Discrete measures and strict topologies

Abstract

Let X be a topological Hausdorff space and Bo denote the σ-algebra of Borel sets in X. Then the Banach algebra B(Bo) of all bounded Bo-measurable complex-valued functions on X is a left Banach co(X)-module. The strict topology βd induced on B(Bo) by co(X) coincides with the mixed topology γ[τp,τu], where τu and τp denote the topologies of uniform convergence and pointwise convergence in B(Bo), respectively. A characterization of relative σ(Md(X),B(Bo))-compactness in the Banach space Md(X) of all complex-valued discrete measures is given. We obtain that (B(Bo),βd) is a strongly Mackey space, and hence βd coincides with the Mackey topology τ(B(Bo),Md(X)). It is shown that the space Md(X) has the Schur property, that is, every σ(Md(X),B(Bo))-convergent sequence in Md(X) is norm convergent. Moreover, we study Banach space-valued discrete measures. In particular, we derive a Nikodym type theorem on the setwise sequential convergence of Banach space-valued discrete measures.