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INTEGRATION with VECTOR VALUED MEASURES M. M. Rao

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INTEGRATION with VECTOR VALUED MEASURES M. M. Rao DISCRETE AND CONTINUOUS doi:10.3934/dcds.2013.33.5429 DYNAMICAL SYSTEMS Volume 33, Number 11&12, November & December 2013 pp. 5429{5440 INTEGRATION WITH VECTOR VALUED MEASURES M. M. Rao Unversity of California, Riverside Riverside, CA 92521, USA Dedicated to Jerome A. Goldstein on the occasion of his 70th birthday ABSTRACT. Of the many variations of vector measures, the Fr´echet variation is finite valued but only subadditive. Finding a `controlling' finite measure for these in several cases, it is possible to develop a useful integration of the Bartle-Dunford-Schwartz type for many linear metric spaces. These include the generalized Orlicz spaces, L'(µ), where ' is a concave '-function with applications to stochastic measures Z(·) into various Fr´echet spaces useful in prediction theory. In particular, certain p-stable random measures and a (sub) class of these leading to positive infinitely divisible ones are detailed. 1. Introduction. If (S; S; µ) is a measure space where S is a δ(or σ)-ring of subsets of a space S, and (X ; k · k) is a linear metric space, then a mapping Z : S!X is termed a vector valued measure (or simply a vector measure) if it is σ-additive in the sense that for any disjoint sequence fAn; n ≥ 1g ⊂ S whose union is in S, then one has: X1 [1 Z( n=1An) = Z(An); (1) n=1 the series on the right converging in the metric topology of X . Note that by a classical result, originally due to S. Kakutani for general metric groups and extended by V. L. Klee for abelian groups admitting a metric under which it is complete having an invariant metric, one can always consider the analysis for functionals to be (metric) translation invariant on X . Such a linear space is termed a Fr´echet space hereafter. (See RolewiczP (1972) for details.) If f : S ! R(C) is a simple n function, represented as f = i=1 aiχAi , then define as usual an `integral': Z Xn fdZ = aiZ(Ai \ A); (2 X ;A 2 S) (2) A i=1 where Ai 2 S, disjoint. Observe that the left side is uniquely defined (does not depend on the representation of f above) and so (2) is unambiguous. An extension of it for a more general f which is a pointwise limit of simple functions (by the structure theorem of measurable functions for S) with the σ-additivity of Z, is desired. This means if fn; gn; n ≥ R1 are such that fn ! f gn; n ! 1 so that − ! − ! ! 1 fn gn 0, both pointwise, then A(fn gn)dZ 0 as n must hold under standard conditions. If X is not finite dimensional there is no `Lebesgue type' limit 2010 Mathematics Subject Classification. Primary: 46G10, 60H05; Secondary: 28B05, 28C20. Key words and phrases. Vector measures in Fr´echet spaces, controling measures, independently valued measures, stochastic measures in generalized Orlicz spaces, prediction theory. 5429 5430 M. M. RAO theorem to guarantee this, but one needs it to extend the work for all such functions. This is solved by Bartle-Dunford-Schwartz when (X ; k · k) is a Banach space who showed that there exists a `controlling' finite measure λ for Z on S, implying that Z ≪ λ using the fact that X ∗ separates points of X . But this result on existence of such a λ needs a substitute in the Fr´echet case, covering X = Lp(µ); p ≥ 0. The Vitali and Fr´echet variations are now recalled in the following: Definition 1. If Z :Σ !X is a vector measure, then the Vitali variation, denoted jZj(·) is given by (k · k being the metric in X ): Xn jZj(A) = supf kZ(Ai)k : Ai 2 S(A); disjointg; (3) i=1 where S(A) = fA \ B 2 S; A; B 2 Sg, is the trace of S on A. The Fr´echet variation kZk is introduced, with Ai disjoint again, by: Xn kZk(A) = supfk aiZ(Ai)k : ai 2 C; jaij ≤ 1;Ai 2 S(A)g: (4) i=1 Here on the right k · k stands for the norm or metric of X respectively. It is clear that jZj(A) ≥ kZk(A), and if X is the space of scalars then it is well-known that jµj(A) ≤ C0kµk(A) < 1 holds where C0 = 4, so that both are equivalent. Considering coordinate-wise inequalities one sees that the same equiv- alence statement (with a constant Cn in place of C0 above) holds in n-dimensions, but in infinite dimensions the constant Cn ! 1, and so a different approach is needed. In fact it can be shown that kZk(S) < 1 always, and one may have jZj(S) = 1. However the Vitali variation is σ-additive but the Fr´echet variation is only σ-subadditive. Since X can be a Banach or a Fr´echet space, a finer analysis is needed to define integrals in infinite dimensional X , for instance if X is any of the spaces Lp(P ); p ≥ 0 on a probability triple (Ω; Σ;P ). A classical observation due to I. Kluv´anekimplies that a (vector) measure Z : A!X , a Banach space, where A is a σ-ring actually has its support in a set S0 2 A so that it is effectively defined on the (trace) σ-algebra A(S0) and so a vector measure into a Banach space defined on a σ-ring or algebra makes no difference and hence the words σ-ring and algebra may be used without distinction and this will be done below. Also it follows that a vector measure Z : S!X , a Banach space, defines a signed measure x∗(Z); and hence is bounded for each x∗ 2 X ∗, the dual k k space, implying that (by the uniform boundedness theorem) supkx∗∥≤1 Z(A) < 1;A 2 S which shows that the Fr´echet variation of a vector measure is always finite on a σ-ring or -algebra. It is seen that this (Fre´chet)variation is σ-subadditive but generally not countably additive in infinite dimensional (Banach) spaces. If the range space of the vector measure is of the form B(X ; Y), the vector space of bounded linear operators, which is a Banach space under the uniform (operator) norm the above definitions of Vitali and Fr´echet variations apply, but using the special structure of the range space one may consider an analog of Fr´echet's variation (suggested by Dinculeanu) replacing the scalars of (4) by some vectors as follows. Observe that in Definition 1 if the multiplying scalars ai are replaced by the vectors aix; x 2 X and the range space is Z = B(X ; Y) so that kZk(Ω) is still ∗ finite, but if the range Y of Z, is the adjoint space X , then changing ai to xi; i = INTEGRATION WITH VECTOR VALUED MEASURES 5431 1; ··· ; n alters the situation and the resulting Fr´echet and Vitali variations (finite or not) become equal, so that jZj(Ω) = jjZjj(Ω). This is seen as follows. Since kZk(A) ≤ jZ(A)j;A 2 Σ is always valid, for the opposite inequality consider for " > 0 and integer n, with the Hahn-Banach theorem, an xi 2 X ; i = 1; ··· ; n such 2 S k k ≤ h i " that for disjoint Ai (A), one has Z(Ai) xi;Z(Ai) + n , so that Xn Xn kZ(Ai)k ≤ hxi;Z(Ai)i + " ≤ kZk(A) + ": (5) i=1 i=1 This implies, on taking the supremum suitably, that kµk(·) = jµj(·) as asserted. It will now be shown that this strengthening excludes one of the most important applications of the subject, namely the Brownian motion (BM). This (known) result will be sketched here for a clarification of the ensuing discussion. 2 Let fXt; 0 ≤ t ≤ 1g be a BM-process with variance parameter σ = 1. Let Yin = Xi=2n −X(i−1)=2n be the increments which are independent Gaussian random variables with means zero and variances 2−n. Then a simple computation shows that P n P n 2 2 −n+1 j 2 2 − j ≥ V ar( i=1 Yin) = 2 . If An;" = [ i=1 Yin 1 "], then the Gaussian property n 2 of the Xt easily implies that P (An;") ≤ 2=2 " using the Cebyˇshevinequality.ˇ Hence by the Borel-Cantelli lemma P (An;"; i:o) = 0 where i.o. stands for infinitely often which is `limsup' of sets. Thus the complementary event satisfies X2n j 2 − j ≥ P [ Yin 1 < "; n n0] = 1: (6) i=1 But this means that, on using another key property of the BM namely the almost sure uniform continuity of its paths on each interval, one has the following inequality with probability one: X2n 2 1 = lim [Xi2−n − X(i−1)2−n ] n!1 i=1 ≤ lim max jXi2−n − X(i−1)2−n j n!1 1≤i≤2n X2n × jXi2−n − X(i−1)2−n j: i=1 Note that the first factor tends a.e. to zero (the uniform continuity of the BM process) and consequently the second factor must tend to infinity, with probability n n one. This implies that the set function Z defined by Z([(i−1)=2 ; i=2 ]) = Xi2−n − X(i−1)2−n , which is seen extendable to a vector measure on the Borel σ-algebra of [0; 1] denoted again by Z cannot have a finite (Vitali) variation. But note that this Z actually takes values in a Hilbert space and the thus defined (vector) measure has (as Banach valued) a finite Fr´echet variation.
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