Vector measures on topological spaces. (English) Zbl 1154.46025

Summary: Let \(X\) be a completely regular Hausdorff space, \(E\) a quasi-complete locally convex space, \(C(X) \) (resp., \(C_{b}(X)\)) the space of all (resp., all bounded), scalar-valued continuous functions on \(X\), and \({\mathcal B}(X) \) and \({\mathcal B}_{0}(X) \) be the classes of Borel and Baire subsets of \(X\). We study the spaces \( M_{t}(X, E)\), \( M_{\tau}(X, E)\), \(M_{\sigma}(X, E)\) of tight, \( \tau\)-smooth, \(\sigma\)-smooth, \(E\)-valued Borel and Baire measures on \(X\). Using strict topologies, we prove some measure representation theorems for linear operators between \( C_{b}(X)\) and \(E\) and then prove some convergence theorems about integrable functions. Also, Alexandrov’s theorem is extended to the vector case and a representation theorem for order-bounded, scalar-valued, linear maps from \(C(X)\) is generalized to vector-valued linear maps.


46G10 Vector-valued measures and integration
28B05 Vector-valued set functions, measures and integrals
28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
46E27 Spaces of measures