A Complete Space of Vector-Valued Measures

Abstract

Let X be a Hausdorff uniform space and E a Fréchet space (or more generally an LF-space) with dual F. Let ${U^c}(X,E)$ denote the uniformly continuous functions from X into E which have a precompact range, and let ${U^c}(X,E)$ have the topology of uniform convergence. Let $L(X,F)$ be the space of all F-valued measures on X with finite support, and let $L(X,F)$ be given the topology of uniform convergence over the uniformly equicontinuous subsets of ${U^c}(X,E)$ having a common precompact range in E. The main result in the paper is a characterization of the completion of $L(X,F)$ under this topology.