Sequence in the range of a vector- value 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 dimen- sional vector integral and obtained results similar to those of Lia- pounoff 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 non- atomic 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 addi- tive function F defined on 2 with values in X. F is of bounded varia- tion if var(F)(Q) =sup E ll F(En)|| < a W n where the supremum is taken over all partitions r = { En }=C2 consisting of a finite collection of disjoint sets in z whose union is U. A set EE2 is an atom of F if F(E) $0 and E'C2, E'CE imply F(E') = 0 or F(E') = F(E). F is nonatomic if F has no atoms. The following theorem is the main result of this note. THEOREM 1. Let X be a Banach space which is either a reflexive space or a separable dual space. If F: 2;- X is a measure of bounded variation, then the range of F is a precompact set in the norm topology of t. More- over, if F is nonatomic, then the closure of the range of F is compact and convex.