Vector measures and Mackey topologies

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