Mackey topologies on vector-valued function spaces

Let E be an ideal of L-0 over a sigma-finite measure space (Q, E, p), and let (X, // center dot //x) be a real Banach space. Let E(X) be a subspace of the space L-0(X) of mu-equivalence classes of all strongly Sigma-measurable functions f : Omega -> X and consisting of all those f epsilon L-0(X) for which the scalar function //f (center dot)//x belongs to E. Let E(X)(n)(similar to) stand for the order continuous dual of E(X). We examine the Mackey topology tau(E(X), E(X)(n)(similar to) in case when it is locally solid. It is shown that T(E(X), E(X)(n)(similar to) is the finest Hausdorff locally convex-solid topology on E(X) with the Lebesgue property. We obtain that the space (E(X), tau(E(X), E(X)(n)(similar to))) is complete and sequentially barreled whenever E is perfect. As an application, we obtain the Hahn-Vitali-Saks type theorem for sequences in E(X)(n)(similar to). In particular, we consider the Mackey topology tau(L-phi(X), L-phi(X)(n)(similar to)) on Orlicz-Bochner spaces L-phi(X). We show that the space (L-phi(X), tau(L-phi(X), L-phi(X)(n)(similar to))) is complete iff L-phi is perfect. Moreover, it is shown that the Mackey topology tau(L-infinity(X),L-infinity(X)(n)(similar to)) is a mixed topology.