Remarks on ranges of charges on $\sigma$-fields

Abstract

In this paper we present the following results about ranges of charges on a $\sigma$-field $\mathcal{A}$ of subsets of a set $X$.

• (1) For any bounded charge the range is either a finite set or contains a perfect set, contrary to an assertion made by Sobezyk and Hammer [8].

• (2) If $\mu_{1},\mu_{2},\ldots,\mu_{n}$ are strongly continuous bounded charges on $\mathcal{A}$ then the range of the vector measure $(\mu_{1},\mu_{2},\ldots,\mu_{n})$ is a convex set and need not be closed.

• (3) There is a positive bounded charge, on any infinite $\sigma$-field, whose range is neither Lebesgue measurable nor has the property of Baire.