Let (Ω, Σ,μ) be a finite atomless measure space, and let E be an ideal of L 0(μ) such that ({L^infty(mu) subset E subset L^1(mu)}). We study absolutely continuous linear operators from E to a locally convex Hausdorff space ({(X, xi)}). Moreover, we examine the relationships betweenμ-absolutely continuous vector measures m : Σ → X and the corresponding integration operators T m : L ∞(μ) → X. In particular, we characterize relatively compact sets ({mathcal{M}}) in ca μ (Σ, X) (= the space of allμ-absolutely continuous measures m : Σ → X) for the topology ({mathcal{T}_s}) of simple convergence in terms of the topological properties of the corresponding set ({{T_m : m in mathcal{M}}}) of absolutely continuous operators. We derive a generalized Vitali–Hahn–Saks type theorem for absolutely continuous operators T : L ∞(μ) → X.
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