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 ) {U^c}(X,E) denote the uniformly continuous functions from X into E which have a precompact range, and let U c ( X , E ) {U^c}(X,E) have the topology of uniform convergence. Let L ( X , F ) L(X,F) be the space of all F-valued measures on X with finite support, and let L ( X , F ) L(X,F) be given the topology of uniform convergence over the uniformly equicontinuous subsets of U c ( X , E ) {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 ) L(X,F) under this topology.