MEASURES OH BANACH SPACES ' lNVERSlON- FORMULAE FOR HE PROBABILITY Thesis for the Degree of Ph. D‘ MIC'iGAN STATE UNIVEHSWY GH‘OLAMHOSSEIN GHARAGOZ HAMED‘ANE 41 7 9 all A __ gig? £3 5%.“?2 A J I. I B P A I? Y Mi’iligm {State Umversxty |' -. .7 u . C n - Int”' This is to certify that the thesis entitled INVERSION FORMUIAE FOR THE PROBABILITY MEASURES ON BANACH SPACES presented by Gholamhosse in Gharagoz Hamedani has been accepted towards fulfillment of the requirements for Ph.D. Statistics and Probability Major professor degree in Date May 72 1971 0-7639 é 70$ /5 the fun 0n Nil inv. WOr' Mai] 18} whh ABSTRACT INVERSION FORMULAE FOR THE PROBABILITY MEASURES ON BANACH SPACES By Gholamhossein Gharagoz Hamedani Let B be a real separable Banach space, and let u be a probability measure on .BKB), the Borel sets of B. The char- acteristic functional (Fourier transform) m of u defined by My) = j‘Bexp{i(y.x)ldu(x) for y e 13* (the topological dual of B) uniquely determines p. In order to determine p, on 6(3), it suffices to obtain the value of IBG(s)du(s) for every rea1.valued bounded continuous function G on B. Hence an inversion formula for u in terms of ¢ is obtained if one can uniquely determine the value of IBG(s)dp(s) for all real valued bounded continuous functions G on B in terms of m and G. The main efforts of this thesis will be to prove such inversion formulae of various types. For the Orlicz space Ea of real sequences we establish inversion formulae (Main Theorem II) which properly generalize the work of L. Gross and derive as a Corollary the extension of the Main Theorem of L. Gross to Ed Spaces (Corollary 2.2.12). In Chapter One we prove a Theorem (Main Theorem I) which is Banach space generalization of the Main Theorem of L. Gross which differs from the Main Theorem 11 in the sense that the class of pi have Theo: and Main using Theor of probability measures for which inversion formulae hold is smaller than that of the Main Theorem II. Finally in Chapter Three we assume our Banach space to have a shrinking Schander basis to prove inversion formulae (Main Theorem III) which express the measure directly in terms of m and G without the use of extension of m as required in the Main Theorems I and II. Furthermore this is achieved without using Lévy Continuity Theorem and hope that one can use this Theorem to obtain a direct proof for the Iévy Continuity Theorem. INVERSION FORMULAE FOR THE PROBABILITY MEASURES ON BANACH SPACES BY Gholamhossein Charagoz Hamedani A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF HII IDSOPHY Department of Statistics and Probability 1971 TO MY PARENTS ii advi diss and effc reac for of t bil: COnt no. Mic} ACKNOWIEDGEMENTS I would like to thank Professor V.S. Mandrekar, my thesis advisor, for his patient guidance during preparation of this dissertation. His comments and suggestions led to way to theorems and simplified proofs. Above all this, I deeply appreciate his efforts in teaching me many aspects of probability theory. I also wish to thank Professor H. Salehi for his careful reading of the thesis. Special thanks are due to Mrs. Noralee Barnes for her excellent typing and cheerful attitude in the preparation of the manuscript. I am grateful to the Department of Statistics and Proba- bility, Michigan State University, the National Science Foundation contract no. GP-23480 and the National Science Foundation contract no. GP-11626 (Summer 1970) for partial support during my stay at Michigan State University. iii TABIE OF CONTENTS Chapter Page 0 mTRODumION .00...0.000....COCOCOCOOOI00.0.0.0... 1 I INVERSION FORMULAE OF THE CHARACTERISTIC FUNCTIONAL OF A PROBABILITY MEASURE 0N BANACH SPAmS WITH A SWER BASIS . O O O C O O O O O O O O O O O O O O O O 8 1.0 Introduction 0.0ICOCOCCCCOCOCOCOOOOOOOOOOO... 8 1.1 BaSic Definitions OCOOOOCCOOCCOO0.0.0.0000... 9 1.2 Measures on Banach spaces with a Schauder baSis 00.00....OCOOOCOOOOOOCOOOOOOOO00.0.0... 15 1.3 Extensions of characteristic functional ..... 22 1 .4 General invers ion fem lae O O O O O O O O C O O O O O O O O O 24 II OPERATOR THEORETIC CONDITIONS FOR THE INVERSION FORMULAE ON F-SPACES POSSESSING A SCI-IAUDER BASIS AND A QUASI-NORM WHICH IS ACCESSIBLE IN BOTH DIREflIONS OOOOOOOOOOOOOOOOCOOOOO00.0.0000... 34 2.0 IntrOdUCtion ococo-000000000000.ooooooooooooo 34 2.1 Preliminaries and Definitions ............... 35 2.2 Associated Hilbert space .................... 39 2.3 Inversion formulae for Orlicz space of real sequences ................................... 44 III INVERSION FORMULAE OF THE CHARACTERISTIC FUNCTIONAL OF A ROBABILITY MEASURE ON BANACH SPACES WITH A SHRINKING SCHAUDER BASIS ........... 53 3.0 IntrOdUCtion oooooooooooooooooooooooooooooooo 53 3.1 Preliminaries and Definitions ............... 54 3.2 Main.The0rem III loo-000000000000000000000000 62 REFERENCES O0..OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 70 iv {b get tré <90 Spa the fun of on be t Bans fOnn a fir and a NEre abSOL 0. INTRODUCTION Let (3,“ouB) be a real Banach space with Schauder basis {bn}° Let 6(B) denote the Borel sets of B, that is, the a-field generated by the open sets. The characteristic functional (Fourier transform) q) of a probability measure u on 8(3) defined by (p0!) = j‘Bexp {i(y,x)}dp.(x) for y e 3* (the topological dual Space of B) uniquely determines p. In order to determine u on 18(B), it suffices to obtain the value of IBG(s)du(s) for every real valued bounded continuous function G on B. Hence an inversion formula for u in terms of m is obtained if one can uniquely determine the value of IBG(s)dp(s) for all real valued bounded continuous functions G on B in terms of m and G. The main effort of this Thesis will be to prove such inversion formulae of various type for different Banach spaces B. The Main Theorems 1, II, III give inversion formulae which express JBG(s)dp(s) in terms of m and G. In order to motivate these formulae let us consider first a finite dimensional space Rk’ a probability measure u on RR and a real valued bounded continuous function G on Rk. If G were not only bounded but also in L](Rk,dx) and if u were absolutely continuous with respect to Lebesgue measure with an L2 derivative then putting (( (o In ne ex 8X] ob! put The UPC] dis: whe: with ing diff in t1 (0.1) as) -- (211)-k/ 2 ijcexp[-idu = (2n>""2 kacpcymowy . In the absence of the validity of the two assumptions made above neither of the two right hand integrals in (0.1) and (0.2) need exist. However, upon inserting the convergence factors exp[-“xH2/2a2] and exp[-“yH2/2t2] in these two integrals one may obtain an inversion formula in the following well known manner. Assuming merely that G is real valued, bounded, continuous we put Ga(y) = (211)”k/2 IR G(x)exp[-“xH2/Zaz]exp[-i(x,y)]dx . k Then it can easily be shown that for any probability measure p -k/2 a 2 2 In G(8)du(8) = 1m (Zn) in cp(y)G (y)exP[-\\yH /2t jdy - a k cut-coo k Upon rewriting these equations in terms of the canonical normal distribution (cf., Remark 1.1.8) on Rk (with variance parameter one) we obtain by a change of variables k k IR G<8)dp(8) = lim. 0 t EyEx{¢(ty)G(ax)exp[-ita(x,y)]} k a,t-co where Ex and By denote expectation in the indicated variable with respect to the canonical normal distribution. Before extend- ing this formula to a real separable Hilbert space H three difficulties must be overcome in its formulation. First of all in the limit as k a m, (at)k approaches either zero, one or m. us ex Fu eq 3P va f0: (O. the Fun See: and Secondly the expectation with respect to the canonical normal dis- tribution on H will not make sense for a general bounded continuous function G ([8], Theorem 1) and thirdly even if C were, for example, uniformly T-continuous (cf., Definition 1.1.11) so that EX{G(ax)exp[ita(x,y)]} makes sense the resulting function of y need not define a measurable function with respect to the canonical normal distribution so that Ey{...} will not be defined. These problems were first pointed out and handled by L. Cross in [7] by modifying the finite dimensional formula as follows. Instead of using the convergence factor exp[-Hx“2/2a2] one uses exp[-“A-1x“2/2a2] where A is an invertible operator on Rk' Furthermore, let us observe that (ta)k‘det Al is asymptotically equal to [det (I + (ta)2AAI)]% in the sense that their ratio approaches one as t and a 4.”. After an obvious change of variables one can then obtain the following correct inversion formula in RR (0.3) kac(s)du(s) = lim [det(I + (ta)2AA*)]§£EyEx{cp(ty)G(an) t,” expE-ita(AX.y)]}- In formulating this equation in a Hilbert space one notes that the determinant factor makes sense provided AA* is trace class operator, that is, provided A is a Hilbert-Schmidt Operator. Furthermore, if A is a Hilbert-Schmidt operator then it may be seen in view of Theorem 1 and Corollary 3.2 of [7] that the second and third difficulties mentioned above also vanish, since EX{G(an)exp[-ita(Ax,y)} may also be written as tt ca of is Do. def prc (cf on cam fum I + cano IH exp[-ita(x,y)]-G(ogA)dnoAm1 where n is the canonical normal distribution on H. Since noA-1 is a measure on H ([7], Corollary 3.2) the last integral not only makes sense for an arbitrary bounded continuous function G but also defines a uniformly T-continuous function of y ([7], Theorem 1). A fourth difficulty now arises. An estimate of the dif- ference between the left hand side of (0.3) and the expression under the limit sign in (0.3) shows that although this difference goes to zero for each dimension k as t and a go to 9 independently, the rate at which the difference goes to zero de- pends more and more critically as the dimension k gets larger on the relative manner in which a and t go to a. In the limiting case of a Hilbert space it results that the relative growth rates of a and t must be restricted. In the Theorem below which is a restatement of Theorem 4 of [7], this is effected by putting a = f(t) as is done in [7] following earlier work of Cameron and Donsker mentioned there. 0.4. Theorem. Let A be a Hilbert-Schmidt operator with dense range on a real separable Hilbert space H. Let p be a probability measure on H and f(t) a positive admissible function (cf., Definition 1.4.2) on (0,m). let h(t) be a positive function on (0,») and denote by v the measure noA"1 where n is the canonical normal distribution on H. Let m be the characteristic function of p and denote by Ct the positive square root of I +-t2f(t)2AA*. Let En denote expectation with respect to the canonical normal distribution. In order for the inversion formula (0.5) AH G(s)dp(s) = :im‘h(t)(det Ct)En{m(ty)"(IHG(f(t)x) —cco exp[-1tf(t) (x .mdv (x) )“l to hold for all real valued bounded continuous functions G the following two conditions are necessary and sufficient (0.6) f(t)2 trace (c;2 AA*) —. o as c -. an (0.7) The measures h(t)exp[-t2ucglsH2/2]dp(s) converge weakly to p. as t-0m. Furthermore if (0.6) and (0.7) hold then (0.3) also holds for any bounded measurable function G which is strongly continuous almost everywhere with respect to p. The condition (0.6) of L. Gross although valid for Hilbert space seems to depend heavily on the symmetry structure of the space. We re-interpret this condition for a general Banach space in terms of convergence of certain Gaussian measures (cf., Lemma 2.3.4). In terms of this re-interpretation the Theorem can then be extended to a Banach space with Schauder basis as follows. Using the fact that B has a Schauder basis, we can, following ideas of J. Kuelbs [12], imbed B measurably in a real separable Hilbert space HA, whose norm is weaker than the Banach norm H “B. We then treat the probability measure u on B as a probability measure on H . This enables us to get the necessary and sufficient A conditions for the inversion of u regarded as a measure on H A using essentially ideas of 1" Gross [7]. However this method allows one to obtain such a formula only for G bounded and con- tinuous on HA’ which is a proper subclass of the G's required. TC 01 E) ir ba 0% Or of [1: in for The ass to! inve main Spac To circumvent the problem we have to use essentially the notion of x-family introduced by J. Kuelbs and V. Mandrekar [13], which exhibits the detailed structure of the probability measure H on HA which is actually supported on B. In Chapter I, such an inversion formula is obtained for any Banach Space with Schatder basis. Our initial objective in Chapter II is to prove a Theorem (Main Theorem II) which generalizes Theorem 4 of L. Cross [7] in his form. For this purpose we will need to restrict ourselves to Orlicz Space Ea of real sequences since in this case the form of characteristic functional of a Gaussian measure is known (See, [13]). The Main Theorem II is stronger than the Main Theorem I, in the sense that, in case of Ba Spaces, the class of measures for which the inversion formulae can be obtained from the Main Theorem II is larger than that of the Main Theorem I. We further assume that the function a(o) associated with Ea possesses a particular prOperty relative to one-dimensional Gaussian measures to get Corollary 2.2.12 which gives us analytic condition for the inversion formulae and also gives precise generalization of the main inversion formulae of L. Gross ([7], Theorem 4) to Orlicz Spaces of real sequences. Finally in the third Chapter we let (B,H-HB) be a real Banach space with shrinking Schauder basis {bu}. Since {bn} is shrinking, the coordinate functionals on B form a basis for * * B , and hence we may consider B as a Borel measurable subset of L, the vector Space of all sequences of real numbers with topology of coordinatewise convergence. Also we Shall let n be the canonical normal distribution on HA so that for each x 6 HA’ n(x) is a random variable on B*, and let Pk be the countably additive (will be shown) cylinder set measure on 3* induced by the above family. Then we shall prove a Theorem (Main Theorem III) which gives a class of inversion formulae different from that of the Main Theorem I. In the Main Theorem I we have extension of characteristic functional to L whereas in the Main Theorem III we have extension of characteristic functional to B*. Hence (3.2.2)(a)t is stronger than (1.4.4)(a) Since for A 6 L1; PA is countably additive on B*. Furthermore since {bn} is shrinking we are able to give a proof for the Theorem without using levy Continuity Theorem and hope that one might be able to use this Theorem to obtain a proof for the Lévy Continuity Theorem. CI‘IAPI‘ERI. INVERSION FORMULAE OF THE CHARACTERISTIC FUNCTIONAL OF A PROBABILITY MEASURE ON BANACH SPACES WITH A SCHAUDER BASIS §l.0. Introduction. Let (B’H’HB) be a real Banach Space with a Schauder basis denoted by {bu}. Let B(B) be the a-field generated by the open subsets of B. Every element of 6(3) will be called a Borel set. In order to determine a probability measure .3 on B(B), it suffices to obtain the value of IBG(s)dp,(s) for every real valued bounded continuous function G on B. Hence an inversion formula for p. in terms of its characteristic functional, (p is obtained if one can determine uniquely the value of IBG(s)dp.(s) for all real valued bounded continuous functions G on B in terms of q) and G. The main effort of this Chapter will be to prove such inversion formlae for p. on 6(B). Following [12], we shall first define a particular inner product on B which generates a norm weaker than the Banach norm ““3. Upon completing B with respect to this norm we will obtain a real separable Hilbert Space H with the prescribed inner product. A Since “xHB is measurable with respect to the norm “.“l on H)‘, it follows that 6(B) is contained in the Borel subsets of H)‘ which we denote by 6(Hx). Thus any probability measure p. on (B,B(B)) induces a probability measure on (HX,B(H>‘)) by defining subsets of HA4; to be of u-measure zero. Now if p. is a probability measure on (B,/3(B)) with characteristic functional, m, then u can be defined to be a probability treasure on (3)35me with characteristic functional, ¢('). Note that ¢(°) is actually the restriction of m to H:. 1" Gross describes various inversion formulae for t(-) ([7], §4). We will use Gross' result ([7], Theorem 4), the notion of "x-family" and the idea of "stodhastic linear functional" first occurring in [13] and [14], to prove a class of inversion formulae for m. We start by introducing some preliminaries required in the remainder of this Chapter. §l.1. Basic Definitions. In this Section we present for the sake of completeness some Standard concepts and definitions. For further details the reader is referred to [1] and [16]. 1.1.1. Definition (a). Let S be a complete separable metric Space and let. W2 be the Space of positive finite measures defined on the a-field generated by the open subsets of S. A sequence pn of measures in. W; is said to converge weakly to a measure u in. WI if Isfdun a Isfd“ for every bounded continuous function f on S. We will denote this convergence by “'n g u. If {pt : t E (0,m)] is a family of measures in, ”b then we say pt 2.“ as t a.m, if for any sequence {tn} approaching infinity, 118 (b) A sequence pn of measures in.'Wz is said to be conditionally compact (tight), if for every 6 > 0 there exists a compact set 6 K in S such that pn(Ke) > 1 - e for all n. 10 1.1.2. Definition. Let (0,33) beaprobability Space and let x be a random variable on 0 taking values in S. Then x is said to be distributed as v if v = Pox-1. A family of S-valued random variables {Xt : t 6 (0,ao)] is said to converge in distribution to an S-valued random variable X as t‘n a if Poxgl 3 Fox.1 as t .-. on. We will denote this convergence by Xt ‘90 X as t _. an. The following definitions are due to I. Segal and are taken here from [7]. 1.1.3. Definition. A weak distribution on a topological linear Space L is an equivalence class of linear mappings F from the (topological) dual space L? to real-valued random variables on a probability space (depending on P) where two such mappings F1 and F2 are equivalent if for every finite set of vectors yl,...,yk in L? the sets {Fi(y1),...,F1(yk)} have the same distribution in k Space for i B l or 2. Here L* denotes the Space of continuous linear functionals on L. In a finite dimensional Space a weak distribution coincides with the notion of a measure, that is, if L is finite dimensional then for any given weak distribution there exists a unique Borel probability measure on L such that the identity map on If is a representative of the given weak distribution ([9], p. 372). 1.1.4. Definition (a). A weak distribution m on a Banach Space B is said to be continuous if for any sequence {yk} : B*, l “yku l'* 0 implies m(yk) converges to zero in probability. B 1) For y e 3*, My“ * - Sup \y(x)| (See, e.g. [18], p. 160). B “x“BSl -.—-‘ tc 0n tr: is in V6);- can depé 11 (b) A weak distribution m on a topological linear space L is a measure if there exists a probability measure p defined on the o-field S generated by weakly open Subsets of L such that the identity map on L? is a representative of m. 1.1.5. Definition. If m is a weak distribution on a locally convex topological linear space L and A is a continuous linear operator on L with adjoint A*, then the weak distribution y a m(A*y) will be denoted by moA-l. 1.1.6. Definition. A measure u on a locally convex topological linear Space L is defined to be Gaussian if, for every continuous linear functional T on I” T(x) has a Gaussian dis- tribution. u is called Gaussian with mean zero if, in addition T(x) has mean zero for every T. 1.1.7. Definition. The characteristic functional (Fourier transform) of a probability measure H on the Borel subsets of a * linear topological Space L is the function m(-) on L (the topological dual of L) given by (P0!) = jL €XP{i(y,X)}du-(X). for each y e L* . 1.1.8. Remark. One Special example of a weak distribution on a real separable Hilbert Space H is the canonical normal dis- tribution (with variance parameter one). This weak distribution is that unique weak distribution which assigns to each vector y in H* a normally distributed random‘variable with mean zero and variance Hynz. It follows from the preceding property that the canonical normal distribution carries orthogonal vectors into in- dependent random variables ([7], p. 4). It is known that some of 12 the theory of integration with reSpect to a measure can also be carried out with reSpect to a weak distribution. For details we refer the reader to [9] and the bibliography given there. We shall also need the following definition from ([20], p. 190). 1.1.9. Definition. An operator from a real separable Hilbert space H into H, which is, linear, symmetric, nonnegative, compact, and having finite trace is called an S-operator. If T is an S-Operator on H, then it is well known that T has the representation a (1.1.10) T): = nil >.n(x.¢=:n)en where {en} is some orthonormal Subset of H, An 2 0, and Q E A < ”- n=1 n The S-operator T on {,2 has a representation as an infinite symmetric, nonnegative-definite matrix T - {tij} where by non- negative-definite it is meant that 2 t x x i,k=1 ik i k integer n and any (x1,...,xn) E Rn' Furthermore, tik = (Tfi’fk) 2 0 for any where f is the vector in L2 of all zeros except one in the th 1 on .. 1 position and hence z t = 2 l .< a where 1 's are as in (1.1.10). From the representation in (1.1.10) it is easy to verify that ('I'cx,cx))5 - |c‘(Tx,x)% for any real number c and 55 is 3! (T(x+y),x+y) s (TX,x) + (Ty,y) . Thus (Tx,x)% is a semi-norm on L2. Let 2 be the class of all S-operators. 1.1.11. Definition of T-topology, The r-topology on L2 is the smallest locally convex tepology generated by the family of semi-norms pT(x) B (Tx,x)!5 on L2 as T varies through E ([18], p. 172). tit fa 0? th a1 tiOr Schm is a 0n! tribe fOr t Spat 13 1.1.12. Definition. Let H , H be Hilbert spaces with l 2 orthonormal Systems {en}, {fn} reSpectively. Then a continuous linear operator A from H1 into H is called Hilbert-Schmidt 2 operator if there exists an orthonormal system {gn} in. H1 such that E1HAgnHIiz is finite ([5], p. 34). n 1.1.13. Remarks. (a) Let H be a real separable Hilbert space, then H is isomorphic to' L2. If T is an S-operator on H then T possesses a unique nonnegative, symmetric square root, which we denote by T35 ([17], Theorem, p. 265). Now using the fact (See; [5], Theorem4, p. 39) that the square roots of S- operators are Hilbert-Schmidt operators one can easily Show that the topology 7 on H is the weakest topology on H for which all Hilbert-Schmidt Operators are continuous from T to strong topology on H. Thus a basic open neighborhood of x is 0 {x : “A(x-xo)“ < 6} whenever A is a‘Hilbert-Schmidt operator on H. Therefore our definition of T-topology coincides with that of L. Cross ([7], p. 5). (b) By Corollary 3.2 of [7], if m is a continuous weak distribu- tion on a real separable Hilbert Space H and A is a Hilbert- Schmidt operator, then moA-1 is a measure on H. Hence noA- is a measure on H if n is the canonical normal distribution on H, since in this case n is clearly a continuous weak dis- tribution on H. We will use the same notation, namely, noA-1 for the weak distribution noA"1 and its correSponding probability measure u (See; Definition 1.1.4). 1.1.14. Definition. A tame function on a real Hilbert Space H is a function of the form fo) = §(Px) where P is a l4 finite dimensional projection on H and Q is a Baire function on the finite dimensional space PH. For such a function we have [f(x) 8 ¢((x,x1),...,(x,xk)) where x1,...,xk is a basis of PH and V is a Baire function of k real variables. If F is a representative of a weak dis- tribution then the random variable f~ = ¢(F(x1),...,F(xk)) depends only on the function f and the mapping F while integration pro- perties of f~ such as the integral of f", the distribution of f",-convergence in probability of sequences f;, etc. depend only on f and the fn's and on the weak distribution of which F is a representative. Let us denote by 3 the directed set of finite dimensional projections on H directed under inclusion of the ranges. For a given continuous function f on H and a given weak distribution one may consider whether the net (foP)~ of the random variables where P ranges over directed set 3, converges in probability as P approaches the identity through 3. If so then the limit which we Shall denote by f~ is a random variable whose integration properties are completely determined by the function f and the weak distribution. In [8] and [9] classes of continuous functions are described for which the limit defining the random variable f~ exists when the weak distribution in question is the canonical normal distribution, and some explicit evaluations are also given. The part of this integration required for our purpose is given below and is directly taken from ([9], p. 374). It is clear that a function f on H is a tame function if and only if there is a finite dimensional projection P on H n1 f1 t< §1 We St; 15 such that f(x) = f(Px) for all x and such that f restricted to the finite dimensional Space PH is a Baire function. Then f is said to be based on PH. If f is based on PH then it is clearly also based on. QH whenever Q 2 P and Q is a projection. If f is based on the finite dimensional subSpace PH then we note that its expectation with reSpect to the canonical normal dis— tribution (with variance parameter one) is given by E(f ) = (2n) Im£(x)exp[-\\x\\ /2]dx when the integral exists where k is the dimension of PH and dx is Lebesgue measure on PH. 1.1.15. Remark. Let H be a real separable Hilbert Space, and let {P1} be any sequence of finite dimensional projections converging strongly to the identity operator. If a complexdvalued function f on H is uniformly continuous in the tapology T then lim in prob. (foP J 340 normal distribution and equals f~ ([7], Theorem, p. 5). Now if )" exists with reSpect to the canonical for each j, E[(fon)”] exists and if lim E[(fon)"] exists then following L. Cross [8], we say that f lSmintegrable with respect to the canonical normal distribution. That is, for a uniformly T-continuous function f on H which is integrable we denote by En(f") = ]i: E[(fon)"] . §1.2. Measures onéganach Spaces with a Schauder basis. We shall study here Banach Spaces with Schauder basis. We need some preliminary results for measures on such Spaces. We start with the following definition from ([3], p. 67). 16 1.2.1. Definition. Let B be a Banach Space. A Schauder basis {b1} in B is a sequence of elements of B such that for each x in B there is a unique sequence of real numbers {a1}, depending on x, such that n le Hx - 2 a b,“ = 0 ; i=1 i 1 B n—aoo (D the series 2 aib1 is called the expansion of x in the basis i=1 {bi}, and the coefficient ai = ei(x) is the 1th coordinate of x in the basis {b.}. 1 Throughout this Chapter B will denote a real Banach space with Sdhauder basis {bu} such that without loss of generality “anB = l ([3], p. 68). We will write the expansion of x as Q 2 Bn(x)b‘n and this emphasizes that the coefficients generate n-l coordinate functionals on B. It is clear that these coordinate functionals are linear and it is well known that they are continuous as well ([3], p. 68). Further it is possible to assume without loss of generality ([3], Theorem 1, p. 67) that k (1.2.2) “an = Sup H z an(x)anB . k n=1 Following ideas in [12], we introduce a Hilbert Space associated with B. For A 6 L1, and x,y E B define a (1.2.3) (x,y) = 2 ann(x)Sn(Y) (convergence follows from n=l page 17, line 2) where Bn's are coordinate functionals. Then (,) is an inner product on B and B is a pre-Hilbert space with the norm M, = (mi and 17 2 2 2 \lek S HAHI 5:? lenool = Cl 8:? lanml . We know ([3], p. 68) that Sup ‘B:(x)‘ s Czuxug, and hence n unisqgnu§=chfi- This implies that the tepology on B induced by “.“A is weaker than the norm topology on B. Let HA denote the completion of B under H “A. Then clearly B C HA. Upon replacing y by bk (x,bk) = xk5k(x). Since xk > 0, Bk(x) is uniformly continuous in (1.2.3) we get in x in “'Hk-topology on B, and since B is dense in HA’ Bk(°) can be extended uniquely to a continuous linear functional §k(-) on HA. Furthermore it can easily be seen that for x,y E H , A (1.2.4) (x,y) = 21 An ans) Sum . n: From (1.2.2) and the fact that an is a “-“x-continuous function on B, it follows that “XHB is a measurable function in “-“x-topology, and hence B is a “on-measurable Subset of HA. Therefore if u is a measure on B, it can be regarded as a measure on HA via p,(A) 8 “(A n B) for all A 6 5(3)). Let v be a Gaussian measure with mean zero on (BgBCB)). Then by argument similar to Lemma 2.2.of [12], v is a Gaussian measure on H , and therefore there exists a nonnegative, symmetric A trace class operator (that is, an S-operator) Tv on HA such that (whenA = fo2dv(y) for x 6 HA’ and that v is uniquely determined on H by the A operator Tv’ These results are well known and appear, for example, 18 in [20]. Furthermore, Tv has the representation (1.1.10), that is, a (1.2.5) Tv(°) = 1.21 1115381931, on HA where {8k} is an orthonormal sequence in H and A le 2 o, E “k < a. 1.2.6. Remarks. (a) Since B is separable, 5(B) is the Same as o-field generated by the weakly open sets and the latter one is the same as o-field generated by the field of the cylinder sets . is v is a Hilbert-Schmidt operator (b) Since Tv is an S-operator, T 35 on HA, and hence noT; is a measure on H . But Tv uniquely A determines v, so by definition 1.1.4, v is the probability measure ’5 on HA corresponding to the weak distribution noT; . (c) Since cylinder sets in HA and B are the same, from (a) is and (b) above, it follows that noT; is countably additive on (B ,B(B)) and noTj’ (d) We note that (c) could also be obtained from the fact that (B) - 1. the Borel subsets of B are also Borel subsets of HA (because “XHB is “-“x-measurable) and therefore every countably additive measure on HA is countably additive on B. Following ideas are motivated by [13]. 1.2.7. Definition. If A 6.Lm and [pt : t E A] is a family of probability measures on B such that 2 ut{x E B : “E1 Aan(x) < a] a l for each t G A we say A is sufficient for the family [pt : t E A]. 19 Now for each x 6 B we have Sup ‘Bn(x)‘ < a, thus it follows that any A 6 L1. is sufficientnfor any family of proba- bility measure on B. 1.2.8. Definition. A family of probability measures {pt : t E A} on B is a x-family for some A 6 g:' if A is sufficient for {ut : t E A] and for every 6,6 > 0 there is a sequence {6N} such that a) (1th: 6 B : E Anfiibc) < 6} > l-e n=N+1 implies ut{x e B : “n=§+15n(x)bn“n < k(6)} > 1-(e+,N) for all t where lim 6N = 0 and k is a strictly increasing continuous functionyon [0,m) with k(0) = 0. A family of probability measures {pt : t E (0,m)] on B is said to be a x-family as t a m, if for any sequence {tn} approaching infinity the family {pt : n = 1,2,...] .is a x-family. It is quite Clea: that any family of probability measures on a real separable Hilbert Space is a x-family with A = (1,1,...) and k(6) - 5%. For x E B, N = 1,2,... we define N P x . 2 B (x)b N k=1 k k Qx= z s(x)b N k=N+1k 1‘ * and for y E B , N = 1,2,... we define 20 N PNY(°) B 8 Bk(°)Y(bk)° k=l 1.2.9. Definition. If {pt : t E A} is a family of probability measures on B and x E 1,: then we say the family * of c.f.'s {(pp‘ (.); t E A} is x-continuous at zero in B if: t (i) for every integer N the family [qu (-): t E A} are equi- t * continuous at zero in PN (B ), and (ii) lim Sup lim J [1 - Re (p (-)] = 0 where N t k “’1‘ “'t JN,k[...] = f(PN-l-k-PNm" [...]§x(N,k.dy) * (Pu-H: ' Pu)B with each coordinate y(bi), N+l s i S N+k, Gaussian with mean and §X(N,k,-) is the Gaussian product measure on zero and variance xi' 1.2.10. m. Let (1) x e L: . (ii) {pt : t 6 (0,00) be a x-family as t -+ an of probability measures on B , (iii) pt 3 p. on HA as t -+ on where p. is a probability measure on B. Then ptgp, on B as t—oao. M. let {tn} be a sequence in (0,012) such that tn .. an as n .. on. Then (iii) implies that pt 3 p. on HA, that is, {utn: n - 1,2,...} is compact on HA. SinZe {pt : n - 1,2,...} is compact on HA and Ph is continuous it follows tfiat {pt oPg1 : n = 1,2,...} is compact on HA for all N 8 1,2,..., which is equivalent to saying {pt OBI;1 : n - 1,2,...} is compact n on B for all N = 1,2,... . Thus the c.f.'s {:p 0P-1(-):n = 1,2,...} ”tn N 21 * are equicontinuous at zero on Ph(B ) ([15], Corollary 2, p. 193) and since cp 0P_1(X) = (PH: (PNx) ”’t n n n * the equicontinuity at zero on Pk(B ) of [mh (.) : n = 1,2,...} tn follows. Hence condition (1) in Definition 1.2.9 is satisfied. Let A 6 LI. and let S >»0 be given and choose a compact set R6 in HA such that “t (Re) > 1 - e/2 for n = 1,2,... . Since n m (-) is c.f. of pt and §x(N,k,-) is symmetric about zero I"‘t n n it follows that N+k 2 1 - = 1 - - O t n Since pt (B) = 1 and Bi = B1 on B, it follows that n N+k .2 J [1 - Re <9 (3')] [1 - exe{-’5 2‘. A B (x)}]dp. (X)- N’k “'tn In), i-N+1 i 1 ‘:n Since 1 - eC s l for g 2 0 we have N+k .2 S [1 - exp -’5 z x a (x) d0 (x) + e/2. Ike { i=N+1 1 1 l] tn We note that Sup Sup |§1(x)‘ < M Since Sup |§1(x)\ s Cux“ e i i A x€K ([3], p. 68) and “x“x is a continuous function. Hence 2 m M JN,k[1 - Re a; (-)] s 1 - exp{- 5" 2 A1} + 6/2 t i¢N+l n for all n = 1,2,..., and letting N approach infinity and using the fact that A 6 LI. we have the right hand term dominated by 6/2. Thus condition (ii) of Definition 1.2.9 is Satisfied, and 22 hence {H1 (-); n = 1,2,...] is A-continuous for A E L+' This 1. together wlgh the assumption that [pt : t E (0,m)} is a x-family as t a m imply ([13], Lemma 3.2, p. 11) that [pt : n = 1,2,...} is conditionally compact on B. Hence pt g u on B Since [ut : n = 1,2,...} is conditionally compagt on B and “-1w -1 pt oPfi = “DEN for all N = 1,2,... ([1], p. 35). Since this is true for any sequence [tn] with tn a m as n a a we get ut g H on B as t S,m, 1.2.11. Remark. We remark that from the proof of Lemma 1.2.10 one can derive the stronger statement: Let [pt : t E A} be a x-family of probability measures on B. Then ut conditionally compact on HA implies that [mt : t E A} is conditionally compact on B. However, since we shall be needing only the statement in Lemma 1.2.10 for further easy reference we have not stated the Lemma in all its generality. §1.3. Extensions of characteristic functional. Suppose L is the space of all real sequences with the topology of coordinatewise convergence and PA(.) is the product probability measure on L Such that the ith coordinate is Gaussian with mean zero and variance *1 where A = [xi] 6 L: . If u is a probability measure on B then A 6 L1. is sufficient for p and for y 6 L we define the "stochastic linear functional" on B in the following manner: ~ N . N “= lim 2 eiyi = 11m x 610‘”.- N i=l N i=1 The following Lemma is proved in [13], page 22. 23 1.3.1. Lemma. The stochastic linear functional N N A (1-3-2) (y.X)" = 11m )2: 81(103'll N 1-1 is Borel measurable on L x B and if F = [(y,x) : (y,x)z exists and is finite], 1.3.3. Definition. If p is a probability measure on the Borel subsets of B with the c.f. m, we define the extended characteristic functional $z(.) on L as follows f(y) = f}, exp{1(y,x)~}d,(x) (y e z.) ([13], p. 23). Then $§(-) is a Borel measurable function on L which is defined almost everywhere with respect to the measure P Furthermore, A. since each y E B* generates the unique sequence of real numbers {a = [y(b1),...,y(bk),...} we may consider B* as a linear subset of L under J. Kampe de Fériet map ([11], pp. 123-127), and hence the terminology extended c.f. Since for y E B* and x E B, (y,x)” .- (y,x) which implies that cp(y) = (f(yz). 1.3.4. 539355. Let u be a probability measure on B with c.f. m, then as was shown earlier, u can be regarded as a probability measure on H . Let t be the c.f. of H when u A is regarded as a measure on HA, then * * My) = In exp{i(y.x)}du(x) v y 6 H)\ c B c L . A '1: By Theorem 1 of ([7], p. 7), V is uniformly T-continuous on HA’ 24 and hence by Theorem of ([7], p. 5), the random variable t" (that is, the Gross extension of V) is well-defined with respect to the canonical normal distribution n on H1. Finally from (1.3.2), the fact that p.(B) '- 1, and lemma 4.3 of [14], it follows that ¢(y)" - cp~(y) almost everywhere with respect to P1. §1.4. General inversion formlae. Let v be a Gaussian measure on (B,B(B)) with mean zero. Then v(A) - v(A n B), A 6 501)) is a Gaussian measure on (111360113), and there exists a nonnegative, symmetric trace class operator Tv on H1 corresponding to v. Let f(t) be a real valued function defined on (0,00) and denote by Ct the positive square root of I + t2f(t)2Tv on H). (See; 1.1.13 (a)). Let p, be a probability measure on (B ,B(B)), and define 1 2 -1 2 ptm) -= 3- IA h(t)exp[-t “C: IBM/21‘1”“) A 6 6(3) t where h(t) is a positive function on (O,co), and 2 ‘1 2 (1.4.1) at = $3 h(t)exp[-t “Ct s\\x/2]dp,(s) . 1.4.2. Definition. A real valued function f(t) defined on (0,eo) will be called admissible if t f(t) _. so as t _. on. We are now ready to state and prove the Main Theorem 1. 1.4.3. Main Theorem I. let (i) B be a Banach space with Schauder basis {bi}’ (ii) p be a probability measure on (B,B(B)) with the c.f. (p, (iii) f(t) be a positive admissible function on (O,oo), and h(t) be a positive function on (0,»), 25 (iv) X be a B-valued random variable distributed as v where v is a Gaussian measure on CByB(B)) with mean zero and the property that Tv is positive-definite, (v) Yt and Y be B-valued randomrvariables distributed as pt and p respectively, (vi) EP denote the integral with respect to PX on L. 1 Then for all real valued, bounded, “'uB-continuous functions G on B the following are equivalent: r(a) chdu = :3: h(dec ct)EPx{cp”x)exp[-itf (y,x)“Jdvmn l R“; m = h(J‘B eit(y’3) du.(s))(jB c(f(t)x)e'itf(t>(y’x) dv(x))}dPx(y). . as a: We note that elt(y’8) e-itf(t)(y,x) G(f(t)x) is jointly measurable since it is a product of a jointly measurable function of s, x and y with a “-“B-continuous function of x namely G(f(t)x). Since it is bounded and all the measures are probability measures by Fubini's Theorem ([19], p. 140) we have 26 EP {¢~(ty) qB c<£x>expL-1tfexpi- %' 12:1 K13:(3'f(t)X)}dv(x)du(s) a IBIB c(f(t)x)exp[-t2ns-f(t)x":/2]dv(x)du(8)o Now from 1.2.6 (c) and the fact that ”(B) = 1, it follows that 2 2 -5 = IH.IH. G(f(t)x)exp[-t Hs-f(t)xux/2]dnoTv (x)du(s). x x We note that G is H°Hx-measurable (that is, measurable in the “-“x-topology) since the norm “x“B is “-“x-measurable and G is 5 “-“B-continuous; and T is a Hilbert-Schmidt operator on Hl' Hence by Lemma 4.1 of [7] we have 1 -1 -2 - 2 -1 2 I'-' WEEK-[11x G(f(t)Ct X + 8--Ct s)dnoTvk(x)h(t)exp[-t “Ct 3ux/21dH-(8)° Again from 1.2.6 (c) and the fact that ”(B) = 1 we get = Eg—C—t). Mn c(£(t>c;1x + s-c;Zs)dv(x)hexpt-t2\\0;18\\:/21du. 27 We may now start with the assumption that for all real valued, bounded, “'“B-continuous functions G on B we have r(a) IBG(s)dp.(s) = if: IB[IBG(f(t)C;1x + s-C;28)dv(x)]h(t) 4 exp[-t2HC£18“:/2]du(s) (1.4.6) (b) {pt : t E (O,m)} is a x-family as t a'm . L Putting G E 1 in (1.4.6)(a) We get 1 = lim In h(t)exp[-t2“C;13H:/2]dp(s). t-cco From (1.4.1), it follows that (1.4.7) 1 = lim a t-ccn t 0 Using (1.4.7) we obtain (1.4.8) In G(8)du(8) = 1m MB ccglx + (I-c;2>s)dvdut(s) t-m for all real valued, bounded, H-uB-continuous functions G on B. From-(1.4.8), it follows that (1.4.9) vo(£(c)c;1)'1 * ”tea - of)“: u on B as c 4.. . Since G is bounded on B and v is a probability measure, the measure G(f(t)x)dv(x) is a measure of bounded variation on B, and hence a measure of bounded variation on H . Therefore by 1 Theorem 1 of ([7], p. 7) the Fourier transform of G(f(t)x)dv(x) is uniformly T-continuous, and hence the Gross extension of its Fourier transform is well-defined ([7], Theorem, p. 5). Similarly the Gross extension of the Fourier transform (c.f.) ¢(-) of p 28 when regarding p as a measure on (HX“B(H1)) is well-defined. Now from Remarks 1.3.4 and 1.2.6 (c), it follows that (1.4.10) EP {$“(ty>x>exp[-1tf>} 1 k = priw(tY)"(IHk0(f(t)x)eXP[-itf(t)(Y,X)]dnoT; >"} - let {Pj} be a sequence of finite dimensional projectiom on H}. converging strongly (that is, in H-“x-topology) to the identity operator. Then using the fact that P 's are continuous together 1 with Lebesgue Dominated Convergence Theoremnwe obtain % e E (2‘)) } PX = 31:: 11wa(tPij'(IHXG(f(t)X)exp[-itf(t)(Pjy .X)]dnoT;%(X))"] - {w (ty)"'(J‘H G. Now using the fact that the integral of a tame function with respect to the product measure P1 is the same as its integral with respect to the canonical normal distribution n on H1, we get % prw(ty)"(j’HxG(f(t)x)exp[-itf(t)(y,xndno'r; (20)") k = f: EnHx)expt-1tf“]dv(x>>} 1 = EDHOJYYQ‘H)‘ G‘-t0pology is weaker than H'HB-topolosy on B we obtain foo(s)du(s) a :4: h(t)(det Ct)En{¢(ty)"(IHxé(f(t)x) .% ~ exp[-itf(t)(y,x)]dnoTV (x)) } for all real valued, bounded, u-uA-continuous functions G on BX. Therefore by Theorem 4 of [7] we have r f '2 (a) f(t trace (Ct Tv) -0 0 8.8 t -+ on (1.4.11) < (b) The measures h(t)exp[-t2“0;18“:/2]du(s) converge K weakly to p. on H)‘ as t-ocn . Now (1.4.11) (a) implies ([7], Corollary 3.4) that (1-4-12) f(t)C;1X'-q 0 on H}. as t .4 co , and (1.4.11) (b) together with (1.4.7) imply (Definition 1.1.2 (b)) that (1-4-13) Yt £1? on H)\ as t —. co . By the assumption {pt : t E (0,a)} is a x-family as t a«a. This together with (1.4.13) satisfy the hypotheses of Lemma 1.2.10, and hence the conclusion of the Ienma which is Y ‘3 Y on B as t t -+ on gives us (1.4.5) (b)° 30 To get (1.4.5)(a) we note that, it is easy to verify using the fact that Tv is positive-definite (and therefore Tv has positive eigenvalues) and tf(t) «'m as t «(a that 0:2 con- verges strongly to zero operator as t 4.”, One needs only express “CEZXH: in terms of an orthonormal basis in H1 which diagonalizes Tv' Hence I - 0:2 converges strongly to I, and clearly -2 (1.4.14) “{x E B . (I - Ct )xn’A>x when xn « x] 0 . From (1.4.14) and (1.4.5)(b), it follows ([1], Theorem 5.5, p. 34) that (1.4.15) (I - C;2)Yt QY on B as t -v as . Let f(t)C;]X be distributed as Vt' then (1.4.9) and (1.4.15) imply ([16], Theorem 2.1, p. 58) that for any sequence tn approach- ing infinity, [vt : n 8 1,2,...} is conditionally compact on B. Now by lemma 3.1 2f [13], {Vt } is a x-family for any 1 €.q: which is sufficient for {Vt n: n 8 1,2,...}. But by assumption 1 6 LI“ which is sufficient for any family of probability measures on B, and hence {Vt } is a x-family on B. Since this is true for any sequence [tn] approaching infinity we conclude that {Vt : t E (0,m)] is a x-family as t «»m of probability measures on CBJB(B)). From this and (1.4.12) it follows (Lemma 1.2.10) that f(t)C;]X'-QO on B as t-°°°s which is (1.4.5) (a)' 31 We now proceed to the proof of that (1.4.5) implies (1.4.4). (1.4.5)(b) flmplies (Definition 1.1.2) that for any sequence tn'd a, {at : n 8 1,2,...} is compact, and hence it is a x-family for anyn X 6 LI, ([13], lemmm.3.1). Thus [pt : t E (0,o)} is a 1' family as t «in, hence (1.4.4)(b) holds. Furthermore, from (1.4.5)(b), it follows that (1.4.16) lim at = 1 . t-m Let G be a real valued, bounded, “-“B-continuous function on B, and let at = h(t)x>epritf(y,x)”jdwxm - 1 ch(s)du(s). From (1.4.16) we get at 1 11m Bt Balitn — 8 11m :- h(t) (det C 1:13)]; xmfip (ty) (IBG(f(t)x) t-tco t-cco at; t-uco t expE-icfm(y,x)“Jdvmn - ch(s)duCt1x + (I-c;2)s)ds(x)dp,t(s) - t-«n t-eco f3 G(s)du(s) . We note that (1.4.15) and (1.4.5)(a) imply ([16], Lemma 1.1 and Theorem 1.1, p. 57) v o (f(t)C;1)-1*u o (I - C 22) 1.W =u on B as t a a, which can equivalently be written as 32 (1.4.18) IBIBG(f(t)C;1x +-(I-C;2)s)dv(x)dpt(s) a ch(s)du(s) as ta~ m for all real valued, bounded, “-HB-continuous functions G on B. From (1.4.17) and (1.4.18) we get lim at . 0 which completes the t—u proof. 1.4.19. Corollary. Let (i) B be a Banach space with Schauder basis {bi}, (ii) p be a finite positive measure on (BgB(B)) with Fourier transform m, (iii) f(t) be a positive admissible function on (0,o), (iv) x be a B-valued random variable distributed as v where v is a Gaussian measure on B with mean zero and the property that Tv is positive-definite, (v) pt be a probability measure on (ByBCB)) where l pt(A) B 2 2 IBexp[-t “Cglst/2]dp(s) IAexp[-t2uCE18“:/2]du(3) for all A 6 5(8), (vi) EP denote the integral with reapect to P1 on L. 1 Then for all real valued, bounded, “-HB-continuous functions G on B the following are equivalent: (”(a) chds>} (1.4.20) Eb) [pt : t E (0,m)] is a x-family as t a m (a) f(t)C;lX'-D;0 as t—bm (1.4.21) 2 _2 (b) t (Ct s,s) w 0 in measure with respect to u 88 tfimo 33 2:22;, Theorem 1.4.3 is clearly true for a finite positive measure u. Putting h(t) - l in Theorem 1.4.3 we see that (1.4.5)(b) implies IB(1 - exp[-t2(C;Zs,s)/2]dp(s) a 0 as t‘~ a, that is, exp[-t2(C;28,s)/2] converges to one in L1(B,u) and in fact this is clearly equivalent to (1.4.5) (b) when h(t) - 1. It is also equivalent to (1.4.21)(b). (1.4.20) and (1.4.2l)(a) are restatement of (1.4.4) and (1.4.5)(a) respectively. CHAPTER II OIERATOR THEORETIC CONDITIONS FOR THE INVERSION FORMULAE ON F-SPACES POSSESSING A SCHAUDER BASIS AND A QUASI-NORM WHICH IS ACCESSIBLE IN BOTH DIRECTst §2.0. Igtroduction. Let E be an F-space with Schauder basis {bu} and a quasi-norm “on which is accessible in both directions ([13], p. 39). In Section 2.1, we will observe that, if the function a(-) given in the definition of accessible quasi-norm “o“ possesses a particular property, then E and the Orlicz space Ea of real sequences are homeomorphic and isomorphic ([13], Theorem 6.3). In Section 2.3 we reduce for the case of Orlicz space Ea condition (1.4.5)(a) in the form similar to that of L. Gross ([7], (10), p. 36). This restriction on the space is needed since in this case the form of characteristic functional of a Gaussian measure is known (See; [13]). We shall first give a Theorem (Main Theorem II) which is Orlicz space generalization of Theorem 4 of Gross [7] and is stronger than the Main Theorem I, in the sense that, in case of Orlicz spaces, the Theorem holds for 1 belonging to a set con- taining L11 Then we assume that the function a(-) possesses another particular property to be able to get Corollary 2.3.12 which is our main objective. From Corollary 2.3.12 we get in- version formulae for (LP, p > 2) spaces. Finally as a special 34 35 case when Ea = L2, we take 1 = (1,1,...), then H = L2, and 1 Theorem 4 of [7] follows from Corollary 2.3.12. §2.1. Preliminaries and Definitions. Let a(s) be a convex function on [0,m) such that a(0) = 0, a(s) > 0 for s > 0. Further, assume (2.1.1) 0(23) S M a(s) for all s 2 O and some M < m. Now we define Ea as the Space of all sequences satisfying .;1 a(x:) < m. Since a(°) is convex, it follows that Ea is a vector Space over the reals ([13], p. 49). Let T(s) = 0(32), then F(-) has the same properties as a(°) and, it follows that Ms) = j: pdx where p(0) = 0 and p(s) is nondecreasing on [0,m). We assume, without loss of generality, that p(s) is left continuous. 2.1.2. Definition. By u = §(v) the inverse function of p(u) is defined, on the understanding that if p(u) makes a jump at u = a, then §(v) = a for p(a-) < v s p(a+), while, if p(u) = c for a < u s b, but p(u) < c for u < a, then §(c) = a. Furthermore §(0) = 0, and, if lim p(u) = L is finite, then Q(v) = +w for v >.L. With these co:::ntions u = @(v) is evidently nondecreasing for v 2 0, and left continuous for those values of v at which §(v) is finite. It follows that 6(v) is also Borel measurable on [0,m). 36 We now define s A(s) = $0 Q(x)dx . Then F and A are complementary in the sense of Young ([21], it p. 77), and by 6T we mean all real sequences {xi} such that (2.1.3) 2: F(‘xi\) = z 1 i= 2 . 01(xi) < °° - 1: 1 * Similarly, 6A is all sequences such that A(‘xi‘) < w . u't38 1 l * From (2.1.3), it follows that 6? contains the same sequences If x = {xi} is a sequence we define nxu, =Sup {3 \xim = 33 MM!) s 13 y i=1 i=1 and HA = 511M; lxiyj.L : i: r(\yi\) s 1} . y i=1 i=1 2.1.4. Definition. The Orlicz space 6F (5 ) is the A collection of all real sequences such that “X“? (“x“A) is finite. 2.1.5. Remarks. (a) Since a(s) satisfies (2.1.1), it follows that r(2s) = a(4s2) s M a(282) s Mza(sz) = M21“(s), * and hence we know ([21], Corollary, p. 81) that 6F (and, therefore Ea) contains the same sequences as or. Further, it is known that 6P is a real separable Banach Space in the norm “x“? and since 37 I"(23) s M2r(s) for s 2 0 we also have ([20], Lemma 0,, p. 33) that {pn} : 6F converges to p E 6F in norm provided m G 2 lim 2 F(|x, - x,‘) = lim 2 a[(x. - x.) ] = O n i=1 1,n 1 n i=1 1,n 1 where by xi n’ xi we mean the ith elements of pn, p respectively. 9 (b) By Theorem 6.2 of [13], (Ea’ H'Hr) is a Banach space with a Schauder basis {bu} where bn is the vector with one as the nth coordinate and other coordinates zero. Following [13], we now introduce the notion of F-Space with a Schauder basis and an accessible quasi-norm in both directions. Let E be an F-space with quasi-norm “-H (See; [13], p. 2) and Schauder basis {bu}. We assume further that the following assumptions (A) are satisfied: (A.1) the basis elements {bu} can be adjusted so that “bu“ s 1 (this is always possible), (A.2) if K is any compact subset of E then Sup \Bn(x)‘ < w n,xE K where Bn's are the coordinate functionals on E, (A.3) the a-field generated by the weakly open subsets of E is identical with the Borel subsets of E. It can easily be seen that in case “-H is actually a norm and E is then a Banach space assumptions (A) are always satisfied. Let a(-) be a convex function on [0,m) such that o(0) = 0 and a(s) > 0 if s > 0. Further, assume for every compact subset K of E there exists an r > 0 such that x 6 K Q hnplies A(x) = Z a[6i(x)] < r, and for every r > 0 there i=1 exists an M > 0 such that A(x) < r implies 38 331 swim] a M wuxu) where y(-) iS a.continuous function on [0,m) such that v(0) = 0. 2.1.6. Definition. If the quasi-norm H’“ on E admits the existance of functions a(°) and y(-) having the above pro- perties we will say that it is accessible. We also note that if a(°) and y(-) satisfy the conditions indicated then a(') is continuous and strictly increasing on [0,m), y(s) > O for s > O, and that y(o) can be taken to be increasing on [0,m). 2.1.7. Definition. The quasi-norm “-H on E is said to be accessible in both directions if there exist functions a, Y1’ Y2 such that H.“ is accessible with reSpect to o and Y2 , and for every x E E °° 2 Y1(“XH) 5 .21 34:51:00] 0 1: Here y1(-) is an increasing continuous function on [0,m) such that y1(0) = O, y1(s) > 0 for s > 0, and a, v2 satiSfy the conditions required in the Definition 2.1.6. In this Section we shall be concerned with an F-Space E with Schauder basis satisfying assumptions (A) for which the quasi- norm “-H is accessible in both directions and the associated a(-) satisfies (2.1.1) We now recall the following Theorem from ([13], p. 57). Theorem. If E has a quasi-norm which is accessible in both directions with reSpect to the functions 0, Y1’ Y2 and a(-) satisfies condition (2.1.1) then E and the Orlicz Space 39 Ed are homeomorphie and isomorphic. 2.1.8. Remark. In view of the above Theorem we can (and will) identify the F-Space E with Ea and restrict ourselves to sequence Space Ea' §2.2. Associated Hilbert Space. Following [13], we Shall denote by Ed the Hilbert space L2 or an Ea Space where o(°) satisfies (2.1.1). We assuue that ac(-), the complementary function of a(-) in the sense of Young ([21], p. 77),satisfy (2.1.1). Notice that if Ea = L2 then a natural choice for the function a is a(s) E 5. Hence ac(s) = O on [0,1] but qc(s) = o for s > 1. Thus qc(-) does not satisfy (2.1.1) when Ea = L2 and this is a Special case which is easily handled. In terms of the notations we have used in Section 2.1, Ed is equivalent (isometrically isomorphic) to the Orlicz Space or where F(s) = a(32). We will let Sq, Sa -denote the Orlicz c Spaces given by a(-) and ac(-), reSpectively. Then the dual space of SQ can be identified as Sq and Since aC(-) also c satisfies (2.1.1), except when Ea = L2, it follows that the dual of Sac is 30 ([21], p. 150). For each vector 1 = (x1,x2,...) in the positive cone of $0 , we define the Space H1 as all sequences x = (x1,x2,...) c m such that 2 xix: < a. Then H1 is a Hilbert Space with a i=1 “x“ = ( 2 )\.x2)!5 and the inner product x i=1 1 i a 40 In the Special case Ea = L2 we have 80 = LOD and for simplicity c we take x = (1,1,...). Then H = L and we Shall assume without 1 2 loss of generality that a(s) E S. The following Lemma is proved in ([13], p. 62). 2.2.1. Lemma. Ed is a Borel Subset of H1 for each h in the positive cone of Sa . Furthermore, every Borel subset of c Ed is a Borel subset of “1'. We note that from Lemma 2.2.1, every probability measure u on Ea can be regarded as a probability measure on H Furthermore x. every countably additive measure on H is countablly additive 1 on Ea. Now having this observation in mind we shall prove the following Lemma, which is similar to Lemma 1.2.10. 2.2.2. Lemma. Let . + (1) 168 s 0[c (ii) {pt : t 6 (0,0)} be a x-family as t a m of probability measures on Ea’ (111) ”t :1” on “X as t a m where u is a probability measure on Ea- Then ptgp. on Ea as taco. Proof. Let {tn} be a sequence in (0,m) such that . W tn a»m as n a m. Then (iii) implies that utn = p on HA’ that is, {pt :n.= 1,2,...} is compact on H1. Using exactly n the same argument given in Lemma 1.2.10 the equicontinuity at * zero on FN(Ea) of {¢Lt (°) : n = 1,2,...} follows. Hence con- dition (i) in Definition n1.2.9 is satisfied. + . Let A 6 Sa . Since ¢h (.) ls c.f. of “t and §1(N’k’.) C t n n is symmetric about zero it follows that 41 N+k 2 JN k[1 - Re T] (Y)]= [E011 ' eXP{' '% Z xixi}]dut (x). n i=N+l tn By Young inequality ([21], p. 77) 2 2 xixi s ac(xi) +-a(xi) for each i . Hence [ N+k N+k 1 - Re mu (y)] s [1 - exp{- -% 2 ac (l )- -% 2 0(x2 )} du (X) N, k IE0 i=N+1 i=N+1 ] tn tn 5 1 - exp{- 35 z o (1.1%”E exp{- J: z o(x. 2nd,).tn (x)- +1c i=N+ +1 We note that as N 4.x, 2 “C(xi) a 0 Since 1 E S , and hence i=N+1 ac exp[-% 2 :iac(:i)} a 1. Also for x 6 Ed we have {xi} E Sa, 1% that is E a(x 2) < m. From this it follows that the functions i=1 f (x) = exp[- -% Z a(x.)] converge pointwise to one. Further- N i=N+l 1 more fN'S are nondecreasing, so by Monotone Convergence Theorem we get a: [E exv{-% 2 04X?) )ldu-t (JO-+1 as N-°°° . a i=N+1tn Therefore J l - Re a 0 as N A.” for n = 1 2 ... . N,k[ qht (Y)] a a n Thus condition (ii) of Definition 1.2.9 is also satisfied, and hence {m (°) : n = 1,2,...} is x-continuous for A 6 8+ . ”t 0’c n . This together with the assumption that [pt : t E (0,m)} is a x-family as t a m imply ([13], Lemma 3.2, p. 11) that {pt : =1 ,2,...} is conditionally compact on E1. Hence ”tn a p on Ea since {pt : n = 1,2,...} is conditionally n 42 9&1 g u o pgl for all ‘N = 1,2,... compact on Ea and pt 0 ([1], p. 35). Since thgs is true for any sequence {tn} approach- ing infinity we get pt 3 p on Ea as t a m. We use the fact that every linear operator on E: into Ea can be represented as an infinite dimensional matrix to give the following definition. 2.2.3. Definition. A linear operator from E: into Ea is an a-Operator if the matrix of the operator, {tij} is symmetric, nonnegative definite with .;1 aii) a O as t a w i=1 -2 -2 h = o w ere (Ct T)ii (Ct T bi’bi)Hx Proof. We shall first prove that (2.3.6) implies (2.3.7). let {tn} be a sequence in (0,m) such that tn e m as n a a. Let f(tn)Ct:X be distributed as vt ,,then Vt 13 define] on (EagB(Ea)) n n and by (2.3.6), [Vt : n = 1,2,...} converges weakly on Ea to n 50. Since Ea is a Borel subset of HA’ {vtn : n = 1,2,...} can be regarded as probability measures on HA, and Since topology of H1 is weaker than that of Eo’ vt converges weakly on HA to n 60. The c.f. of Vt when regarded as a probability measure on n . 2 -2 HA is exp{-%(f(tn) CtnT -,-)H }. Hence vtn g 50 on HA as n a m implies ([7], Theorem 2, p. 8) that °° 2 -2 2 -2 z x.(f(t ) (C T)..) = trace (f(t ) C T) a 0 as n a w. ,_ i n t 11 H n t 1—1 n A n From this, it follows that (f(t )2(C-2T) ) a O as n a a £0 all i 4i n tn ii r ' Since xi > 0. we get ‘ 2 -2 , f(tn) (CtnT)ii a O as n a m for each 1, Hence, for every N N 2 -2 (2.3.8) 121 a(f(tn) (CtnT)11) --9 O as n —o m . 48 Under condition (2.3.5) and the fact that {Vt } is compact on n Ea we get ([13], Theorem 9.1) that for each s > 0 there exists an N0 such that on (2.3.9) 2 a(f(t )2(C-2T),,) < e for all t . n t 11 n 1=N n 0 Therefore (2.3.8) and (2.3.9) imply that m 2 -2 11:1 Q2<0;2T)ii} s Hind Hff(tn)2(0;2T)ii}Ha . C n 1 l n From (2.3.7) and Lemma 0 of ([21], p. 83), it follows that 2 -2 “{f(tn) (CtnT)ii}Ha "" O as n —+ a) . Hence, 2 -2 11(f(tn) (ct T)ii) 4 o as n 4 m . 2 -2 traceH (f(tn) Ct T) — l n 1 n i "MB Therefore, by Corollary 3.4 of [7] we have (2.3.10) v 49 In order to complete the proof it suffices to Show that [Vt : n = 1,2,...] is a x-family for 1 E 8:». First observe n + c that 1 E S is sufficient for {vt : n = 1,2,...] Since ac n °° 2 2 xi a(xi) < a for x E E . i=1 a Let 3,6 > 0, and let Y1 be any continuous Strictly increasing function on [0,m) satisfying (6.6) of ([13], p. 50). Then from ([13]; (6.6), (b.7), p. 51) it follows that thix 9 HQNXHF > 5} = thix ’ YIKHQNXHF) > Y1(5)} °° 2 S vtn[x : i=§+ia[Bi(x)] > v1(5)} where Bi(x) = x1 is the ith coordinate functional on Ea. Using (2.3.5) we get m 2 2 of B (X)dv (X) i=N+1 an l tn 1 C y1(6) S c y1(6) °° 2 -2 2“ a(f(tn) (Ct T)ii). i=N+l n From (2.3.7), it follows that, for each s > 0 there exists an No such that for all n = 1,2,... m f 2 -2 Y1(6) 13s s( “.9 (ctn'r)fi> < C e . 0 Therefore, for all n = 1,2,..., and all N 2 NO (2.3.11) vtn{x : HQNXHF > 6} < e . Hence, [vt : n = 1,2,...} is a x-family for any 1 E sg'. Since n c (2.3.10) and (2.3.11) hold for any sequence {tn} approaching 1.55;... ....l ...-.511 _ t: 50 infinity, it follows (Lemma 2.2.2) that 5 on E as t a m, which is (2.3.6), and hence the proof is completed. 2.3.12. Corollary. If, in addition to (i) - (vii) of Theorem 2.3.1, the function a(-) satisfies (2.3.5), then the following are equivalent: Ra) jE C(s>ds(det ct)H EP {¢z(ty)(j‘E mom or t-m 1 1 oz (2.3.13) é exp[-itf(t)(y,x)z]dv(x))} (b) {H : t E (0333)} IS a x-family aS t -+ G) K t [0a) 1 (2.3.14) L(b) Y Q Y 33 t -—v 0.3. t IIP18 oz2(c'21)..) .. o as t .. a 1 t 11. Proof. By Lemma 2.3.4, condition (2.3.14) is equi- (a) valent to (2'3‘3)(a)’ hence the proof follows from Theorem 2.3.1. 2.3.15. Remark. In case the matrix {tij} of the a- operator T is diagonal, condition (2.3.14)(a) would be replaced by 2 f(t) tii (2.3.16) 2 (y( 2 2 ) —.o as t:—~m . i=1 1+t f(t) tii 2.3.17. Definition. If 2 s p < m l l T from —'+ — Lq(P q then a linear operator 1) into LP is an Sp-Operator if T can be represented as an infinite Symmetric, nonnegative definite matrix as {tij} such that 2 (tii)p/2 is finite. Here, by nonnegative i=1 n definite, we mean that Z t .x,x, 2 O for all , (x,x ,...,X)ER i,j=1 1] i J l 2 n n 51 and all integer n. Thus for p = 2, S -operator is S-operator. 2 For p > 2, LP is an Ea Space (Orlicz Space) with a(S) = 813/2 ([21], p. 78). Furthermore, for this d(.), con- * * dition (2.3.5) is satisfied. The a-operators on Ea = Lp are +- * + the same as S -0perators, and S = (L ) . Now Corollary P ac p/2 2.3.12 gives us a proof for the following Corollary. 2.3.18. Corollary. Let p > 2, and let * (i) T be a positive-definite Sp-Operator on Lp, (ii) p. be a probability measure on (LP,B(LP)) with c.f., cp. (iii) f(t) be a positive admissible function on (0,m), and h(t) be a positive function on (0,m), (iv) v = noT-% where n is the canonical normal distribution on H, 1 (v) EP denote the integral with reSpect to PA on L. A Then for all real valued, bounded, LP-continuous (that is, con- tinuous in Lp norm) functions G on Lp the following are equivalent: as) h G(s)dp.(s) = lim h(t)(det ct)H EP {tf(tyuh G(f(t)x) p t-m i i p (2.3.19) + exp[-itf(t)(y,x)%]dv(x))} (b) {pt : t E (0,m)] is a x-family as t a m a -2 Re) 2: [f(t)]"icct ThilP/z —' i=1 0 as t a m (2.3.20) < (b) The measures {pt : t E (O,m)] converge weakly K tomon LP as t—oco. 52 2.3.21. Remark. (a) If the matrix {tij} of the Sp- operator T is diagonal, then (2'3'20)(a) would be replaced by tii )p/2 (2.3.22) O as t -" Q o IIM8 [f(t)]p ( l 2 2 1 1+t f(t) tii (b) It can easily be Shown that in Theorem 4 of L. Cross [7] one can, without loss of generality, assume that the Hilbert- Schmidt operators in (10) and (11) of ([7], p. 36) to be diagonal. (c) In the spacial case Ea = L2 we have 30 = LOD and for simplicity we take 1 = (1,1,...), then H1 = :2. Now using Lemma 4.3 of [14], and the fact that TJ5 is a Hilbert-Schmidt operator on L2 whenever T is an Sz-operator on L2 we get Theorem 4 of [7]. (d) From (b), it follows that in case of Hilbert Space condition (2.3.22) is restatement of condition (10) of ([7], p. 36). CHAPTER III INVERSION FORMULAE OF THE CHARACTERISTIC FUNCTIONAL OF A PROBABILITY MEASURE ON BANACH SPACES WITH A SHRINKING SCHAUDER BASIS §3.0. Introduction. In the first Chapter B was taken to be a Banach Space with Schauder basis {bn}. In this Chapter we assume that the Schauder basis {bu} is also shrinking. Since {bu} is shrink- * ing, the coordinate functionals Bn's form a basis for B and 9 hence we can use results of J. Kampe de Feriét [11] to identify B* with a Borel subset of L. Thus any probability measure on B* can be defined to be a probability measure on L through this identification. We shall let n be the canonical normal distribution on HA so that for each x E HA, n(x) is a random variable on B*, and let Pl be the cylinder set measure on B* induced by the above family. Then we Shall Show P1 is countably additive on the o-field of tame sets of 8*. Finally we prove a Theorem (Main Theorem III) which gives a class of inversion formulae different from that of the Main Theorem I. In the Main Theorem I we have extension of characteristic functional to L whereas in the Main Theorem III we have extension of characteristic * functional to B . Hence (3.2.2) is Stronger than (1.4.4) (8) (a) * since for A E LT, P1 is countably additive on B . Furthermore 53 54 * since [bu] is shrinking (and, therefore B has a basis) we are able to give a proof for the Theorem without using Lévy Continuity Theorem and hope that one might be able to use this Theorem to obtain a proof for the Lévy Continuity Theorem. §3.1. Preliminaries and Definitions. A tame (cylinder) set in a real separable Hilbert Space H can be described as a set of the form C = P-1(E) where P is a finite dimensional orthogonal projection on H with range £3 say, and E is a Borel set in Eh The cylinder set measure v (See; [6], p.32) associated with the canonical normal dis- tribution is called Gauss measure on H, and for the above tame set C we have -k/2 -“xH2/2 v(C) = (211) j‘E e dx where k is the dimension of fi’ and dx is Lebesgue measure on Rk' 3.1.1. Definition. A semi-norm qul on H is called a measurable semi-norm if for every real number s > 0 there exists a finite dimensional projection Po such that for every finite dimensional projection P orthogonal to P6 we have (3.1.2) Prob(\\qu'i > e) < a where “Bx“; denotes the random variable on the probability Space (O,m) corresponding to the tame function “Ple and Prob. refers to the probability of the indicated event with reSpect to the probability measure m associated with the canonical normal dis- tribution. 55 Observe that the condition (3.1.2) can also be written v({x : HPle > e}) < e where v is Gauss measure on H (See; [6], p. 33). We note that a measurable norm is a measurable semi-norm which is a norm. 3.1.3. Definition ([3], p. 69). A Schauder basis [bi] in a Banach Space B is called Shrinking basis for B if for each * B in B , 1im pn(B) = 0 where pm(B) = norm of B restricted n-ioo m to the range of x - 2 Bi(x)bi; that is, mi=1 Pm<6> = SUP{B = igleiba = 0 and HXHB s 11. Throughout this Chapter B will denote a real Banach Space with a shrinking Schauder basis {b } such that Hb u = 1. As 11 m n B before we will write the expansion of x as E Bn(x)bn, and k “=1 . +' . HXHB — 11m H E Bn(x)anB. For 1 E L1, H1 denotes the completion k—uoo n—l of B under the inner product (1.2.3). * Let n be the canonical normal distribution on HA into * the set of all random variables defined on B , that is, for each * * x E H1, n(x) is a random variable on B which is distributed normally with mean zero and variance HxH:*. * We identify HA by H1, hence for each x E H1, n(x) is a random * variable on B distributed normally with mean zero and variance 2 x O n u, The basis elements bi's can be considered as coordinate * functionals on B ([3], Lemma 1, p. 70). Then “(bi) = (bi,°) * is a random variable defined on B which is distributed normally 0 . 2 — — With mean zero and variance ubin - (bi’bi) — xi. 56 Let PA be the cylinder set measure on the field c, generated * by tame sets of B induced by the above canonical normal dis- tribution on H . 1 3.1.4. lemma. P1 is countably additive on Ca Proof. Without loss of generality we assume “Bi“ * = l. * By lemma 1 of ([3], p. 70) {Bi} is a basis for B , hence we uyn.=nmu,§bse.u- a. Hyun=u§bsmeaua B new i=1 B i=1 B and observe that Pxfy = HYHBa < e} = Pxiy = ii: Hyun < e} n > Px{y : 11m 2 lbi1 < e} - Ham i=1 n * lim 2 ‘bi(y)‘, then X is a random variable on B Let X(y) = n “A“ 1:12 n m since E{ 2 [b.(y)] } = 2 A. < 2 x < m, and the series is a 1 _ i , i 1'1 1-1 i=1 series of independent random variabled ([15], p. 234; and [2], Theorem 9.5.5). In view of the property of Laplace transform, Theorem 6.6.2 of [2], we observe that the distribution of x is Q absolute normal with mean 2 E[‘bi(y)‘] and variance 2 xi. i=1 i=1 Hence the distribution of X puts mass around zero, and therefore (3.1.5) Px{y : Hy“ * < e} > PA{X < e} > 0 . B From the definition of HA we see that H1 is a dense subset of B*, and hence B* is the completion of H1 in the Banach norm H “B* on 3*. Furthermore, “y“n is a tame function on HA and hence it is a measurable norm (See; Definition 3.1.1). From (3.1.5) and the fact that “y“n is a nondecreasing 8eQuence of measurable norms on HA, it follows that “y“ * is 57 a measurable norm on HA ([9], Corollary 4.4). So far we have shown that “y“ * is a measurable norm on H1 and 8* is the completion of BHx in this norm. Hence by Theorem 1 of [6], P1 is countably additive on CL Let x E B, y E 3*, then a z ei i=1 (y.X) = y(X) = y( 2161(X)bi) i: 2 51(X)bi(y) i=1 m It follows that (',x) = 2 51(x)bi(-) is a random variable de- fined on B which is distributed normally with mean zero and . °° 2 variance 2 1-5 (x) under P . i=1 1 i A * Now let x E H1, y E B and define the "stochastic linear func- a. tional" (y,x)A' as follows a N . (3-1~6) (y.X) = 11m 2 81(x)b.(y) 1 N-cas 1"]. where as before Bi is extension of Bi to H1. 2. * From (3.1.6) we have (y,x)~'= y(x) for x E B, y E B . 3.1.7. Lemma. The stochastic linear functional a N A N—Ioo i=1 * is Borel measurable on B X H1 and if F = {(y,x) : (y,x)R‘ exists and is finite], then P1 X n(F) = l where n is a probability measure on HA. Proof. That (y,x)R‘ is jointly Borel measurable follows easily since it is the limit of jointly Borel measurable functions. 58 Fix x and consider Fx (that is, the x-section of F; See [10]). For this fixed x, (y,x)é‘ is the limit of sums of independent random variables, and the variance of the Sum ; 1 is: (x) is finite. This implies ([15], p. 234; and [2], T6e6rem19.5.5) that (y,x)fig is finite almost everywhere with reSpect to P1' Thus Px(Fx) = 1 for each fixed x E H1. Hence Pk X n(F)= [H1 P A(Fx)duu(X) = In 1 (11100 = A Let H be a probability measure on B, then u can be regarded as a probability measure on HA. Let W be the c.f. of H when p is regarded as a probability measure on H , then V(Y) = Ink ei(y’x)dp(x) for all y E H: . * * Now I is a function defined on H1 9 B , and we would like to * extend W to be defined on B . * For each x E H ,(y,x)& is defined on B a.e. P , and is 1 * equal to y(x) with H measure one for each y E H1. We call In ei(y’x)~du(X) (y 6 3*) the extension of W to 8*. Clearly on 11* we have I ei(y’x)~du(X) =J‘ ei(y’x)du(X) = M ) 1 H H y ' Since H is actually defined on B we have J“ ei(y’x)~d (x) =J" ei(y’x)£’dp(x) a.e. P . Hx 1" B 1 e * But (y,x)~'= (y,x) for all x E B, y E B , hence (3.1.8) IH ei(y,x) dp(x) = $3 ei(y’x)dp(x) = ¢(y) a.e. P1 1 59 where ¢ is the c.f. of u when u is regarded as a probability measure on B. Thus c.f. of a probability measure p on B when considered as a random.variable on B* is equal almost everywhere P1 to the extension of the c.f. of p when p is regarded as a probability measure on Hx. 3.1.9. Remarks. (a) Let L be as before. Since {bu} is a shrinking basis for B, the coordinate functional Bn's form a basis for B* ([3], Lemma 1, p. 70). Hence there exists an isomorphism U* from 8* to a Borel measurable subset of L, say 0* ([11], Section 2, pp. 123-127). Therefore 8* can be identified with a Borel measurable subset of L, and hence Pk can be regarded as a countably additive cylinder set measure on L through this identification. (b) By (a), Leanna 4.3 of [14] and the fact that P). sits actually on B* we get $(y)~ = ¢(y) a.e. P1 where ¢(-)~ is Gross extension of the uniformly T-continuous function ¢(-) with respect to the canonical normal distribution n on H1. We will close this section by proving the following Lemma. The hypotheses of this Lemma are the same as of Lemma 1.2.10, however the proof is completely different. In the proof of Lemma 1.2.10 we used Le'vy Continuity Theorem which we will not use in the proof of the following Leanna. Instead we use the fact that {bu} is shrinking and the ideas of [4]. 3.1.10. m. Let (1) x be in 61‘. (ii) {pt : t E (O,m)} be a x-family as t a m of probability measures on B, 60 (iii) p be a probability measure on B, (iv) {pt : t E (0,m)} converges weakly to p on H1 as t-aao. Then {pt : t E (0,m)} converges weakly to u on B as t-oao. Proof. Let {tn} be a sequence in (O,o) such that tn « m as n d’m. Then (iv) implies that {pt : n = 1,2,...} n is tight on H1 ([1], Theorem 6.2, p. 37). Hence, for each s > 0 there exists a compact subset of H1, 88y K6 such that pt (K6) > 1 - e for all n. n K6 is compact implies that K6 is bounded, that is, for each Q .2 5 > 0 there exists an N such that 2 X B (x) < 6 uniformly . i i a 1fN+1 in x e K6. Thus pt {x e Hx : z x18§(x) < 5} > 1 - e for n i=N+1 all n. Since 11's are positive, for all N' 2 N .2 w .2 {x 6 H : Z 1.8.(x) < 6} lex E H : 2 l B (x) < 6}. X 1=N+1 1 1 )1 1=N '+1 1 1 Hence, for all n and all N' 2 N m .2 z X18100} > 1 ' e 0 p. {xEHz tn 1 i=N'+l But for each n, pt is defined on (Bg6(x)), so we get n Q (3.1.11) pt {x e B : z xiB:(x) < a} > 1 - e for all n n i=N'+l and all N' 2 N. Since {pt : n = 1,2,...} is a x-family, it follows that there n exists {6N} such that ptn{x e B : “i=§+191(x)bius < k(6)} > 1 - (e + 3N). 61 Now using (3.1.11) we have, for all n and all N' 2 N z 191(x)b1“3 < k(5)} > 1 - (e + eN)~ '+ ptn{x E B : “1:“ Let NO be sufficiently large so that, for all n thfx E B 3 “i=N2+lei(X)bi“B < k(5)} 2 1 _ e . We now let S be the subSpace of B generated by {b1,...,bN }, km ° and let 8 = {x : inf{“x - zHB : z E S} S k(6)}- For X E B N 0 we have P x = Z B.(x)b. E S, hence No i-l 1 1 3k“) 3 {x e B : H z 81(x)biHB < k(5)} . i=N +1 Thus pt (Sk(6))2 1 - e for all n. n * Let T be the Subspace of B generated by {51,...,BN }. o * Then SJ'C>T = B since Bi's are coordinate functionals and * form a basis for B (See; Remark 3.1.9 (a)). We now show there exists an r > 0 such that for all n pt {x e B : ‘Bi(x)| < r, 1 = 1,2,...,NO} 2 1 - e . n Let r1 = sup ‘§i(x)|, then r1 is finite since K6 is a compact 1,K' . subset of HR, and Sup ‘ai(x)‘ S MHXHx ([3], p. 68). Now let i r > r1, then “tn” 6 B : ‘Bi(x)‘ < r, 1 = 1,2,...,N0} = utn{x E B : ‘§i(x)‘ < r, i = 1,2,...,No} 2 pt {x E B : Sup ‘§i(x)‘ < r, i = 1,2,...,No} n 1 2 ”t (Kg) n 2 1 - e for all n. 62 Therefore by Theorem 2.1 of ([4], p. 11), {pt : n = 1,2,...} n is conditionally compact on B. Since {pt : n = 1,2,...} is compact on H1 and RN n is continuous, it follows that {pt 0 Ekl : n = 1,2,...} is n compact on H1 for all N = 1,2,... . Hence [pt 0 Pgl : n = 1,2,...} is compact on B for all N = 1,2,... n and pt 0 Phl'g>p o Phl. This together with the fact that n [utn : n = 1,2,...} is conditionally compact on B imply that pt ‘E’p on B ([1], p. 35). Since this is true for any sequence n {tn} approaching infinity we have pt 3.” on B as t a a. §3.2. Main Theorem Til. The following Theorem (Main Theorem III) gives inversion formulae for a probability measure on a Banach space B with a shrinking Schauder basis. It differs from the Main Theorem I in the sense that (3°2'2)(a) is stronger than (1.4.4)(a). This can easily be seen since P1(B*) = l for 1 E LI. Furthermore, we will not use Lévy Countinuity Theorem in the proof, and hope that one might be able to use this Theorem to get a proof for the Levy Countinuity Theorem. and a be as in Section 1.4. Lat V, TV, Ct, “gt t 3.2.1. Main Theorem III. Let (i) B be a Banach Space with shrinking Schauder basis {bi}, (ii) '- (v) be as in Theorem 1.4.3, * denote the integral with reSpect to P on B (See; (vi) E 1 p Lama 3.1.4). Then for all real valued, bounded, “-HB-continuous functions G 63 on B the following are equivalent: r2a) jBG(s)du(s) = lim h(t)(det Ct)EP {e(ty)(ch(£(t)x) 1 t—m (3.2.2) 4 exp[-itf(t)(y,x)]dv(x))] (b) [pt : t E (O,m)] is x-family as t a m K [(3) f(t)C;]X '2 0 as t -o on (3.2.3) [(b) Yt‘QY as 12—91:). We note that the proof is similar to the proof of Theorem 1.4.3. Proof. Suppose (3.2.2) holds, and let G be a real valued, bounded, H-HB-continuous function on B. Then EP wry)(jBG(f(t)x>exp[-itf]dv>} l = I *{cp(ty)(IBG(f(t)x)exp[-itf(t)(y,x)]dvb‘nydPXQ’L B Using the fact that G(f(t)x)dnoT;%(x) is a measure of bounded variation together with Remark 1.2.6 (c) and (3.1.8) we get _ n(mfi -itf"“ -.5 - [Signx e du(s))qnxcx)e dnoTv (x))dP>\(y). A ~ The function e1t(y,s) e-ltf(t)(y’x) G(f(t)x) is jointly measur- able and all the measures are probability measures so we may use Fubini's Theorem ([19], p. 140) to interchange some integrals to obtain 64 A A 't . “’ -'t£(t)< . r“ J: IBe32/2}dnor;%(x>de i=1 ‘13-‘11 c(£(c)x)exp{-t2 1 1 = G(f(t)x)exp[-t2Hs-f(t)XH2/2]anT-%(x)dU(S)- H H 1 v 1 k % Since G is “-Hx-measurable and T is a Hilbert-Schmidt operator on H1, it follows ([7], Lemma 4.1) that —.___L____ -1 -2 -g - (det Ct) IHAJHX§(f(t)Ct x + s-Ct s)dnoTv (x)b(t) exp[-tzuc;ls“:/2]dp(s) -—————-— ‘ ' - 2 (de: G ) IBIBG(f(t)Ct1x +’S'Ct28)dv(x)h(t)exp[-t2uct18Hx/2]du(s) t where we have used Remark 1.2.6 (c) and the fact that n(B) = 1. Hence we may start with the assumption that for all real-valued, bounded, “-HB-continuous functions G on B we have - -2 ((a) IBG(s)du(s) = 2::1IB[IBG(f(t)Ct1x + s-Ct s)dv(x)]h(t) (3.2.4) 4 exp[-tzncgls“:/2]du(s) K(b) {Ht : t E (O,m)] is a x-family as t a m . Putting G E l in (3.2.4)(a) and using the same argument given in the proof of Theorem 1.4.3, condition (3.2.4)(a) can be written as follows -1 -2 (3.2.5) J‘Bc(s)du(s) = t1}: fBj‘Bcuumt x + (I-ct )s)dv(x)dut(s) for all real valued, bounded, H°“B-continuous functions G on B. 65 From (3.2.5), it follows that (3.2.6) v o (f(c)c't'1)'1 * pt 0 (I-ng) E u on B as t s»«. Since G(f(t)x)dv(x) is a measure of bounded variation on H1, the Fourier transform of G(f(t)x)dv(x) is uniformly T-continuous ([7], p. 7). Hence the Gross extension of its Fourier transform is well-defined ([7], Theorem, p. 5). Similarly the Gross extension of the Fourier transform (c.f.) V(-) of u when regarding p as a probability measure on (H1“B(H1)) is well-defined. Now from Remarks 3.1.9 (b) and 1.2.6 (c), it follows that (3.2.7) EP may)(J‘Bc(fx>exp[-itf(y,xndvmn 1 k = EPX{$(tY)~(fH G(f(t)x)exp[-itf(t)(y,x)]dnoT; (x))"]. 1 Let {P1} be a sequence of finite dimensional projections on H1 converging strongly to the identity operator. Then it is easy to see using lemma 3.1.5, Lebesgue Dominated Convergence Theorem and the fact that P 's are continuous that J % prfw(ty)~(IHxS(f(t)x)9XP['itf(t)(y,x)]anT; (x))"} % s (X)) }~ J- P1 = lim E H(tPjy)"(IHXG(f(t)x)exp[-itf(t)(Pjy,x)]dno'l‘v Now as before, using the fact that the integral of a tame function with respect to the probability measure P1 is the same as its integral with reSpect to the canonical normal distribution n on H1, we get 66 % EPX{¢(ty)"(IHxa(f(t)X)exp[-itf(t)(y.X)]dn0T; (X))"} k = lim En[¢(tPjy)"(IHxC(f(t)x)exp[-itf(t)(Pjy,x)]dnoT; (x))"} j—tco where En is the integral with reSpect to n on HA. From Remark 1.1.15, it follows that lim En{¢(tpjy)”(fH G(f(t)x)exp[-itf(t)(P y.x)]dnoT;% (x>)"} jam 1 j e = En{¢(ty)"(foC(f(t)x)exp[-itf(t)(Yax)]dndT; (x)>"} . Hence from (3.2.7), we have EP {e(ty)(fBG(f(t)x)exp[-itf(t)(y.x)]dv(x))} 1 k = En{¢(ty)"(foC(f(t)x)exp[-itf(t)(y,x)]dnoT; (x))”} - Now from (3°2'2)(a) and the fact that “-“B-topology is stronger than “-“x-topology on B we obtain IH)é(S)dp(S) = 2:: h(t)(det Ct)En[¢(ty)"(foé(f(t)x) % exp[-itf(t)(x,y)]dndT; (x))"} for all real valued, bounded, H-Hx-continuous functions G on H . 1 Therefore by Theorem 4 of [7] we have ’(a) f(t)2 trace (cgzrp -. o as t .. .. (3.2.8) 4 2 _1 2 (b) The measures h(t)exp[-t “Ct sux/2]du(s) converge L weakly to p on H1 as t d a . Now (3'2'8)(a) implies ([7], Corollary 3.4) that 67 (3.2.9) f(t)C;1X'-€0 on H). 38 t-*°° 3 and (3.2.8)(b) implies (Definition 1.1.2) that .3 (3.2.10) Yt a Y on H1 as t ~>w . From (3.2.2)(b) and (3.2.10), it follows (Lemma 3.1.10) that Yt‘QTY (H! B as t aim . Thus condition (3.2.3)(b) holds. To get (3'2'3)(a) we note that (3.2.11) u{x E B : (I-C;2)xan'x when xn a x] = 0 . From (3.2.11) and (3.2.3)(b), it follows ([1], Theorem 5.5, p. 34) that (3.2.12) (I-C;2)Yt£Y on B as taco . Let f(t)C;1X be distributed as Vt’ then (3.2.6) and (3.2.12) imply ([16], Theorem 2.1, p. 58) that for any sequence [tn] approaching infinity, {vt : n = 1,2,...] is conditionally compact on B. Now by Lemma 3.1 :f [13], [vt } is a x-family for any 1 E L:' which is sufficient for [vt n: n = 1,2,...]. Since 1 E L13 it follows that {Vt : n = 1,2,...] is a x-family, and since this is true for any szquence {tn} with tn 4 m as n a m we conclude that {Vt : t E (0,m)] is a x-family as t a a of probability measures on (B,B(B)). From this and (3.2.9) it follows (Lemma 3.1.10) that f(t)C;]X'-QO on B as t-oao. 68 Hence condition (3'2'3)(a) holds. We now prove the converse. From (3°2°3)(b)’ it follows that for any sequence {tn} approaching infinity, {pt : n = 1,2,...] is compact. Hence it is a x-family for any x EnL: ([13], Lemma 3.1). Thus {pt : t E (O,m)] is a x-family as t34 m, and hence condition (3.2.2)(b) is satisfied. Further- more, from (3.2.3)(b) we get (3.2.13) 1im at = 1 . t-m Let G be a real valued, bounded, H-HB-continuous function on B, and let at = h(t)(det Ct)EPx[m(ty)(IBG(f(t)x)exp]-itf(t)(y,x)]dv(x))] - IBG(x)dp(S). Then from (3.2.13), it follows that li = li '52 a l‘ l—-h(t)(det C E (t )( G(f(t) m8t ma 1ma 91,993,113 ’0 t—aoo 12-03 t t—m t X expi-itf(y.x>3dv(x>>} - [Bc