The range of a vector-valued measure

Abstract

Liapounoff, in 1940, proved that the range of a countably additive bounded measure with values in a finite dimensional vector space is compact and, in the nonatomic case, is convex. Later, in 1945, Liapounoff showed, by counterexample, that neither the convexity nor compactness need hold in the infinite dimensional case. The next step was taken by Halmos who in 1948 gave simplified proofs of Liapounoff's results for the finite dimensional case. In 1951, Blackwell [I] considered the case of a measure represented by a finite dimensional vector integral and obtained results similar to those of Liapounoff for these measures. Various versions of Liapounoff's theorem appeared in the 1950's and 1960's, and in 1966, Lindenstrauss [8] gave a very elegant short proof of Liapounoff's earlier result. Finally, in 1968, Olech [9] considered the case of an unbounded measure with range in a finite dimensional vector space. The purpose of this note is to demonstrate that the closure of the range of a measure of bounded variation with values in a Banach space, which is either a reflexive space or a separable dual space, is compact and, in the nonatomic case, is convex. To this end, let Q be a point set and z be a a-field of subsets of U. If X is a Banach space, then an p-valued measure is a countably additive function F defined on 2 with values in X. F is of bounded variation if