 => \%(z°) - Xiz1)| < e/4 . ' Since £ - WN is an open nbnd of 0 in XN , there exists V e nbnd 0 in X s.t. (v) -| V e W , (vi) x + VUctWn . For any y e x + Vn 'and F e F with F^ u F^ c F , let y > y Fx 1 x F y • x A (5) = {f E F* : \v (f) - V (f) I > 5} y y x 1 -and Then, k=n-l B (t) = U I (t) v v * k=0 "k . yF(A (6)) = yF(kU 1 I (6)) J k=0 k k k=n-l. k=n-l " k=o yF(V6)(xk-yk)(1)) - k^0 ^i/^k-yk)(1li/6Xxk-yk)> < e/16M by (ii) and (v). From (vi), we have that y e tW Hence, — v, e W for each ken , and therefore, t k by reasoning as above, y (B (t)) < e/I6M . b y In particular, yF(Bx(t)) < e/16M . Consequently, if B = B (t) U B (t) V A (5) , x y y then, y (B) < 3 e/16M < e/4M , and, by (iv), f e F* ~ B => \X(V (f)) - (f))| < e/4 y x 1 Hence, 1/ Xdyy - / XdyJ = 1/ % 0 yyF - / % 0 ^d,^} < /| X, V - % 0 ¥ dy + / |X o ¥ - X 0 ¥ |d,i ~~ B y X F F*~B Y X F < 2M.e/4M + -f . U(F" ~ B) < e . — 4 F i.e. y e x + Vn => |J Xdy^ - / %dyj < e . 3. Limits of Continuous Cylinder Measures Let X be a Hausdorff, locally convex space, and y be a finite cylinder measure .over X . In the previous section we have seen that conditions under which y is E-tight are important for determining when y has a c - or an s -Radon limit measure" on X In terms of the one-dimensional subspaces of X Proposition 2.4 gives a necessary such condition. In seeking some kind of converse to that proposition, we are led to introduce the concept of a weighted system in X , which is defined below. We shall use the following notation. 3.1 Notation For any vector space X , absorbent absolutely convex V c X , and F E F , ker V={xeX:xetV for every t > 0} , Fv = F cv ker V , (V c\ F)° = {f e F'C : |f (x) | <_ 1 for all x e V f\ F} = {f E F" : f(x) =0 for all x c 7^} For any t > 0 , I(t) = {(x,f) E F x F* : |f(x)| i.t} , I = KD • So for any x e F , f e F , If = {x e F : |f (x) | >_ 1} , I = {f E F* : If(x)| > 1} . • •v 1 — Remarks We note that (1) Fv e F ; and since (x,f) e F x F f(x) e C the product topology on F x F , (2) for any t > 0 , I(t) is closed. 3.2 Definitions is continuous with respect to c Let X be a locally convex space. (1) (v,F»f) is a system of S-weights in X if 6 > 0 ; [/ is a family of absolutely convex neighbourhoods of the origin in X , F c F is directed, by C and Li F is dense in X and v:Vel/,FeF->Vyp , a probability Radon measure measure on F for which f e F* ~ (V n F)° => vVjF(If) >• 6 . When f is a singleton {V} , we shall write (v,F,V) instead of (v,F,l/) • (2) W is weighted by such a system (v,F,f) iff W is a family of neighbourhoods of the origin in X , for each W e W there exists V c V such that ker V c W , and v„ „(F ~ tW C\ F) -* 0 as t -> » , uniformly for F e F, 66. (3) (!) is a weighted system in X iff W is weighted by some system of 6-weights in X We shall now state and prove the fundamental results of this section. 3.3 Theorem Let X be a Hausdorff, locally convex space, and fx be a finite cylinder measure over X If y is W-continuous for some weighted system in X , then y is E-tight. Corollary Let X be a Hausdorff, locally convex space, and y be an arbitrary cylinder measure over X If y is W-continuous for some weighted system in X , then y is E-tight, and therefore, y has a c -Radon limit measure on X' Proof of Corollary By Thms. 3.3, I. 5.6.2, and 2.1. We shall need the following lemmas in the proof of Theorem 3.3. They are proved at the end of the section. Lemmas (1) Let F be a finite dimensional space, and z• > 0 If £ is a finite Radon measure on F such that x e F => ^(I ) < e x then £(F* ~' {0}) ±e . (2) Let X be a locally convex space and p a continuous finite cylinder measure over X For any dense subspace Y of X , and e > 0 , if V is an absolutely convex neighbourhood of. the origin in X such that yp(p" ~ (V A F)°) <_ e for every F e Fr , then p„(F~ ~ (Vf\ F)°) < e for all F e Fv r — A Proof of Theorem 3.3 By the Hahn-Banach extension theorem, we see that for any F e F . , and absolutely convex V e nbnd 0 in X , (1) rp[V0] = (V A F)° . Hence, by Lemma 1.5.1.2, we need only prove that, for any e > 0 , there exists V e nbnd 0 in X s.t. (2) y_(F* ~ (¥A F)°) < e for all F e F . r — We assume that p is normalized. Let W be a weighted system in X with respect to which p is W-continuous, and let (v,F,l/) be a system of 6-weights in X by which W is weighted. For any e > 0 , let 0 < e' < min(<5e,e) , W e W s.t. x e w => y (D ) < e'/4 , xx — V e V and t > 0 s.t. ker V C W and vv F(F ~ tW r\ F) < e'/4 for every F e F Let U = V/t . Suppose that * 0 y (F ~ (U c\ F) ) <_ E for every F e F Since F is directed by c , then U F is a subspace of. X , and for any finite dimensional subspace E of UF there exists F e F with E c F . Hence, by (1) , (3), and L.2.3 of Proofs 1.4, yE(E* - (U r% E)°) <£ . Since W is a family of neighbourhoods of the origin in y is necessarily continuous, and by hypothesis, U F is dense in X . Hence, from Lemma (2) and the foregoing remarks we conclude that (2) holds. It remains for us to establish (3). For any F e F , yF(F* ~ (UAF)°) = yp(F*.~ t(Vn F)°) • = yp(Fa ~ t(V n F)°) + yp(F* ~ F^) since (V A F) ° c F?T . We show that each of the last two terms given in (4) is less than e/2 We estimate the first term. Since (v,F,l/) is a system of 6-weights in X , the f £ ~ t . (V ft. F)° => f e F* ~ (Vft F)° => vVjF(If(t)) = vyjF(If/t) > 6 . Consequently, 6.uF(F^ ~ t.(V f\ F)°) f{vv>F(If(t)): f £FJ~ t.(V A F)°}.pF(F^ ~ t.(VftF)0) <. Vy F * vF(Kt)) by Sion [44] Ch. Ill, Thm. 1.2.6, = JF/F;V1I ^dUpdVy F by Fubini's theorem, = /pF(Ix(t))dvVjF(x) = /SF(Ix(t))dvVjF = ^F(Ix)d\,F(x) = /y°x)dVV,F(x) t t t XtW F-y_x(Dx)dvV,F(x) + f VtW F'yx(Dx)dVV,F(x) t t  t -f~ "vV,F(tWA F) + 1,VV F(F ~ tWA F) <f . l + l.f < 6.f . Hence, yp(F* ~ (U ft. F)°) < e/2 . We now estimate y„(F " F ) F V We have that x e FT => x E ker V => x E W => y (D ) < e'/4 V  x => y_ (I ) < e'/4 since F c F1T e F . Fy x x V Hence, by Lemma (1), y (F* ~ {0}) < e'/4 . V 70. Since rp [F* ~ FJ] C F* ~{0}' , FV,F V it therefore follows that Uf(F"'? - ?p < u^r'^fF* ~'{0}]j = Uv (F" ~ {0}) < e'/4 < e/4 . Then, certainly, (6) uF(F* ~ F*) < £/2 . From (4), (5), and (6) we see that (3) holds. Remark. We point out that the theorem still holds when we use a somewhat weaker notion of system of 6-weights, in which F:Vel/->FcF directed by C and [J Fv is dense in X The other definitions remain unchanged. Proofs 3. Proof of Lemma (1). For any x e F and new, and Consequently, for any x e F , . x =)= 0 , 5({f e F* : f (x) f 0}) = U I .(-)) new = lim 5(1 (-)) < e x n — new 71. Hence if (1) there exists y e F s.t. •£({f e F* : f (y) = 0} ~ {0})- = 0 , then 5(F~ ~' {0}) = C({f e F* : f (y) ={= 0}) + ?({-f E F* : f (y) = 0} ~ {0}) We shall establish (1) by induction. For any subspace E of F , let Ea = {f e F" : f(x) = 0 for all x e E} Let dim F = n . If n = 1 then (1) holds. We therefore assume that n >^ 2 . For any k-dimensional subspace G of F with 2 <_ k <_ n , (2) {H3^GA : H is a (k - 1)-dimensional subspace of G} is an uncountable, disjoint subfamily of M Let GQ = F . Then, by (2) and the finiteness of F, , there exists an (n - 1)-dimensional subspace G^ of F s.t. 5(GA -' {0}') = 5(GA ~ GA) = 0 . For any 0 <_ k <_ n - 2 , if there exists an (n - k)-dimensional subspace G^. of F s.t. 5(GA - {0}) - 0 , Then, by (2) and the finiteness of E, , there exists an (n - k - 1)-dimensional subspace G of G1 s.t. ^Gk+i * GV =0 • Consequently, aGa+1 - {0}) = 5(GA+1 - GA) + C(GA ~ {0}) = 0- . Hence, there exists a one-dimensional subspace G ., of F s.t. n-1 5(GA . ~ {0}) = 0 . n-1 i.e. (1) holds. 72. Proof of Lemma (2) For any F e Fv , n e w, X e Fn , and t > 0 , let A k=n-l A* (t) = A {f e F" : |f (x ) | < t} . • X k=0 k We shall assume that u is normalized. Since (V A F)^ = (VQ r\ F)^ , where is the interior of V , we shall further assume that V is open. Let E e Fv . A Since E is separable there exists a countable, dense subset {x } of V A E . Then, n new (V r\ E)° = A {£ e E* : | f (x^) | £ 1} . kew Now, for any new , tv1^1*: |£°vi ^ Acxn.....x • k=0 mew 0 n-1 Consequently, for any 6 > 0 , there exists new and m e w s.t. PE((V^E)VVA* (1+1)) < VE((VAE)°) +{ . 0 n-1 Since y is continuous, there exists U e nbnd 0 in X s.t. (1) u e U => yu(Du) < 5/2n , and since Y is dense in X and V is open there exists {y0'"'"'yn-l} in V S-t" (2) x, - y, e - U for all k = 0, . . . ,n - 1 . k k m Let F e F,, be such that E U {y„,...,y ,} C F and X 0 n-1 let x = (XQ,...^^) , y = (yQ.-.-.y^) Then, VF* " Ax-y(™)} = U 6 F* : |f(Xk " V 1 k=0 k=n-l - E yx,-y (Dm(x -y )} < 6/2 by (1) and (2) * k=0 *k yk nHxk V Further, • AY(1)C-AX(1+7> • Hence, J,,,, . ,,F ,1s . ,F„,, , ,.F,„N „ , F ,1 -y W1}) = Wytt * AY(1)) + W1' ~ Vy?5 < uE((V r\ E)°) + 6 . i.e. (3) yF(A|(l)) < yE((VrN E)°) + <5 . However, if F denotes the linear soan of {y„,...,y , } y J0 Jn-1 we observe that AF(1) 3 r"1 [(VA F )°] . y „ > y y Since F e F , and (V r\ F ) is closed, we have that y Y y (V (\ F ) £ Kp and y VAy(1)) > yF(rFX F[(VA F )°]) = yp ((V A F )°) > 1 - e y' y Hence, by (3), uE((V A E)°) > 1 - £ - 6 . Since 6 was arbitrary, it follows that uE((V A E)°) >_ 1 - £ . Consequently, since (V A E)^ £ M^, , uE(E'f ~ (VA E)°) < e . 74. 4. Induced Cylinder Measures. It can happen that a finite cylinder measure over a Hausdorff, locally convex space X is given indirectly. For example, it may have been induced by a finite cylinder measure u over a vector space Y and a linear map T on X to Y ([11] p. 311). In such a situation we shall be interested in obtaining conditions on ]-~ and T which will ensure that the induced cylinder measure over X will have a limit measure on X1 , Radon with respect to some given topology on X* . This kind of problem seems to have been first mentioned in [H] Ch. IV. It has been studied extensively, by L. Schwartz, 5. Kwapien, and others, in a series of papers ([19], [20], [39] - [42]). In view of the previous theory, our emphasis will be on determining conditions under which the induced cylinder measure will be E-tight, Using the notions of continuity and weighted system we readily obtain such conditions. 4.1 Definition For any vector spaces X and Y , linear map T : X -> Y , and finite cylinder measure y over Y , the cylinder measure £ over X induced by y and T is defined as follows: for each F e Fv > A where Tp is the adjoint of T[F , i.e. T* : f e CT[FJ)* -> f 0 (T | F) e F* . We shall denote this induced cylinder measure K by u D T . We prove below that 5 is indeed a cylinder measure over X Proof For each E e F„ , T is continuous. X E Hence, by §0.4, £ is a finite Radon measure on E E Since all. the maps considered are continuous, then, by §0.4 and R.emark 1.2.1, for any E and F in F with E c F , " X rE,FCr'F] = rE,FtTF[uT[F]]] = rE,F 0 VyT[F]] TE 0 rT[E],T[F][yT[F]] = VrT [E] ,T[F] [yT[F] ] ] VyT[E]] = ?E ' and therefore, again by Remark 1.2.1, 5 is a cylinder measure over X . B We now prove the following important lemma. 4.2 Lemma For any vector space X , family U of subsets U of X with 0 e U , topological vector space Y , and linear T : X if T is U-continuous, then u a T is U-continuous for every continuous finite cylinder measure over Y 76. Proof. Let y be a continuous finite cylinder measure over Y For any x e X , by Lemma 0.4.2, <U a T)X(DX) = TF [y ](DX) = .Tx(T;-1[Dx]) = y: (D ) . x :-. r, x K For any e > 0 , there exists V e nbnd 0 in Y s.t. y e V => y (D ) < e , y y and there exists U e U s.t. T[U] C V . Then, by the first assertion, x e U => TJC e V => (y Q T)X(DX) = PTX(DTX) < e • It follows that y -,3 T is (i-continuous. ©• Our key theorem on induced cylinder measures is now an immediate consequence of Theorems 3.3, 2.1, and the above lemma. 4.3 Theorem. Let X be a Hausdorff, locally convex space, Y be a topological vector space, and T be a linear map on X to Y If T is (^-continuous for some weighted system W in X , then for every continuous finite cylinder measure over Y , y cs T is E-tight are therefore •k y C\ T has a, c -Radon limit measure over X Remark. It is clear that this theorem reduces to the finite case of Corollary 3.3 when X = Y and T is the identity map. 77. CHAPTER III APPLICATIONS We shall apply the theory of the previous chapter to a study of cylinder measures over Hilbertian and 5,^-spaces. Our results on cylinder measures over arbitrary Hausdorff, Hilbertian spaces generalize and clarify many known theorems (Minlos [25], Sazonov [35], Badrikian [1], Fernique [9]). In the case of £^-spaces we obtain significant extensions of formerly known results (L. Schwartz [39], Kwapien [19]). Our main tool is Corollary II.3.3, which requires us to construct weighted systems in the above spaces. In view of Proposition II.2.4, it is the search for such systems which leads us to consider the families S , for r > 0 , defined below. 1. Preliminaries For any vector spaces X and Y , L[X,Y] is the set of linear maps on X to Y For any topological vector space X , CM(X) is.the family of continuous finite cylinder measures over X 78. Remarks. From Appendix 3.1.1 and 3.2.1, we have that for any family C of finite cylinder measures over a vector space X , there exists a coarsest topology on X under which it is a topological vector space, and such that p e C => u is continuous. This topology is called the C~topology. For any topological vector space X , if the topology of X is the CM(X)-topology, then we call X a CM-space (Appendix 3.1.2). For any topological vector space X , 0 < r < ra , and w -Radon measure n on X with supp .n e E , S = {x e X : /|f(x)|rdn(f) <1} . For each r > 0 , Sr is the family of all sets S c X . r, n 1.1 Remarks Let X be a topological vector space. (1) For each r > 0 , there is a unique topology on X under which X is a topological vector space having S as a base for its neighbourhoods of the origin. When r >_1 , this topology is locally convex. We shall call this topology the Sr-topology. t r (2) If 0 < r < t , then S is finer than S , i.e. for every a e S3" there exists 3 e S*" with 3 c a (3) If X is locally convex, then, for each r > 0 S*" is a family of neighbourhoods of the origin in X 79. We prove only 1.1.2. ' Proof of 1.1.2. For -any finite measure space (£2,n) and integrable f : 0 -> £ , if 11 p = t/r and • 1- — = 1 ,' P q then, by >.L;>;H«AJ&#'s inequality, /|f|rdn 1 (/;|f|r)?dn)1/p nCn>1/q- . Hence, (l) (J|f|rdn)1/r K/lfl'dT,)17' n(a)(t"r)/rt • r For any S e S , r ,n n(xx) < <» , since supp n e E C K(w ') and n is w -Radon. Consequently, by (1), if K - n(xA)(t-r)/rt.n then S ,- C s t,5 r,n The assertion follows. To point the significance of the families 5 we note that Proposition II.2.4 can be restated as follows. 80. 1.2 Proposition Let X be a topological vector space. For any finite cylinder measure u over X , u is E-tight => u is S1-continuous for every r > 0 . @ Mien X is Hausdorff and locally convex, the above proposi tion and Theorem II.3.3 yield the following assertion: if u is W-continuous for some weighted system W in X , r then u is S -continuous for each r > 0 In view of this, when searching for weighted systems in X we shall look for suitable subfamilies of S In general, Sr-continuity for some r > 0 does not imply E-tightness. (Example 1, Appendix 4). We shall need the following result on induced cylinder measures. 1.3. Proposition. Let X be a topological vector space, Y be a vector space, and T e L[X,YJ . For any family C of finite cylinder measures over Y , if u a. T is E-tight for every u e C , then, for each r > 0 , T is S -continuous with respect to the C-topology on Y 81. Proof. Let r > 0 . By Prop. 1.2, IT y e C => y a T is S -continuous. Hence, by Appendix 3.2,1, the S -topology is finer than the (C o T)-topology on X . By Appendix 3.2.2, this says exactly that T is Sr-continuous with respect to the C-topology on Y . Q 2. Hilbertian Spaces. Throughout this section, X is a Hausdorff, Hilbertian space ([1]). i.e. X is a Hausdorff, locally convex space, for which there exists a family F of pseudo-inner products on X , such that nbnd 0 in X has as a base the family of all sets {x e X : [x,x] <_ 1} , [.,.] e r . The fundamental theorem of this section is the following. 2.1 Theorem For each 0 < r < 00 , S is a weighted system in X The proofs of this and other assertions will be given at the end of the section. Now, we concentrate on the consequences of the above theorem. 2.2 Theorems. Let y be a cylinder measure over X and .0 < r < 00 Then, (1) y is E-tight <=> y is S -continuous. (2) y is S -continuous => y lias a c -Radon limit measure on X V'' (3) If K(c ) = E , in particular, if X is barrelled, then u is 5 -continuous <=> u is E-tight <=> \x has a c -Radon limit measure on X' Using Theorems 2.2, we can now characterize certain positive-definite functions on X (Appendix 2). 2.3 Theorem. Let i> be a positive-definite function on X and 0 < r < «> Then, i> is ^-continuous => there exists some finite c -Radon measure E, on X' such that ij;(x) = J exp i Re f (x)dg(f) for every x e X If K(c ) = E , in particular, if X is barrelled, then ^ is S -continuous <=> there exists some finite c -Radon measure E, on X' such that i> (x) = /exp i Re f(x)d£(f) for every x e X Remarks We note that Theorem 2.2.2 generalizes a result of Minlos ([25] p. 303 Thm. 1). Theorem 2.3 generalizes results due to Minlos ([25] P. 310), and Badrilcian ([1] p. 16 Cor. 1). The special case when X is a Hilbert space will be discussed below (§2.7). 84. We point out that, with the viewpoint of §1.4,- the assertions of Theorems 2.2.2 and 2.3 for the case r = 2 can be established by using the technique of characteristic functionals ([1], p. 9, 2 Lemma 1, Prohorov [33]). Also, it can be shown that the S -topology is nothing else but the Gross-Sazonov topology on X ([35], [1], [13] p. 65). By means of Proposition 1.2 and Remark 1.1.2 we can deduce the assertions above for 0 < r < 2 from the case r = 2 . We have been unable to give a similar deduction for the case r > 2 . However, in this context, we draw attention to §2.6 below. As consequences of Theorems 2.1, II 4.3, and Proposition 1.3, we have the following assertion concerning induced cylinder measures over X 2.4 Theorem Let Y be a vector space, T e L[X3Y] , and 0 < r < 00 For any family C of finite cylinder measures over Y , y n T is E-tight for every u e C <=> T is S -continuous with respect to the C-topology on Y The above theorem yields immediately the corollaries given below,. Corollary (2) significantly generalizes a result in [11] (p. 349).. 85. Corollaries Let Y be a topological vector space, T e L[X,Y] , and r > 0 (1) If Y is a CM-space, then y D T is E-tight for every u e CM(Y) <=> T is S -continuous. (2) If T is S"-continuous, then, for every y e CM(Y) , y a T is E-tight, and therefore has a c -Radon limit measure on . X 2.5 Remarks Under certain circumstances one can readily strengthen the assertions of Theorems 2.2 - 2.4. Let y be a cylinder measure over X (1) (Theorem II. 2.3) If there exists a base (J for nbnd 0 in X such that for each U e U , the Banach space X^ is separable, then, y is E-tight => p.-has aiv ' 5 -Radon limit measure on X * y Hence, in those theorems involving the existence of a c -Radon /*c k limit measure on X' , we may replace _c by S (2) Let G be a regular topology on X' with w |x'c G If E c K(G) , or E = K(G) , then the foregoing theorems may be modified as indicated by Theorem 1.4.3. 86. In parituclar, we note that when X is a Montel space, E = K(s*) . (cf. Thm. II.2.2.1) The theorems above allow us to make some interesting assertions about the S -topologies. 2.6 Theorems. (1) For all 0 < r < oo } the families of 5 -continuous cylinder measures coincide. (2) For all 0 < r <_ 2 , the S -topologies coincide. (3) Let Y be a topological vector space, and for each r > 0 , = {T z L[X,Y] : T is ^-continuous} . If Y is a CM-space, then, for all. 0 < r < oo , the families T coincide, r Remark. In general, the S "-topologies do not coincide for r > 2 (Example 3, Appendix 4). S Clearly, we may interpret all of our results for the special case when X is a Hilbert space. In particular, we have the following theorems. Theorems. Let X be a Hilbert space, Let 0 < r < oo . For any cylinder measure y over X , nr u is o -continuous <=> y is E-tight <=> y has a c -Radon limit measure on X' Let 0 < r < oo y and \p be a positive-definite function on X I/J is S -continuous <=> * for some finite c -Radon measure £ on X' , iKx) = / exp i Re f(x)d£(f) for all x e X . Let Y be a Hilbert space, and T e L[X,Y] •k , y a T has a c -Radon limit measure on X' for every y e CM(Y) <=> T is a Hilbert-Schmidt map ([36] p. 177). ([42] VIII, Pietsch [31], Petcynski [28]). Let Y be a Hilbert space. For all 0 < r < 03 , {T'.E L[X,Y] : T is r-summable} = {T e L[X,Y] : T is Hilbert-Schmidt} . (For the definition of r-summability, see [31], and [42] p. VII. 3). 88. Remark By Theorem II.2.3.2, when X is a separable Hilbert space, every c -Radon measure on X' is s -Radon. Hence, in Theorems 2.7, we can replace c by s x<rhen X is separable. We point out that Theorems 2.7.1 and 2.7.2 are equivalent (Cor. 1.4.3, Thm. 1.5.6.2, Appendix 2.5 and 2.6). We observe that even when X is a Hilbert space our work extends previously knoxm results. Sazonov in [35] discusses the case xtfhen X is separable, obtaining Theorem 2.7.2 for the case r = 2 Waldenfels in [48] extends Sazonov's theorem to the non-separable case. Theorem 2.7.3 extends a result given in [11] (p. 349), where X is assumed to be separable and r = 2 significantly generalizes the Pietsch-Peieynski theorem given above (Theorem 2.7.4). Proofs 2. From Appendix 3.5 and Proof 2,7.4 we see that theorem 2.6.3 We shall need the following lemma. Lemma Let X be a locally convex space, r > 0 and S = S Let P = P(M jsupp n) directed by refinement, and for each P £ P S' = {x £ X : P I inf |f(x)|r.n(B) > 1} BeP feB 89. Then, for any F e , Radon measure E, on F and t > 0 , £(F ~ tS) = lira £(F A tS'j . PsP Proof of Lemma We first make the following observations. (1) If P E P, Q e P, with Q finer than -P , then P Q * (2) For any u > 0 , X ~ uS = U u S' PEP (3) For every P E P , Sp is open in X . We prove only (3). Let P E P . We have that (4) S' = U {x £ X : E inf |f(x)|r.n(B) > 1} , B BeB feB where the union is taken over all finite B c P . Hence, since supp n e E is equicontinuous, for every B e P , x £ X •> inf | f (x) | £ R is continuous. feB Hence, for any finite B Q P , (5) x £ X -> E (inf | f (x) |)r. n(B) is continuous, Be8 feB and therefore, by (4), (3) holds. From (3) we deduce that tF A is open for every P E P . Consequently, as P is directed by refinement, from (2) it follows that for any compact C in F with C c F ~ tS , there exists P E P s.t. 90. C c tF ft. Sp ' . . Hence, since £ is Radon and F ~ tS is open in F , £(F ~ tS) = sup {5(G) : C C F ~ tS is compact} = sup U(tF ft. Sp : P e P } = lim 5(tF AS'). PeP 2.1 Let f be a base for nbnd 0 in X s.t. for each V e 1/ there exists a pseudo-inner-product [.,.] on X for which V = {x e X : [x,x]v £ 1} . Let ^2 r k£ as §iven in Appendix i.l. For each V'e V , let F = F . For each V. e f and F e F , let p be a probability Radon measure on F related to [.,.] |F x F as in Appendix 1.3. Then, from Appendix 1.3 we see that (v,F,lO is a system of 62~weights in X By Remark 1.1.3, (1) Src nbnd 0 in X . Let S = S e Sr . Since \l is a base for nbnd 0 in X and supp n e E , there exists V e 1/ with supp ri C " 91. Then, • x e ker V => sup |f(x)| = 0 => J |f>(x)|rdri = 0 => x e S feV° i.e. (2) ker V c S . Let t > 0 . For any B c X' , let fB e B and §B = p r, (B)1/r. f g . Using the notation of the above Lemma, for any PeP, FotS'CtxeF: E | f (x) | r .n(B) > tr} BeP Hence, v (F o tSl) < v _({x e F : E |g(x)|r>l}) V' V' BeP a <_ C E sup |g (x)|r by Appendix 1.3.2, ' BeP xeV F <C E sup |gR(x)|ri^-C Z sup |f (x)|r.n(B) ' BeP xeV ° t ' BeP xeV < — C„ n(X') since f e for every BeP — r 2,r B 3 From the above lemma it now follows that vVjp(F ~ tS) l^C2>rr,(X') . Since C. n(X') < 00 , we conclude that (3) v„ _(F ~ tS) -> 0 as t -* °° uniformly for F e F. V ,h From (1), (2), and (3) we see that S is weighted by (v,F,l/) 2.2.1 By Cor. II.3.3, Thm. 1.5.6.2 and Prop. 1.2 92. 2.2,2 and 2.2.3 By Thms. II.2.1 and 2.2.1. 2.3 By Remark 1.1.3 and Appendix 2.2.5, r i/j is 5 -continuous => is continuous at 0 = > ijj is continuous =>(jjJF is continuous for all F E p. Hence, by 2.4 and 2.5 of the Appendix, there exists a finite S -continuous cylinder measure y over X s.t. i|>(x) = J exp i Re f(x)dy (f) for all x e X . By Thm. 2.2.2, y has a c-.Radon limit measure E, on X' Then, for every x e X , <Kx) = /F exp i Re f(x)dy (f) = J , exp i Re f(x)d£(f) . Suppose now that E = K(c ) , and for some finite Rad<5n measure E, on X' , i>(x) = / exp i Re f(x)d£(f) for all x e X . We note that for every F e F and Borel subset H of F , rFXtH] E M . If, for each .F £ F ,' yF = rpm > then, by Lemmas 0.4 and Remark 1.2.1, y is a finite cylinder measure over X Further, by Lemma 0.4.2, g is a limit measure of y , and therefore it follows that \p is the characteristic functional of y 93 7S y 7\ Since E, is c -Radon and K(c ) = E , then, from Lemma 1.5.1.2 and the definition of y we see that y is E-tight. Hence, by Prop. 1.2, y is S -continuous, and therefore, by Appendix 2.5, r \p is S -continuous . 1. By Thm. 2.2.1. 2. By Thm. 2.6.1, Cor. 2 of Appendix 3.5, and Appendix 3.2.1. 3. By Cor. 1 of Thm. 2.4. 1 and 2.7.2. are consequences respectively of Thms. 2.2.3 and 2.3, since Hilbert spaces are barrelled. 3. Since X and Y are Banach spaces, by [31] p. 339, Thm. 1, 2 T is S -continuous <=> T is Hilbert-Schmidt. The assertion is now a consequence of Cor. 1 of 2.4, and Cor. 3 of Appendix 3.5. 4. Since X and Y are Banach spaces, by [42] p. VII. 3, §2, for any r > 0 , T is S -continuous <=> T is r-absolutely summable. The assertion now follows from Thm. 2.6.3 and Cor. 3 of Appen dix 3.5. 94. 3. Nuclear Spaces. Nuclear spaces comprise one particularly important family of Hausdorff, Hilbertian spaces (Grothendieck [14], see also [36] and [47]). We shall therefore interpret the results of the previous section for the case when X is a nuclear space. As a consequence of the special structure of nuclear spaces, we shall be able to strengthen considerably the theorems concerning cylinder measures over arbitrary Hausdorff, Hilbertian spaces. We point out that many of the common spaces of distributions are in fact nuclear (Treves [47] Ch. 51). For our definition of a nuclear space we shall use a characteri zation due to Pietsch ([29], [36] p. 178). 3.1 Definition. X is a nuclear space iff X is a Hausdorff, locally convex space with the following property: for any neighbourhood U of 0 in X , there exists another neighbourhood V of 0 in X , and a w -Radon measure n on X with supp n C , such that {x e X : J|f (x) |dn(f) <_ 1} C U . 95. Remarks If X is a Hausdorff, locally convex space, then, from Remark 1.1.3 and the above definition, we see that (1) X is nuclear iff is a base for nbnd 0 in X For any nuclear space X , from (1) above, Remarks 1.1.2 and 1.1.3, it follows that r (2) the S -topologies on X coincide for r >_ 1 2 In particular, taking 5 as a base for nbnd 0 in X , we deduce that (3) X is a Hilbertian space ([36] p. 102). As in Treves [47], p. 519, we can prove that (4) E c K(s*) . Hence, if X is barrelled, then (5) E = K(s*) • We point out that coincidence of all the S -topologies for r > 0 is a consequence of (2), (3), and Theorem 2.6.2. The theorems given below in 3.2 are direct consequences of the above remarks, and assertions from the previous section }specifically, Theorems 2.2, Theorem 2.3, and Remark 2.5.2. 3.2 Theorems. Let X be a nuclear space, and u be a cylinder measure over (1) u is continuous <=> p is E-tight. 96. (2) y is continuous => y has an s -Radon limit measure on X' k (3) If K(s ) = E , in particular, if X is barrelled, then, y is continuous <=> y is E-tight <=> y has an s -Radon limit measure on X' (4) Let -ty be a positive-definite function on X i> is continuous => there exists an s -Radon measure E, on X' such that i>(x) = / exp i Re f(x)d£(f) . for all x e X . If E = /((s ) , in particular, if X is barrelled, then >jj is continuous <=> there exists a finite s -Radon measure 5 on X such that ifj(x) = / exp 1 Re f (x)d£(f) for all x e X . Theorem 3.2.2 extends a result of Minlos ([25], p. 303, Thm. 1), who considered finite cylinder measures over countably normed nuclear spaces ([11] p. 56). Vilehkin extended that result to the case of countable strict inductive limits of such spaces ([11] Ch. IV 2.4). Theorem 3.2.4 extends results due to Minlos ([25] p. 310) and Badrikian ([1] p. 17). We note that the theorems of 3.2 completely resolve a conjecture of I. Gelfand ([25] p. 310, [18], p. 222), that every finite continuous cylinder measure over a nuclear space X has a limit measure on the continuous dual X' Theorem 3.2.1 has a partial converse which extends a result of Minlos ([25]. Thm. 4). 97. 3.3 Theorem Let X be a Hausdorff, locally convex space. If X is a CM-space and y E CM(X) => y is E-tight, . then X is nuclear. Proof By Prop. 1.2, y e CM(X) => y is S^-continuous. Hence, by Appendix 3.2.1, the S^-topology is finer than the CM(X)-topology. On the other hand, by Cor. 2 of Appendix 3.5, and Remark 1.1.3, the CM(X)-topology is finer than the S^-topology. Consequently, the CM(X)-topology = the S^-topology. Since X is a CM-space, it follows from Remark 3.1.1 that X is nuclear. © Remark. . We note that a Hausdorff, locally convex space is not neces sarily a CM-space (Example 4.3, Appendix 4). When X is not a CM-space we see from the above proof that the best assertion possible is the following. If y e CM(X) => y is E-tight, then, the S^-topology and CM(X)-topoIogy coincide. 98. Theorems 3.3 and 3.2.1 lead to the following new characteri zation of nuclear spaces (Remark 3.1.3, Cor. 3 of Appendix 3.5). 3.4 Theorem Let X be a Hausdorff, locally convex space. X is nuclear iff X is a CM-space and y e CM(X) => y is E-tight. Concerning induced cylinder measures, Remark 3.1.4 enables us to strengthen Corollary (2) of Theorem 2.4. In view of Remark 2.5.2, the following assertion is immediate. 3.5 Theorem Let X be a nuclear space, Y be a topological vector space, and T e L[X,Y] . If T is continuous, then, for every y e CM(Y) , y a T is E-tight, and therefore has an s -Radon limit measure on X We observe that an infinite-dimensional normed space cannot be nuclear ([47J, p. 520). As a consequence of this fact we can assert that certain cylinder measures over such a space X cannot have a limit measure on X' 3.6 Proposition Let. X be an infinite-dimensional normed space If u is a finite cylinder measure over X such that the topology of X is the {y}-topology, then y does not have a limit measure on X' Proof. By Cor. 1.4.3, Prop. 1.2, and Appendix 3.2.1, y has a limit measure on X* => y is E-tight 1 => y is 5 -continuous 1 => {y}-topology is coarser than the S -topology => X is nuclear, by Remarks 1.1.3 and 3.1.1. Since X is an infinite-dimensional normed space the last assertion cannot hold, and therefore y cannot have a limit measure on X' Corollary Let A be an index set. For any 1 < p < 2 , if is the finite cylinder measure over (A) with characteristic functional (Remark, Appendix 2.4) exp - ( l\ x \\ P x e £P(A) -> ex  (It x \ )P e £ , then y does not have a limit measure on (£P(A))' 100. Proof See (1) in Proof of Example 4.2, Appendix 4, and Proof 3.1.1 of Appendix 3. Remark For p = 2 the above corollary is well known (Gross [13]). We have not seen a treatment of the case 1 <_ p < 2 in the literature. 101. 4. & -spaces. Applied to j^-spaces, 1 <_ p <_ °° , the theory of the previous chapter yields results analagous to those for Hilbertian spaces. Since 2 I -is a Hilbert space this case has already been discussed in §2.5. The results given there are stronger than those we shall obtain here for an arbitrary £^-space. Notation Let A be an index set. For any 0 < r <_ °° , {x'e <?• : E |x(c0 | < °°} when r <_ aeA i £r(A) = \ A i <• {x E C : sup lx(a) | < °°} when r = 00 asA We give £ (A) the usual topology, i.e., when r < 1 , the topology generated by the quasi-norm (Appendix 3.3) b : x E £r(A) -> E |x(a) |r e R+ ; aeA when r >^ 1 , the topology generated by the norm : x e £r(A) + (• E |x(a)Ir)1/r e R+ , ' aeA where we ( E |x(a)|r)1//r = sup |x(a) | if r = °° . aeA. a£A For any 1 < p < 2 , U = {x E £P(A) : .E |x(a)|2 < 1} . P. aeA 102. For any outer measure n on a space ft £v(n) = lim I n(B)|lnn(B)| PeP(M ) B£P n where tot C<x. ulnu = 0 when u = 0 The heart of this section is the following group of results, which p assert that certain families of subsets of £" (A) , 1 <_ p <_ « , are weighted systems in £P(A) 4.1" Theorem. Let 1 <_ p <_ oo and 1/p + 1/q = 1 For any r > 0 , let —r r r 5 C S consist of those sets S e S for which r ,n satisfies the added condition £v(n) < •» , • and when 1 <_ p j<_ 2 , let ~ r r r S S> consist of those sets S e S for which T\ satis-r,n 1 fies the added condition / (sup | f (x) | rdn (f) < «> . xeU P (1) If 2 < p <_ oo and 0 < r < q then Sr is a weighted system, in £P(A) (2) If 2<p<_oo and r = q then —r p S is a weighted system in £ (A) (3) If 1 < p < 2 and 0 < r < oo , then Sr is a weighted system in £P(A) (We note that Sr = Sr when p = 2 .) 103. The proof of the above theorem will be given at the end of the section. Now, we point out its immediate consequences when taken together with Corollary II.3.3. 4.2 Theorems Let 1 < p < oo and — + — = 1 - - p q (1) If 2 < p <_ oo and 0 < r < q , then, for any cylinder measure y over £P(A) , „r . p is • i -continuous s=> p is E-tight •k . L\»vX p has a c -Radonimeasure on (£P(A))' (Here, we also use Prop. 1.2. and Thm. II.2.1, noting that £P(A) is a Banach space and is therefore barrelled.) (2) If 2 < p <_ « and r = q , then, for any cylinder measure p over £P(A) , —x p is S -continuous => p is E-tight => •k _ Liy*>'^ p p has a c -Radonimeasure on (£ (A))' (3) If 1 <_ p <_ 2 and r > 0 , then, for any cylinder measure p over £P(A) , p is S -continuous => p is E-tight => •k _ p p has a c -Radonimeasure on (£ (A))' 104. Using Theorems 4.2 we can represent certain positive-definite functions on £P(A) as Fourier transforms of measures on (£P(A))' The proofs of the assertions given below are similar to the proof of Theorem 2.3, and are therefore omitted. 4.3 Theorems Let l<p<°°,-- + — =1 , and \j> be a nositive-def inite — — p q function on £P(A) (1) _ If 2 < p <_ «> and 0 < r < q ', then, r * < is S -continuous <=> for some finite c -Radon measure E, on (£P(A))' , U>(x) = / exp i Re f(x) d£(f) for all x e £P (A) . (2) If '2 < p < » and r = q , then, —r * /• ^ is S -continuous => for some finite c -Radon measure E, on (£P(A))! , iKx) = / exp i Re f (x)d£(f) for all x e £P(A) . (3) If 1 <_ p <_ 2 and r > 0 , then, ~ r ^ f ^ is 5 -continuous => for some finite c -Radon measure E, on (£P(A))' , ip(x) = / exp i Re f(x)d£(f) for all x e £P(A) . Concerning induced cylinder measures, Theorem 4.1 yields the following results when taken together with Theorem II.4.3. 105. 4.4 Theorems. Let 1 < p < 00 , — + — = 1 , Y be a vector space, C be - - P q a family of finite cylinder measures over Y , and .T e [£P(A),Y] (1) If 2 < p <_ <*> and 0 < r < q , then, T is Sr-continuous with respect to the C-topology on Y <=> for every y e C , y u T has a c -Radon limit measure on (£P(A))' (Here, as for Thm. 4.2, we also use Prop. 1.3 and Thm. II.2.1.) (2) If 2 < p <_ 00 and r = q , then, T is ^-continuous with respect to the C-topology on Y => for every y e C , y • T is E-tight => for every y e C , y p T has a c -Radon limit measure on (I (A))' (3) If 1 <_ p <_ 2 and r > 0 , then, T is S1-continuous with respect to the C-topology on Y => for every y e C , y D T is E-tight => for every y e C , pa T has a c -Radon limit measure on (&P(A))' As consequences of Theorems 4.4 we have the following extensions of results due L. Schwartz 139] and Kwapien [19]. They consider only the case when r = q and A is countable. 106. Corollaries. (1) If 2 < p <_ «>, 0 < r < q , y e £r (A) and T : x e £P(A) •+ (x(a)y(a)) . e /(A) , aeA then, for every y £ CM(£ (A)) , u a T has a c -Radon limit measure on a" (A))* (2) If 2 < p < co, r = q , y e £r(A) with £ | y(ct) |11 ln| y(a) [ | < co , aeA and T : x e £P(A) (x(a)y(a)) e £r(A) , - x then, for every y e CM(£ (A)) , y n T has a c -Radon limit measure on (£F(A))' (3) If 1 £ p <_ 2 , r > 0 , y e £r(A) , and T : x e £P(A) -»• (x(a)y(a)) A e £r(A) , aeA then, for every y e CM(£ (A)) , * P y a T has a c -Radon limit measure on (£ (A)) We give here the proof of onl\T Corollary (1) . The other proofs are similar. Proof of Corollary (1). For each aeA , let e e (£P(A))? : x e £P (A) ->• x(a) e C , a and n De the discrete measure on (£p(A))r with i i r n({e )) = v(a) for each aeA a 107. Then, supp n e E , and for any x e £P(A) , £ I (T.*)Jr = J|f 00 |rdn(f) • It follows that T is S -continuous, and the corollary is now an immediate consequence of Thm. 4.4.1. # 4.5 Remarks. (1) " If A is countable, then (£P(A))' is separable. Consequently, by Theorem II.2.3.2, every c -Radon measure on (£ (A))! is in fact s -Radon. The foregoing theorems may then be suitably modified. (2) We point out that Corollary 4.4.2 is the best result possible when 2 < p <_ <» and r = q . If \ ye £q(A) with T, | y (a) | q | In | y (a) | | = » aeA and T is as given in the corollary, then by Example 4.2 of Appendix 4, there exists p e CM(i>5(A)) such that p Q T fails to be E-tight. (3) With the notation of (2) above, as in the proof of Corollary 4.4.1, we see that T is S^-continuous, • and therefore, by Lemma II.4.2, pp T is Sq"-continuous. From Remark (2) above, and Theorem II.3.3, it now follows that 1 1 for any 2<p<°° ,if 1 =1 , then - p q Sq is not a weighted system in £P(A) 108. However, Proposition 1.2 suggests that when searching for a weighted system in £P(A) we ought to look for a subfamily of Sq Remark (2) then indicates that is in fact an appropriate subfamily of Sq for us to consider. When 2 <_ p < °= , the construction which we use for producing a system of 5-weights in £P(A) depends on the fact that for any finite set K , x z (EK -> exp - Z |x(a) | q e ® aeK K 11 is a Dositive-definite function on <p , where — + — =1 p q (Remark (1) of Proofs (4), and Proof 2.2 of Appendix 2.) If 1 <_ p < 2 , then, q > 2 and the function given above is no longer positive-definite (Schoenberg [38J p. 532). The construc tion therefore breaks down when 1 < p < 2 . We can show that construction of a system of 5-weights in £P(A) , 1 < p < 2 , would be possible if there were a X : R+ -> £ such that for any finite set K , x e ?K X( I |x(a) |q) e € aeK K was positive-definite on (J . If such a function X existed, then, by Appendix 2.2.4, (i) x e £q(A) X( Z |x(«) |q) e C aeA would be a positive-definite function on £q(A) . However, when q > 2 , one can show as in [5] that there does not exist X : R+ -> <E such that (i) holds. Nonetheless, we can still obtain a system of 6-weights in £P(A) , 1 <_ p < 2 , if we use the system of 5 —weights induced by the canonical imbedding x E iP(A) x £ £2(A) . (Remark (2) of Proofs 4.) 109. (5) Remarks (4), Proposition 1.2, and the proof of Theorem 4.1.1, ~ r r led us to believe that S C S , r > 0 , would be a suitable family to study when searching for a weighted system in £P(A) , 1 < p < 2 1 1 (6) We point out that for p >_ 1 and — + — = 1 , the appearance of q in the hypotheses arises at the finite-dimensional level (Remarks (4) and Appendix 1). Thus, although (£P(A))' may be identified with £q(A) , we have avoided doing this, as carrying out such an identification might have suggested that the relationship between £P(A) and £^(A) was crucial to our argument. Proofs 4. Together with the notations of Appendix 1.1 and Proofs 2, for any p >_ 1 , let p q X = £P(A) , V =' {x e X : Ixl < 1} P P 1 'p -I . I : f e X* -> sup If(x)} . q P xeV P For any finite K c A , let |.| „ : f e (f,V + sup {|f(x)| : x e V ft <CK} , q ,K p r = r K cK,x P Let ^ K F = {€ : K c A is finite} directed by inclusion, 110. For any 2 <_ p <^ » and finite K c A , let K • K Yp be the product measure on £ generated by y on C , D K K and v : £ e F -> Y ? which is Radon. Remarks (1) By Appendix 1.2, for each 2 < p < «> } (vP,F,V ) is a system of 6 -weights in X P P P (2) For each 1 p < 2 , since (V2 f\ CK)°c (V A CK)° for every finite Kc A , then, by Appendix 1.2, for every 1 <_ p < 2 , 2 (v ,F,V ) is a system of 6 -weights in X p 2 p 4.1. We observe that for any p 1 and r > 0 , (1) Src nbnd 0 in X (Remark 1.1.3). P and (2) ker V = {0} e S for every S e . V r Now, for any S =' S e S , and each B c X' , let ' r,n p fB e B , gB =_n1/r(B).fB and s = sup { | f | : f e supp n } . Then, as in Proof 2.2.1, for any PeP , finite K c A ., and t > 0 , we have that (5) vP(CK^ tS') < vP({x e CK : Z \~ ZnM\V > D) , K , P ~ K BeP and, by the lemma of Proofs 2, (6) vP(£K~ tS) = lim VP«EKA tS') . ' PeP(S) p Ill, Case 1. (2 < p <_ °°, 0 < r < q) . Since supp r\ z E , 0 < s < ro Then, for any t > - r/^CX') , s p B c supp n => |1 gj < 1 . Hence, by Appendix 1.2.3, the right-hand-side of (5) is majorized by 1- r /vM j. ±- n^r ,q-r+l, eq^q/r ,v, < C i n(X') + — 2^C P-^sV' (x') .r p,r p fcq pv q-r py (Since q/r > 1 , then E nq/r(B) = nq/r(x') E [p(B)/n(x')]q/r BeP p BeP P < nq/r(x') E n(B)/r,(x') = nq/r(x').) _ BeP P • P ' Whence, by (5) and (6), (7) YK«EK ~ tS) < ±- C Srn(X')'+ i- 2TTC (4^±I)SV/r (X') . P - fcr p,r p tq p q-r p Since the coefficients of 1/t and 1/t are finite and independent of K it follows that K K (8) YP(C ~ tS) -> 0 as t ->• °° uniformly for all finite Kc A . By (1), (2), and (8), 5 is wei ghted by (vP,F,V ) . 112. —V Case 2 (2 < p < », r = q, S e 5 ) . • If c = sup u In u 0<u<s then 0 < C < oo 1 1/r For any t > — n (X') ., by Appendix 1.2.3, the right-hand-side s p of (5) is majorized by (9) C E |r,r(f g) |q „ + 2TTC E | r_,£ g) |q' | In | r (i g) | P'q B£P q' p BeP q' q' The first term of (9) is majorized by • — C sqri(X') . tq psq p The second term of (9) is majorized by 2^Cp ^E^ (!fB|q)qn(B)[|lnt| +l|lnn(B)| + | In | f B | q | ] lint! 2TTC Sqn(X') + — 2TTC Sq - E n(B)|lnn(B)| tq . P P tq P q BeP ' + — 2irC c n(X') . tq P P Hence, by (5), (6) and (9), Y (C tS) P 2TTC < — [c sqn(x') +—E q£v(n) + 2,TC C n(X')] - tq p.q P q P P + Ji^l [2nC q tq p P By the hypotheses, the coefficients of l/tq and |lnt|/tq are finite and independent of K . Hence, (10) vP(CK ~ tS) -> 0 as t -> oo uniformly for all finite KC A K By (1), (2) and (10) Sr is weighted by (vP,F,V ) P 113. Case 3 (1 <_ p <_ 2 , r > 0 , S e Sr) By Appendix 1.2.3, for any t > 0 , the right-hand-side of (5) is majorized by °2,r BE£p lrK(K>l2,K ' Hence, by (5) and (6), v2(£K ~ tS) = lim Yo(CK^ tS') K ' PeP(S) 2 P = C . lim E sup „ |f (x)|)r.r,(B) t PeP BeP xeUr\C P 1~ c2 r lim 1 (sup (sup If(x)l)1) n(B) • t 5 PeP BeP feB xeU p = 7C /*(sup |f(x)|)rdn(f) • t Z'r xeU P By hypothesis, the coefficient .of 1/t is finite and independent of K . Hence 2 K (11) v„(C1 ~ tS) -> 0 as t -> °° uniformly for all finite K C A .' K By (1), (2) and (11), ~r 2 " S is weighted by (v ,F,V ) 114. APPENDIX • In this Appendix we establish a number of results and construc tions which are necessary for the discussions of Chapter III. In the last section we give some counterexamples which complement the considera tions of Chapter III. 1. "Special Measures, on Finite-Dimensional Spaces. In this section we shall construct special measures on finite-dimensional spaces. The existence of these measures enables us to produce systems of <5-Weights in Hilbertian spaces and in &P-spaces, P L 1 . Notation K is a finite set. For any 15.pl03 , p q (x e cK :znrv lx(a)lP 5 l) when p < °°, P (x e C : sup |x(a)| < 1} when p = » aeK For any f e (CK)" , q xeV sup |f(x) P (We note that V° = {f e (tK)" : If.l < 1}) p ' 'q ~ 115. A is the Lebesque measure on <C ' . For any finite dimensional space F , I = {(x,f) e F x F* : |f (x) | >_ 1} . The constructions of this section will be based on the asser tions given below. 1.1 Lemmas Let 2 <_ p <_ «> and r > 0 (1) There exists a strictly-positive, continuous (9 : £ + R+ P such that exp - |w|q = J (exp i Rewz)(9 (z)dw(z) for all w e £ (2) When 2 < p <_ <»• , there exists (i) 0 < C < co P such that (ii) 0 (z) < C /|zl2+q for all z e t . p p 1 Hence, when r < q , for any u > 0 , 2TTC 2 fl + l/uq r] if r < q , (m) q-r n z|r Q (z)dA(z) < <u ? 1< z 2TTC lnu if r = q , and (iv) f Q (z)dX(z) < 2TTC Jul q u<|z| P - P1 1 (3) When p = 2 , 1 Izi2 0n(z) = -— exp LTj— for every z e £ 2 4T( 4 116. Notation and Remarks For each 2 <_ p <_ m , let Y : B c C + f 1J dA £ R+ , 'p ' B u ' 6 = Y ({z e € : Izl > 1}) . P P ii-For any r > 0 , let C9 r = HZ|r <MZ) <*X(z) • For each 2 < p <_ °o > and any 0 < r < q , let C be the constant of Lemma 1.1.2, P and 2TTC + Ji ' |z| ^(z)dA(z) if r < q q-r C q-r |z|<l P P>r 2TTC + / |z|r 0 (z)dA(z) if r = q P |z|<l ' P We note that in view of Lemmas 1.1.1 and 1.1.2, Y is a probability Radon measure on € , 5 > 0 p 0 < C < co, p 0 < C < CO . 117, 1.2 Lemmas Let 2 <_ p <_ oo and K K Yp be the product measure on € generated by the measure y on € , P K K (1) Yp is a probability Radon measure on C K « 0 K f (2) f e («T) ~ V => Y (I ) > <5 P P - P (3) For any sequence {f } C (<C^) , and r > 0 , let n new B = {x e CK : -E If (x) 1 r > 1} 1 n 1 new (i) If p = 2 , then q = 2 and new (ii) If p > 2 , r <_ q , and l^nlq — ^ ^or every new , then, C E If |r + 2^C (q"r+1) E If I2 if r < q , p,r 1 n'q p q-r 'n'q new new C E If lq + 2TTC E If |q|ln|f I I if r = q p,q 'n'q p '.n1q1 1 n1q1 new ^ r new n n 1.3. Lemma. Let F be a finite-dimensional vector space. If [.,.] is a pseudo-inner product on F and V = {x e F : [x,x] <_ 1} , then, there exists a probability Radon measure £ on F such that (1) f e (ker V)3' ~ V° => £(I£) 1 62 • 118. (2) For any sequence {f } in F , n neoo £({x e F : E |f (x)|r > 1}) < C. E (sup|f(x)|)r nea) new xeV Proofs 1. 2 1.1 Let \ be the Lebesque measure on R From Blumenthal and Setoor [3] p. 263, we have the following facts, (See also Levy [21] Ch. VII.) For any 0 < q <_ 2 , there exists a strictly positive continuous 2 + 9 : R R P s.t, (i) exp - |t| = /[exp i(t.u)] 0 (u)dA(u) for all t e R2 , 2 2 where, for any t e R , u z R , t.u = tQu0 + and |t| = /(t2 + t2) . If q < 2 , there exists 0 < c < <» s.t. q lim |t|2+q 9 (t) = c i • i P q Hence, if (ii) C = sup |t|2+q0 (t) , P xeR P then, (iii) 0 < C < P and (iv) 6 (t) < C /|t|2+q for all & P — P 119. Consequently, for any 0 < r <_ q , and u > 0 , /|t|r 6i (t)dx(t) < c / !t|r . —1—dxet) p • ~ p i< t!<u • |t!2+q = 2TTC j • (p^+q ~) ''"dp using polar coordinates. P l£p£u By integrating the last term it follox-Js that (v) 2TTC 2TTC £[l-l/uq r] < ^[l+l/uq r] , if r < q , 1< t q-r - q-r T\Q (t)dA(t) < <U P ~ || — 2TTC lnu if r = q P For q = 2 , using the fact that 2 ,2 (l//2rr)|R (exp i xy)exp - — dx = exp - ~- , a direct computation shows that (vi) <92<t) =^exp Hence, for any r > 0 , •r" (vii) /|t| 02(t)dA(t) < » . Let 2 T : z e <C -> (Re z, Im z) e R , and for each 0 < q <_ 2 •, & = 6 o T . P P The assertions of Lemmas 1.1 now follow from (i) - (vii) above, and the properties of the map T , namely, for any z and w in (C , Re wz = (Tw).(Tz) ' , and T is an isometric, measure preserving, homeomorphism. 120. 1.2. K * Notation For any f e (<S ) , let -f/|f|q > <?f : x e ffK -> f (x) e £ "K,f fr K and let Yp - f [Yp] ~f — ~K f T : w e £ -> (exp i Re wz)dy ' (z) e £ 1 .'2.1 ' This follows from the fact that Yp ls a probability Radon measure on £ 1.2.2 For each aeK , let e = 1 r , e £ a {a} K * Then, for any f e (£ ) > and w e £ , - ^ " K f (1) \p (w) = Jexp i Re wz dy ' (z) = /exp i Re f(wx)dy (x) = II [/(exp i Re wf(e )x(a))dy (x(a))] aeK a P = exp - 1 Iw f(e )|q by Lemma 1.1.1, ' aeK a = exp - lw!q|f|q . 1 q Since a Radon measure on a finite-dimensional space is uniquely determined by its Fourier transform (Bochner's Theorem, Appendix 2.3), it follows that (2) Ifl = 1 => YK'f = Y 121. However, and f =1 . K * 0 II f e (C ) - VU => f > 1 P 1 'q -Hence, = 6 , by (2) above. P 1.2.3. (I) Since E f (x) is a series of positive Yo~measuraDle • n ' -2 new K functions on C , ^(B) = /lB.dY^ < /( Z |fn(x)r)dY^(x) new = E lfni2 /|f;W|rdY|(x) new = E \fjz2 j\z\X dY2 n(.Z) new = C„ E If I* by (2) of Proof 1.2.2. 2, r 1 n'2 new (ii) Let and H = {x e CK : |f (x)I < 1} for each new , n 1 n — H = C\ H n new K K h :' x e € -> 1 if x e C ~ H E |f (x)|r if x e H . n 1 new 122. We have that = I Y„({x e £ : |f'(x)| > l/.|f I }) new P n n q -K,f = Z Y_ n'({z e C : |z| > 1 / I f n I a > > new (2) < 2TTC £ If |q by (2) of Proof 1.2.2. — p 1 n'q J new and Lemma 1.1.2.(iv). f'(x)I is a series of positive v -measurable n 1 • 'p new r functions on , we also have that /iHhdYpwiH[£ igx«r^>> new = £ new < I |£ i„ / 1„ |f'(x)|rdT (x) — 'n'q' H 1 n 1 p new n ~K,f' = Z |f |r / |z|r dY n(z) Hence n'q ' ' ' 'p ,'z|<l/|f j ' 1 — 1 n'q Z If |r / |z|r 6 (z)dA(z) 'n'q 'ii pv' M-^^Jq by (2) of Proof 1.2.2. new new / l17hdY < I If T[ / |z|r 6 (z)dA(z) + / |z|r 9 (z)dA(z)] 'Hp— 'n'q-' p /II pXJ z <1 1< z <1/ f n'q Letting a = J |z|r 6 (z)dA(z) , from P > r p | z| <1 Lemma 1.1.2 (iii), we have that 2TTC Z |f |r(a + P-[l + If iq r]) if r < 1 n1q p,r q-r 'n'q neoj (3) / < E If |r(a + 2TTC |ln|f I I) if r = q 'n'q p,r p1 'n'q' new ^ Consequently, if r < q , then, from (1), (2) and (3), 2TTC Y (B) < 2TTC Z |f|q+ Z If |r(a + P-[l+|f|q1]) P - P new n q new n q P'r q"r n q = C Z |f |r + 2TTC (~q^) Z |f |r , p,r 'n'q p q-r 'n'q new new since all terms are positive. The case r = q is established from (1), (2) and (3) similarly. 1.3 We first suppose that [.,.] is non-degenerate. If so choose a [.,.]-orthonormal basis K for F . Let T:ze(C -> Z z.aeF . „ a aeK and .Then, * , K * feF •+ f = f o T" e (C ) T is a homeomorphism. K K ^ Further, for any x' e C , y' e C , and feF , (1) [Tx'jTy'] = <x',y'> , where <.,.> denotes the inner product in C , and (2) sup |f(x)| = |f| xeV Whence, if h ~ Y2 ° T 124. then and 5l(If) = Y2<LF) C(B) = y^ax1 E CK : Z |fn(x')|r > 1}) new The assertions now follow from (3) and Lemmas 1.2. When [.,.] is degenerate, let F^ be any subspace of F s.t. F is the direct sum of F^ and ker V (Possibly, F = {0}) . We have that [.,.]|F x F^ is non-degenerate, since F^/TN ker V = {0} Let £ be the probability Radon measure on F^ determined as above, and E : HcF+5(H/> F1) e R+ . Then, (3) ^ is a probability Radon measure on F Since F is the direct sum of F^ and ker V , for any x e X there exists a unique representation x = x^ + , with x^ e F^ and x^ E ker V Consequently, (4) f e (ker V)3 => sup |f(x)| = sup |f(x)| xeV xeVKF and therefore, from the non-degenerate case above, (5) f e (ker V)a ~ V° => rp (f) e F* ~ (V A F^)^ rF (f) f => e;(ir) = ^(I ) > 52 . » a If f e F (ker V) for any new, then n sup If (x) | • = » xeV and therefore (2) of the lemma holds. If f e (ker V) for every new, then from the non-degenerate case above, 5(B) = £ '({x e F. : Z |f (x)|r > 1}) 1 1 n 1 new < C2 r Z sup |fn(*)|r ' new XEVAF^ = C Z sup •|f (x)|r by (4). new xeV 126. 2. •Positive-definite Functions on Vector Spaces. In this section we give a number of useful results concerning positive-definite functions on vector spaces. 2.1 Definition Let X be a commutative group. ip is a positive-definite function on X iff if) : X -> C , and for any n e to, {x^,. . . >xn_^} £ X , {z , . . ., <Z C , n-1 _ E ZkZ£ ^(xk ~ X£} - ° • k,£=0 We shall need the following elementary assertions about positive-definite functions on groups. 2.2 Propositions Let X be a commutative group. (1) If if; is a positive-definite function on X , then 0 <_ CO) < °° . (2) Let if) be a positive-definite function on X , and Y be a commutative group. If T : Y -> X is a homomorphism, then if) 0 T is a positive-definite function on X (3) If cf and if) are positive-definite functions on X , then q»ip is a positive-definite function on X (4) If (ilO. , is a net of positive definite functions on J Jed X , and ip : X C is such that 4(x) = lira I|J . (x) for all x e X , jeJ 3 then, is a positive-definite function on X (5) If X is a topological group, and is positive-definite function on X , then \p is continuous on X <=> \p is continuous at 0 e X For finite-dimensional spaces we have the following version of a well known, representation theorem (Rudin [34] p. 19 1.4.3, Bochner [4] p. 58). 2.3 Theorem Let F be a finite-dimensional vector space. is a continuous positive-definite function on F iff there exists a unique finite Radon measure £ on F such that 4<(x) = / exp i Re f(x)d£(f) for all x'e F Using the above theorem, as in [11] (p. 349) one readily establishes its following infinite-dimensional analogue. We omit the proof. (The theorem given in [11] is formulated only for real vector spaces. See also [48]). 128. 2.4 Theorem Let X be a vector space. ij) is a positive-definite function on X with I^|F continuous for every F e F , iff there exists a unique finite cylinder measure u over X such that I|J(X) = / exp i Re f(x)du (f) for all x £ X . Remark When u and i|; are related as in the foregoing theorem, we call i|; the characteristic functional of u (Prohorov [33]). The final theorem of this section is useful for determining continuity properties of cylinder measures. As an adjurxctto Proposition II.2.6, it further motivates the terminology "continuous cylinder measure", introduced in Definitions II.2.5.1. 2.5 Theorem Let X be a vector space, u be a finite cylinder measure over X , and V be a family of balanced, absorbent subsets V of X with uV e V for every u > 0 If i|) is the characteristic functional of u , then u is l/-continuous <=> <jJ 1S ^-continuous. Corollary Let X be a topological space, and p be a finite cylinder measure over X If ii) is the characteristic functional of p. , then p is continuous <=> iji is continuous. Proofs 2 2.2.1 Taking n = 1, x^ = 0 , and = 1 , the assertion follows immediately from Defn. 2.1, 2.2.2 For any new, {x^,...,xn_^}C X , and {z^,...,zn_n} ^® > n-1 _ n-1 1 ZkZ£ * 0 T(X1< " V = E ZkZ£ *(Tx " Tx } ^- ° • k,£=0 k 1 l< £ k,£=0 k * Xk X£ 2.2.3 From [48] we have that Cf(x) = Cp (-x) for all x e X Hence, by Defn. 2.1, for any ne u>, {xA,...,x . } c X , and 0 n-1 {V\'Zn-l}C S ' M = (cpCx, - x ))" }_n is a positive-definite * k £ k, £—U Hermitian matrix. Hence, there exists an n x n-matrix T s.t. M = TT* , where T = (t ) , T* = (t* ) , and t* = I , • 130. Consequently, n-1 _ . I n V^k " V^k ~ V . n-1 _ n-1 1 V* ( E \ sK ?H'(xk _ X£} k,£=0 * s=0 ' b'y' 1 n-1 n-1 E 1 {W s)(zkt£ s^(xk " V -° > s=0 k,£=0 K K'S K l,S K 1 and therefore \p is positive-definite. 2.2.4 For any new, {x^, . . . ,x ,}c X , and {zrt..... z , } C u n-1 U n-1 n-1 _ n-1 _ . E n V* *(xk " X£} = ^ ^ ^ r ZkZ£ VXk " V - ° k,£=0 . jeJ k,£=0 J 2.2.5 From Rudin [34], p. 18, 1.4.1 (4), we have that for any x and y in X , |*(x) - iKy)| 1 2<JJ(0) Re (^(0) - if»(x - y)) . The assertion follows. 2•3 This theorem is a special case of a general theorem in Harmonic Analysis ([34], p. 19 1.4.3). However, it is readily derived from the real case treated by Bochner ([4] p. 58). 2.5 Together with the notations of II.2.4 and II.2.6, for each x e X , let ijj (w) = f exp i Re wzdu (z) for every z e € We note that (1) i|> (w) = IJJ(WX) for every z e (C Suppose that u is (/-continuous. Since z e £ -> exp i Re z e (C is bounded and continuous, by (1) and Prop. II.2.6 we have that (2) <|) is [/-continuous Suppose that I}J is (/-continuous. Since ip/c is positive-definite'for every c > 0 , by Prop. 2.2.1 we may assume that *(0) = 1 • Given any e > 0 , choose o < e« < f<^=h , t > 2//e* , and V e f s.t. x e V => 1 - Re IJJ(X) < e' . By (1) and the fact that V is balanced, (3) x e V , z e <C , |z| <_ 1 => 1 - ^ (z) < eT . i 12 2 2 If z = u^ + iu2 e ' <C ., then |z| = + u2 , and therefore, by (3), for any x e V , (4) u2 + u2 <_ 1 => 1 - i|> (V) < e' . Hence, by the lemma given by Kolmogorov in [17], for any x e V (5) U (D) = y ({z e « : |z| > 1}) < (e' + ~) X X ~^-l t2 < e Since e was arbitrary, it follows that u is (/-continuous Proof of Corollary 2.5 We note that nbnd 0 in X has a base balanced, absorbent sets V , with e.V £ V Hence, by the above theorem and Prop. 2.2.5, /A is continuous <=> u is (/-continuous <=> iff is (/-continuous <=> ij> is continuous. 132. V consisting of for every E > 0 3. CM-spaces. For any family C of finite cylinder measures over a vector-space X , we shall define the C-topology on X and give some of it properties. We shall establish examples of topological vector spaces whose topologies are exactly those determined by the families of continuous finite cylinder measures. We note that' for sets X and Y , topology G on Y , and T : X -> Y , '.{T_1[G] : G e G} is a topology on X . We shall refer to it as the-topology on X induced by G and T 3.1 Definition Let X be a vector space. (1) . For any family C of finite cylinder measures over X , the C-topology is the topology on X having for a base all subsets V of X with V = x + e A {y e X : y (D ) < e} yeH y y for some x e X , finite H C C , and e > 0 (2) X is a CM-space iff X is a topological vector space whose topology is the CM(X)-topology. Concerning C-topologies we have the following assertions. 134. 3 • 2 Propositions Let Y be a vector space, and C be a family of finite cylinder measures over Y (1) Y is a topological vector space under the C-topology, which is the coarsest such topology with respect to which C is a family of continuous cylinder measures. In particular, when Y is a topological vector space, and C = CM(Y) , the C-topology is coarser than the original topology of Y (2) For any vector space X and T e L[X,Y] if C o T = {y a T : y e C> • , then, the C Q T-topology is the topology on X induced by the C-topology and T We shall now show that the class of CM-spaces contains many interesting topological vector spaces. However, not all topological vector spaces are CM-spaces. In Appendix 4 we give an example of a Banach space which is not a CM-space (Example 4.2). 135. 3.3 Definitions (1) Let X be a vector space b : X -> R"*~ is a pseudo-quasi-norm on X iff b(0) = 0 , for any x and y in X , b(-x) = b(x) , b(x + y) < b(x) + b(y) , and z e € -»- b(zx) e R~*~ is continuous at 0 E. (D (For any family {b.}. T of pseudo-quasi-norm on X , as in Yosida [49] p. 31, one can show that X is a topological vector space under the coarsest topology on X making b^ continuous for every j e J ). (2) For any measure space (A,n) , and r > 0 , r,. „A „ . , , r i ,-1 r L (A,n) — {f G £ '• f is n -measurable, / |f| dr\ < 00}' , br : f e ir(A,n) ^ / |f!rdn e R+ , and when r >_ 1 , | - I : f e Lr(A,n) - (/ |f.|rdn)1/r e R+ . Remarks When 0 < r < 1 , b^ is a pseudo-quasi-norm on L1(A,n) , which is therefore a topological vector space under the coarsest topology making b^ continuous. 136. When r >_ 1 , |•| is a pseudo-norm on L~(A,n) > which is therefore a locally convex space under the coarsest topology making |'| continuous. We shall hereafter assume that L (A,n) , r > 0 , carries.the appropriate topology indicated by the foregoing observations. We shall need the following lemmas, which are. of independent interest. 3.4 Lemmas (1) Let X be a vector space, and b be a pseudo-quasi-norm on X . If \p is a positive-definite function on X such that the coarsest topology on X making IIJ continuous coincides with the coarsest topology making b continuous, then, there exists a finite cylinder measure y over X whose characteristic function is , and, the {y}-topology on X is the coarsest topology on X with respect to which b is continuous. (2) Let X be a vector space. If {bv,}T7 .. is a family of V Vey pseudo-quasi-norms on X such that for each V e I/, there exists a positive-definite function I|J on X satisfying the hypothesis given in (1) above, then, X is a CM-space under the coarsest topology making b^ continuous for each V c V (3) Let (A,n) be a measure space. For any 0 < r <_ 2 , f e Lr(A,n) exp - b (f) e £ is a positive-definite function on Lr(A,n) 137. The following theorem and its corollaries indicate that many of the topological vector spaces considered in this paper are in fact CM-spaces. 3.5 Theorem X is a CM-space whenever X is a topological vector space having a family f of neighbourhoods of 0 which satisfies the following conditions: (i) {eV : V e V, c > 0} is a base for nbnd 0 in X . (ii) For each V e C , there exists a measure space (A^,!"^) , rV 0 < rv <_ 2 , and Ty e L[X,L (^,r\ )] ,.such that r V = {x e X : J|Tv(x)| dnv < 1) . Corollaries. (1) Let (A,n) be a measure space. For any 0 < r <_ 2 , IT L (A,ri) is a CM-space. In particular, (A) is a CM-space. (2) Let X be a topological vector space. For any 0 < r <_ 2 , X with the S -topology is a CM-space. (3) Every Hilbertian space is' a CM-space. 138. Proofs 3 Notation For any vector space X , e > 0 , y e X , Ke) = {(x,f) e X x x* : |f(x)| >_ e} , Dy>c = (f e Fj = |f(y)| > , and for any family C of finite cylinder measures over X V(C,E) = H {X e X : p (D ) < e} UEC X X,£ Remark Since D = D for all x e X and e > 0 , x, e x e it follows that V(C,e) = E A {x e X : y (D ) < e} yeC x x 3.2,1 We shall only prove the first assertion. The second then follows immediately from the definition. Let V = {V(H,E) : H c C is finite, e > 0} . In view of the remark above, it will be sufficient if we show that 1/ has the following properties. (i) 0 e V for every V e 1/ (ii) \J is a filterbase. For each Vet/ , (iii) there exists U e l' s.t.U + UcV . (iv) V is absorbent, (v) V is balanced. (Treves [47] p. 21) 139. Proofs of (i) - (v) (i) For any e > 0 , and therefore 0 e V for every V e V (ii) For any 0 < 6 < E , y e C , and y e X , y (D J > y (D ) . y y>o - y y,e Hence, if V(e_.,C'J e V , j = 0,1 , and e = min {e^e.^ ' then. V(e,C„U C\) C H V(e.,C.) . 0 1 j-0,1 2 2 (iii) Let V = V(e,fO , and U = V(e/2,H) . For any x e X , y e Y , and feF, , (II.2.6), (x,y) |f(x) + f(y)| < |f(x)| + |f(y)| , and therefore, Ix+y(e) C Ix(e/2) U It(e/2) . Consequently, for any x e U , y e U , and y e H , y (D ) = y^ (I (e)) x+y x+y,e F, . x+y (x,y) < yF (Ix(e/2)) + yp (I (e/2)) (x,y) (x,y) = y (D , ) + y (D . ) < e . x x,e/2 y y,e/2 i.e.. U + U C V . (iv) For any xeX,e>0,t>0 , and y e C , and 0 < u < t => D , C B , x,e/u x,e/t 140. Consequently, since y^ is finite, lim y , (D . ) = lim y (D ) x/n x/n,e x x,ne rv-K° n-x" = y(A D ) = y (0) = 0 . new x,ne x Hence, for any V e 1/ , there exists new s.t. x/n e V i.e. V is absorbent. (v) For any yeC,xeX,e>0 and z e C . with | z | <_ 1 , y (D ) = y (D, , ) = y (D , .,) zx- zx,e x |z|x,e x x,e/|z| < y (D ) since D , > . C D — x x,e x,e/|z| x,e Hence, for any V e 1/ , zV C V . 3.2.2. As in Lemma II.4.2, for any x e X , e > 0 , and . y e C , Hence, for any e > 0 and finite subfamily H of C , T_1[A ' (y e Y : y (D ) < e}] yeH y y'£ = C\ {x e X : y (DT ) < e) yeh x » = A {x e X : (y a T) (D ) < e} . yeH x X'C It follows that '{T-1[V] : V e V} is a base for the C a T-topology neighbourhoods of 0 in X , where V is as defined in Proof 3.2.1. However, from Proof 3.2.1 we see that \J is a base for the C-topology neighbourhoods of 0 in Y The assertion now follows from Prop. 3.2.. 1 and the linearity of T Lemma Let F be a finite-dimensional space. If b is a pseudo-quasi-norm on F , then b is continuous on F Proof of Lemma. Let K be a basis of F Every x e F has a unique representation £ z (x)a , aeK a and the norm x e F -> E I z (x) I e R+ „ a • aeK generates the topology of F For any net (x.). _ in F , J J £ J x. -> 0 => E I z (x.) I -> 0 => 3 aeK a 3 z (x.) -> 0 for each aeK => a J b(z (x.)a) -> 0 for each aeK => a 3 E b(z (x.)a) -> 0 => b(x.) -> 0 , since aeK a 3 3 b(x.) < E b(z (x.)a) . J aeK J Hence b is continuous at 0 e F . However, for any x and y in F , |b(x) - b (y) | £b(x - y) , and therefore continuity of b at 0 e F implies continuity of b on F 142. 3.4.1 By the above Lemma, 4)|F is continuous for every F e F , and therefore, by Thm. 2.4, there is a cylinder measure y over X whose characteristic functional is ip . From the hypothesis, X is a topological vector space under the coarsest topology making continuous. Hence, by Cor. 2.5 and Prop. 3.1.1, the {y}-topology = coarsest topology making y continuous = coarsest topology making \p continuous = coarsest topology making b continuous. 3.4.2 By Prop. 3.1.1 and Lemma 3.4.1. 3.4.3 Let a : B E M •> a E B . n • B For any P e P(M ) , let d(P) be the family of finite subsets of P directed by inclusion. Then, for any f E L (A,n) , b (f) = lim lim E |f(a )|r*nCB) . r PeP(M ) Ked(P) BeK n Consequently, since t E R exp - t £ R is continuous, we have that i 1/r ,r exp - b (f) = lim lim II exp - |n(B) f (ct ) | r PeP(M ) Q£d(P) BEQ r n 1/r Since f E L (A,n) -> n(B) f(ajj) is linear for every B £ M , we deduce from Lemma 1.1.1 and Props. 2.2.2 - 2.2.4 that f E L1(A,n) exp - b (f) E £ is positive-definite. 143, 3.5 For each V e f , let = 1 when r^ < 1 1/r^ when r^ > 1 bv : x e X ->• (/|Tv(x) | d^ and \J> : x e X exp - b (x) e C For each V e 1/ , we have that by is a pseudo-quasi-norm on X , and for every t > 0 , rveV tV = {x e X : b (x) <_ t } Since the topology of X is completely determined by its neighbourhoods of 0 , it follows that the topology of X is the coarsest topology making b^ continuous for every V e V . However, by Lemma 3.4.3, ifj is positive-definite, and since t e R+ •+ exp - t e (0,1] is a homeomorphism, it follows that the coarsest topology on X making ^ continuous = the coarsest topology on X making b^ continuous. The theorem is now a consequence of Lemma 3.4.2. Corollaries (1) and (2) are. immediate consequences of the theorem. Proof of Corollary (3) Recalling the definition of a Hilbertian space (§111.2), we need only make the following observation. Let X be a vector space. For any pseudo-inner-product [.,.] on X , there exists a measure space (A,n) (A is an 2 index set and n is counting measure on A ), and T e L[X,L (A,n) ] , such that 0 [x,x] =/ |T(x)|idn for all xe X . (Treves [47], p. 115 - 116.) 144. 4. Examples 4.1 Example There exists a Banach space X and finite cylinder measure u over X such that y is S^-continuous but is not E-tight. Proof Let A be a set. Together with Notation 1.1, let X = £^~(A) with the usual topology (Notation III.4), [•',.] : (x,y) e X x X + Z x(a)y(a) e £ aeA 4 : x e X -»- exp - [x,x] e C For any finite K C A , K K * T ; w e £ -> f e (C ) , where K. w K f (x) = Z x(a)w(a) for all x e C aeK Since [x,x] = Z |x(a)|2 for all x e £1(A) , aeA as in the proof of Lemma 3.4.3, we deduce that \p is a positive-definite function on £~^(A) . Since x e X ->- /[x,x] is a norm on £^(A) , we further deduce that \{J|F is continuous for every F e r By Thm. 2.4, there exists a cylinder measure y over £^~(A) whose characteristic function is \\> . Then, for any finite K K C A , and x e C , as in Proof 1.2.2, / exp i Re f(x)d(Y2 o T^) (f) .= <Kx) / exp i Re f(x)du(f) = / exp i Re f(x)dy (f) . (Note that T is a homeomorphism, and therefore y 0 r is Radon.) Hence, by Thm. 2.3, (1) V-Y^,!-1 • Consequently, for any t > 0 , with the notation of Proofs III.4, ycK(rR[tvJ]) = y^KCtO^n £K)°)' = Y^Qw e cK : SUP !w(a)| £ t}) aeK = n / e2 (w(a))dX(w(a)) aeK | / \i |w(a)I<t However, / 62(z)dX(z) < 1 , | z|<t ' since / 62(z)dA(z) = 1 and is strictly positive (Lemmas 1.1). It follows that inf y£K(rK[tvJ]) = 0 , finite K c A and therefore, by Lemma 1.5.1.2, . y cannot be E-tight. On the other hand, by Pietsch [30] p. 82, Prop. 4, there exists S_ e- s.t. 1, u [x,x] <_/| f(x)|dn(f) for all x e X , and therefore x e X ->- [x,x] e R+ is .^-continuous. Hence <JJ is S^-continuous. Consequently, by Thm. 2.5, y is S^~-continuous. 4.2 Example Let A be a set, 2 < p < 00 , and 1 =1 - P q If ye £q(A) is such that E |y(a)|q|ln|y(a)|| = -aeA and T : x e £P(A) (x(a)y(o)) e £q(A) , then there exists y e CM(£q(A)) such that y p T is not E-tight. Notation. Together with the notations of §2.1 and Proofs III.4, for any t _> 0, , let b(t) = tq|lnt|, t > 0 , and b(0) = 0 . We shall need the following lemma ([41] Lemma2].) Lemma. Let w : A -> <L with |w(a)| 1 for all aeA There exists a constant 0 < C < <» such that for every finite K A , YK({z e £K : E |z(a)w(a)|q > 1}) >_ e"1 - exp - E b(|w(a)|) P aeK aeK Proof of Lemma Let 6 be the function of Lemma 1.1.1. p From [3] p. 263, 0 < lim |v|q+2 6 (v) < |v|-*° p 147. Hence, there exists 0 < C' <; «> s.t. (1) 0 (v) > C7|v|^+2 for all v E C with Ivl > 1 . By Taylor's theorem, for any 0 < t < 1 , 1 - exp - t = t exp - t' for some 0 <_ t' < 1 , and therefore (2) 1 - exp - t >_ t e 1 for all 0 £ t <_ 1 . Let (3) C = 27re_1C' . Then, for each aeA , (4) f(l - exp -- | vw(a) |q)0 (v)dA(v) >_ e"1 /|vw(a) |q6 (v)dA(v) by (2), 0<_| w(a) | <1 ie'V /|w(a)|q • 1q+2 dA(v) by (1) l£|v|<_ l/|w(a) | 'V' = 27re~1C,K*»lfy 1/p dp l<p<_l/|w(a) | = C b(|w(a)|) by (3). For any finite K C A , if BR = {z e CK : Z |z(a)w(a)|q > 1} , aeK then, YD(V l/ 1P (z)(1 " 6XP " S lz(a)w(a)|q)dY^(z) P K aeK P = /(l - exp - Z |z(a)w(a)|q)dYK(z) aeK P -/l (z)(l - exp - Z |z(a)w(a)|q)dYK(z) £ ~B aeK P > /(I - exp - Z |z(a)w(a)|q)dYK(.z) - (1 - e"1) aeK P = e 1 - II f exp - | z(a)w(a) | qdY (z(a)) . aeK P 148. However, for each aeK , / exp - |z(a)w(a)|qdYp(z(a)) = 1 - /(l - exp - |z(a)w(a)|q)dY?(z(a)) <_ 1 - Cb(w(a)) by (4) above. Hence, YK0O > e"1 - n [1 - Cb(w(a))] P K _ aeK > e ^ II exp - Cb(w(a)) since 1 - u < e U for all u > 0 . , aeK -1 exp - C E b(w(a)) aeK Proof of Example 4.2 If h : x e £.q(A) -> exp - £ |x(a)|q e £ , aeA then h is continuous. By Lemma 3.4.3 and the lemma of Proofs 3., h is positive definite and h|F is continuous for every finite dimensional subspace F of £q(A) . Hence, by Thms. 2.4 and 2.5, (1) there exists a continuous finite cylinder measure y over £q(A) with characteristic functional h Clearly, (2) y e CM(£q(A)) . Choose (3) t > 0 s.t. |y(a)|/t <_ 1 for all aeA . Let 0 < 6 < e"1 . For any finite subfamily K of A , let K K * h : z e € h (z) e (€ ) , with K K K liK(z)(x) = E x(a)z(a) for all x e C aeK 149, then, as in earlier proofs (Proof 1.2.2 (1), Proof 4.1(1), (4) u - Yp o \ • Hence, for any finite Kc A s.t. a e K => y(a) ^ 0 , (y a I) K(((CK)* - t(V rv £K)°) £K . P = y „((£K)A ~ tT*""1^. r\ CK)°) by Lemma 0.4.2, and the fact £K <EK P that £K = T[£K] ; = yh{z e £K : Z |^z(a)|q > 1}) P aeK >_ e ^ - exp - C E b(|y(a)|/t) by the Lemma. aeK Now, £ b(|y(a)|/t) = °° for any t > 0 , aeA Therefore there exists finite JcAs.t. aeJ=> y(a) =j= 0, and e"1 - exp - C E b(|y(a)|/t) > 6 . ae J Hence, (y G T) T((£J)A - t(V A CJ)°) > 6 . • £J P Since t > sup |y(a)[ , and 0 < <5 < e ^ were arbitrary, aeA and with the notation of Proofs 111,4, rj(tVp°) = t(VpA £J)° , it follows from Lemma 1.5.1.2 that y • T is not E-tight. 4.3 Example There exists a Banach space which is not a C M-space. Proof. Let and CQ = {x e € : lira x(n) = 0} , II | . . , + * : x e c„ -> sup x(n) e R , £ = £ (CO) 2 T : £ CQ be the canonical imbedding As is well known, c^ is a Banach space under the topology generated by the norm | "| From Pietsch [30] p. 83, Remark 2.2, (1) T is not S^-continuous. From Kwapien [19] we have that 2 u e OI(CQ) => y D T has a limit measure on (£ )' , and therefore, by Cor. 1.4.3, u e CM(cQ) => y Q T is E-tight. Hence, by Prop. III.1.3, (2) T is S"*"-continuous with respect to the CM(cQ)-topology on cQ . From (1) and (2) it follows that the CM(CQ)-topology does not coincide with the norm topology, i.e. CQ is not a CM-space. 4.4 Example r 2 For any r >_ 4 , the S -topology on £ (w) does not 2 coincide with the S -topology. 2 2 oof We shall construct a T : £ (to) -> £ («#) which, will be r 2 S -continuous but not S -continuous, from which it follows r 2 that the S -and S -topologies do not coincide. For each new , let n {n} -2/r a = n , n 2 r T : x e £ (w) -> (a x ) e £ (w) . n n new As in the proof of Cor. III.4.4.1, we conclude that (1) T is S -continuous. 2 If T were also S -continuous, then, there would exist a * 2 * w -Radon measure n on (£ (w)) with supp n e E , s.t. (/|f(x)|2dn(x))1/2 < 1 => |Tx|r < 1 . Hence, lTxlr 1 /|f(x)|2dn for all x e £2(w) . Consequently, for any k e w I a2= E |TeJ2 </ E | f (e^ | 2dn (f) n<k n<k n<k <_ f ( sup |f(x)|)2dn(f) , |x|2ll since {e } is a orthonormal basis of the Hilbert space n new £ (w) . Since supp n e E , and k e w was arbitrary, it follows that v 2 L a < 00 n new 152, However, this is impossible, since 2 -4/r ^ -1 , a = n > n , and Y, 1/n n — new Hence m • c2 . T is not o -continuous, Remark. In view of Theorem III. 2.6.3, from the above example we see that for every r >_ 4 , I (to) is not a CM-space. 153. BIBLIOGRAPHY 1. A. Badrikian: Remarques sur les theoremes de Bochner et P. Levy. Symp. on Prob. Methods in Anal. 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Yu. Prohorov: Convergence of random processes ... Theory of Prob. 1 (1956) p. 157-214 (SIAM, Engl, transl.) 33. : The method of characteristic functionals. Proc. 4th. Berk. Symp. Math. Stats, and Prob. 2(1960) p. 403-419. 155. 34. W. Rudin: Fourier analysis on groups. Interscience (1962). 35. V. Sazonov: A remark on Characteristic functionals. Theory of Prob. 3 (1958) p. 188-192 (SIAM, Engl, transl.). 36. H. Schaeffer: Topological vector spaces. Macmillan (1966). 37. C. Scheffer: Sur 1'existence de la lirnite projective ... C.R. Acad. Sci. Paris, A. 269 (28 July 1969) p. 205-207. 38. I.J. Schoenberg: Metric spaces and positive-definite functions. Trans. AMS 44 (1938) p. 522-536. 39. L. Schwartz: Extension du theoreme de Sazonov-Minlos. C.R. Acad. Sci. Paris, A. 265 (18 Dec. 1967) p. 832-834. 40. : Reciproque du theoreme de Sazonov-Minlos. C.R. Acad. Sci. Paris, A. 266 (3 Jan. 1968) p. 7-9. 41. : Demonstration de deux lemmes ... C.R. Acad. Sci. Paris, A. 266 (8 Jan. 1968) p. 50-52. 42. 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