Topological Vector Space Valued Measures on Topological Spaces

If X is a compact Hausdorff space space, E is a complete Hausdorff topological vector space and μ : (C(X),ll.ll) → E a linear continuous exhaustive mapping, we rst give a different proof that there is then a unique reqular, L∞-bounded, exhaustive E-valued Borel measure μ on X such that μ(f) = ∫ fdμ, ∀f ∈ C(X). Then we consider X to be a completely regular Hausdorff space and prove the extension of Alexanderov's theorem: X is a completely regular Hausdorff space and μ : C_b(X) → E a linear, continuos, exhaustive mapping and F is the algebra generated by zero-sets in X. Then there exist a unique nitely additive, exhaustive measure ν : F → E such that (i) ν is L∞-bounded i.e. the absolute convex hull of ν(F) (Γ(ν(F))) is bounded in E; (ii) ν is inner regular by zero-sets and outer regular by positive-sets; (iii) ∫ fdν = µ(f), ∀f ∈ C_b(X).