Complete Spaces of p-adic Measures

Abstract

Let $\mathbb K$ be a complete non-Archimedean valued field and let $C(X,E)$ be the space of all continuous functions from a zero-dimensional Hausdorff topological space $X$ to a non-Archimedean Hausdorff locally convex space $E$. We will denote by $C_{b}(X,E)$ (resp. by $C_{rc}(X,E)$) the space of all $f\in C(X,E)$ for which $f(X)$ is a bounded (resp. relatively compact) subset of $E$. The dual space of $C_{rc}(X,E)$, under the topology $t_{u}$ of uniform convergence, is a space $M(X,E')$ of finitely-additive $E'$-valued measures on the algebra $K(X)$ of all clopen , i.e. both closed and open, subsets of $X$. Some subspaces of $M(X,E')$ turn out to be the duals of $C(X,E)$ or of $C_{b}(X,E)$ under certain locally convex topologies.\\ In this paper we continue with the investigation of certain subspaces of $M(X,E')$. Among other results we show that, if $E$ is a polar Fréchet space, then :\\ 1. The space $\mathcal{M}_{\theta_{o}}(X,E')$, of all $m\in M(X,E')$ for which the support of the corresponding measure $m^{\beta_{o}}$, on the Banaschewski compactification of $X$, is contained in the $\theta_{o}$-repletion of $X$, is complete under the topology of uniform convergence on the family $\mathcal{E}$ of all equicontinuous subsets $B$ of $C(X,E)$ for which $B(x)$ is a compactoid subset of $E$ for all $x\in X$.\\ 2. The space $M_{bs}(X,E')$, of all the so called strongly-separable members of $M(X,E')$ is complete under the topology of uniform convergence on the family of all uniformly bounded members of $\mathcal{E}$.\\ 3. The space $M_{s}(X,E')$ of all $m\in M(X,E')$, for which $ms$ is separable for all $s\in E$, is complete under the topology of uniform convergence on the family of all $B \in \mathcal{E}$ for which the set $B(X)$ is compactoid.