Abstract 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 ess sup ω ∈ 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 ( bvca μ ( Σ , X ∗ ) , L ∞ ( μ , X ) ) -sequentially compact sets in the space bvca μ ( Σ , X ∗ ) of all countably additive measures ν : Σ → X ∗ of bounded variation with ν ( A ) = 0 if μ ( A ) = 0 .