A version of Strassen’s Theorem for vector-valued measures

Abstract

A formulation of is given for measures taking values in a Banach lattice. The main result (Theorem 2) corrects earlier work of the second author. In joint work of M. Marz and the second author [5, 3.7], a version of the theorem known in probability theory as Strassen's Theorem (see [8], [2, 11.6]) was generalized to the context of measures assuming values in a reflexive Banach lattice. In [7, 3.2], the second author announced an extension of the result to all order-complete Banach lattices. However, the proof given in [7, p.816] contains an error, leaving the validity of the result an open question. In 2 below, a result of this general type is proved for measures taking values in Banach lattices of a certain type: the so-called KB-spaces. The KBspaces occupy a position between the reflexive and the complete Banach lattices (reflexive =X KB =* complete), so that 2 is a generalization of [5, 3.7], but is not as strong as what was asserted in [7]. Our technique makes use of a Strassen-type result (given below as 1 and proved in [3]) for finitely additive measures with values in a complete Banach lattice. As a general reference for basic facts about vector measures, we recommend the text of Diestel and Uhl [1]. For Banach lattices, see [4], [6]. If A and B are fields on sets X and Y, respectively, then A x B is the field on X x Y generated by all rectangles E x F for E e A and F e S. The following result is 2.1 in [3]. It replaces the imprecisely stated and incompletely proved 2.2 of [7]. 1. Let A and B be countable fields on sets X and Y respectively and let A: G+ and i : B -G+ be finitely additive measures taking values in the positive cone of a divisible, a-complete, partially ordered group G. We assume that p(X) _= v(Y) = a for some a E G+. Let S be an arbitrary subset of X x Y and let C be the field on X x Y generated by S and the sets in A x B. For an element v E G with 0 < v < a, we consider the following conditions: i) There is a finitely additive measure p: C -+ G+ such that p(E x Y) = /1(E) and p(X x F) = v(F) for all E e A and F E B (i.e. p has marginals ,u and v) and such that p(S) = v. ii) Whenever E x F C S for E e A and F E B, then p(E) + v(F) < c + v. Received by the editors July 24, 1996 and, in revised form, October 28, 1996. 1991 Mathematics Subject Classification. Primary 28B05; Secondary 30C62, 46B42. (D1998 American Mathematical Society