The boundary of the range of a vector measure

Abstract

Let (X, S) be a measure space, /aj (i = I , n) signed measures on (X, S). Then , = (j1A, * * *, jUn) is a (n-dimensional) vector measure on (X, S), u is finite and purely nonatomic if every ,A is finite and purely nonatomic, respectively. Consider the range of a finite ndimensional vector measure as a subset of the n-dimensional Euclidean space En. A. Liapounoff [4] and P. R. Halmos [2] have shown: (1) The range of a finite vector measure is closed, (2) the range of a finite and purely nonatomic vector measure is convex. For any (infinite) vector measure IA call R= {Iu(M): MES and ,u(M) finite } thefinite range of IA. Then it is an immediate consequence of (2) that (3) the finite range of a purely nonatomic vector measure is convex. Two simple examples due to R. Borges [1] however show that there are purely nonatomic as well as purely atomic vector measures the finite range of which is not closed: (a) S is the a-ring of the one-dimensional Lebesgue sets, I,u the Lebesgue measure and Iu2(M) =fM exp( -z2)dz, MES. The positive