Operator measures and integration operators

Let (Ω,Σ,μ) be a finite complete measure space and (X,‖⋅‖X) be a Banach space with the Banach dual X∗. Let L∞(μ,X) denote the space of all μ-measurable functions f:Ω→X such that esssupω∈Ω‖f(ω)‖X<∞. We study the problem of the integral representation of some natural classes of linear operators from L∞(μ,X) to a Banach space with respect to the corresponding operator measures. We characterize relatively σ(bvcaμ(Σ,X∗),L∞(μ,X))-sequentially compact sets in the space bvcaμ(Σ,X∗) of all countably additive measures ν:Σ→X∗ of bounded variation with ν(A)=0 if μ(A)=0.