Spaces of vector measures

Abstract:

Let ${C_{rc}} = {C_{rc}}(X,E)$ denote the space of all continuous functions $f$, from a completely regular Hausdorff space $X$ into a locally convex space $E$, for which $f(X)$ is relatively compact. As it is shown in [8], the uniform dual ${C’_{rc}}$ of ${C_{rc}}$ can be identified with a space $M(B,E’)$ of $E’$-valued measures defined on the algebra of subsets of $X$ generated by the zero sets. In this paper the subspaces of all $\sigma$-additive and all $\tau$-additive members of $M(B,E’)$ are studied. Two locally convex topologies $\beta$ and ${\beta _1}$ are considered on ${C_{rc}}$. They yield as dual spaces the spaces of all $\tau$-additive and all $\sigma$-additive members of $M(B,E’)$ respectively. In case $E$ is a locally convex lattice, the $\sigma$-additive and $\tau$-additive members of $M(B,E’)$ correspond to the $\sigma$-additive and $\tau$-additive members of ${C_{rc}}$ respectively.