Mackey Topologies on Vector-Valued Function Spaces

Abstract

Let $\,E\,$ be an ideal of $\,L^{0}\,$ over a $\,\sigma$-finite measure space $\,(\Omega,\Sigma,\mu)$, and let $\,(X,\|\cdot\|_X)\,$ 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 $\,f: \Omega\to X\,$ and consisting of all those $\,f\in L^{0}(X)\,$ for which the scalar function $\,\|f(\cdot)\|_X\,$ belongs to $\,E$. Let $\,E(X)_n^{\sim}\,$ stand for the order continuous dual of $\,E(X)$. We examine the Mackey topology $\,\tau(E(X),E(X)_n^{\sim})$ in case when it is locally solid. It is shown that $\,\tau(E(X),E(X)_n^{\sim})\,$ is the finest Hausdorff locally convex-solid topology on $\,E(X)\,$ with the Lebesgue property. We obtain that the space $\,(E(X),\tau(E(X), E(X)_n^{\sim}))\,$ is complete and sequentially barreled whenever $\,E\,$ is perfect. As an application, we obtain the Hahn-Vitali-Saks type theorem for sequences in $\,E(X)_n^{\sim}$. In particular, we consider the Mackey topology $\,\tau(L^{\Phi}(X), L^{\Phi}(X)_n^{\sim})\,$ on Orlicz-Bochner spaces $\,L^{\Phi}(X)$. We show that the space $\,(L^{\Phi}(X), \tau(L^{\Phi}(X), L^{\Phi}(X)_n^{\sim}))\,$ is complete iff $\,L^{\Phi}\,$ is perfect. Moreover, it is shown that the Mackey topology $\,\tau(L^{\infty}(X), L^{\infty}(X)_n^{\sim})\,$ is a mixed topology.