Absolutely continuous operators on function spaces and vector measures

Let (Omega, I pound, mu) be a finite atomless measure space, and let E be an ideal of L (0)(mu) such that . We study absolutely continuous linear operators from E to a locally convex Hausdorff space . Moreover, we examine the relationships between mu-absolutely continuous vector measures m : I pound -> X and the corresponding integration operators T (m) : L (a)(mu) -> X. In particular, we characterize relatively compact sets in ca (mu) (I pound, X) (= the space of all mu-absolutely continuous measures m : I pound -> X) for the topology of simple convergence in terms of the topological properties of the corresponding set of absolutely continuous operators. We derive a generalized Vitali-Hahn-Saks type theorem for absolutely continuous operators T : L (a)(mu) -> X.