Vector Measures on Topological Spaces

Abstract

Abstract 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 𝐵(𝑋) and 𝐵 0 (𝑋) be the classes of Borel and Baire subsets of 𝑋. We study the spaces 𝑀 𝑡 (𝑋,𝐸), 𝑀 τ (𝑋,𝐸), 𝑀 σ (𝑋,𝐸) of tight, τ -smooth, σ -smooth, 𝐸-valued Borel and Baire measures on 𝑋. Using strict topologies, we prove some measure representation theorems of linear operators between 𝐶 𝑏 (𝑋) and 𝐸 and then prove some convergence theorems about integrable functions. Also, the Alexandrov's theorem is extended to the vector case and a representation theorem about the order-bounded, scalar-valued, linear maps from 𝐶(𝑋) is generalized to the vector-valued linear maps.