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: Suppose μ: 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.