Strassen's Theorem for Vector Measures

A type of Strassen's Theorem for measures taking values in the positive cone of a Banach lattice is proved. An application is given to metrics for convergence of vector measures. 1. Preliminaries: vector measures Let y be a field of subsets of a set X, and let (B, || • ||) be a Banach space. (All vector spaces we consider are assumed to have real scalars.) Then a(Sr, B) is the set of all additive set functions V : Sf -> B, i.e., V(EX U E2) = V(EX) + V(E2) for disjoint Ex and E2 in F. The elements of a(3r, B) we call charges. Also, ca(^", B) is the set of all countably additive charges in a(£F, B); we call such charges vector measures. Generally, we follow the notation and conventions of Dunford and Schwartz [4] or Diestel and Uhl [2]. Let (B, || • ||) be a Banach space with dual space B*. If F G ca(<^\ B) and tp £ B*, then (p(V) = q> o V is a finite signed measure on & with total variation \tp(V)\. The semivariation of V is the set function ||K||: y -» R defined by \\V\\(E) = suo{\ 0}. Let (X, f?) be a measurable space and V : SF —> B+ a vector measure taking values in B+. Then, for each E £ £F, we have H^IKi?) = ll^(£)l|. Proof. Let tp £ B* be a functional with ||r?|| < 1. Then |H| = ||H|| < 1, so that \tp(V)\(E) = \tp+(V) <p-(V)\(E) < \<p+(V)\(E) + \(p-(V)\(E) = <p+(V(E)) + <p-(V(E)) = \<p\(V(E)) < UHU \\V(E)\\ < \\V(E)\\. Thus \(p(V)\(E) < \\V(E)\\ for each such tp, so that \\V\\(E) < \\V(E)\\. The converse inequality always holds. D Received by the editors February 24, 1993. 1991 Mathematics Subject Classification. Primary 28B05; Secondary 46B42. © 1994 American Mathematical Society 0002-9939/94 $1.00+ $.25 per page