a vector form of aleksandrov’s theorem for normal topological spaces

Let x be a hausdorff normal topological space, e a quasi-complete locally convex space, c(x) (resp. cb(x)) the space of all (resp. all, bounded),scalar-valued continuous functions on x, and f the algebra generated by the closed subset of x. the following form of aleksandrov’s theorem is proved: supposeμ: cb(x) → e a weakly compact linear mapping. then there exists a unique finitely additive, exhaustive measure ν : f → e such that(i) ν is inner regular by closed sets and outer regular by open sets;(ii) ∫ fdν = μ(f), ∀f ∈ cb(x).when x is also countably paracompact some additional results are also proved.