Vector measures and Mackey topologies

摘要

Let Σ be a σ-algebra of subsets of a non-empty set Ω. Let B(Σ) be the Banach lattice of all bounded Σ-measurable real-valued functions defined on Ω, equipped with the natural Mackey topology τ (B(Σ), ca(Σ)). We study (τ (B(Σ), ca(Σ)), ξ )-continuous linear operators from B(Σ) to a quasicomplete locally convex space (E, ξ ). A generalized Nikodym convergence theorem and a Vitali–Hahn–Saks type theorem for operators on B(Σ) are obtained. It is shown that the space (B(Σ), τ (B(Σ), ca(Σ))) has the strict Dunford–Pettis property. Moreover, a Yosida–Hewitt type decomposition for weakly compact operators on B(Σ) is given.