An extension of a result of Liapounoff on the range of a vector measure

Abstract

Liapounoff2 established in 1940 that the range of a countably additive finite measure with values in a finite-dimensional real vector space is bounded and closed and in the nonatomic case convex. A simplified proof of this result was given by Halmos' in 1948. The aim of the present paper is to extend this result to the following case. Let pit, 1 0 there exist a a>0 such that g*(E) <e for all ECS so that v*(E) <8. {Ei}, i= 1, 2, . , k, is said to be a decomposition of F if the Es are pairwise disjoint measurable subsets of X and UiEi= F. Let pit,