On closedness of convex sets in Banach function spaces

Made Tantrawan, Denny H. Leung

Published 2018-08-21Version 1

Let $\mathcal{X}$ be a Banach function space. A well-known problem arising from theory of risk measures asks when the order closedness of a convex set in $\mathcal{X}$ implies the closedness with respect to the topology $\sigma(\mathcal{X},\mathcal{X}_n^\sim)$ where $\mathcal{X}_n^\sim$ is the order continuous dual of $\mathcal{X}$. In this paper, we give an answer to the problem for a large class of Banach function spaces. We show that under the Fatou property and the subsequence splitting property, every order closed convex set in $\mathcal{X}$ is $\sigma(\mathcal{X},\mathcal{X}_n^\sim)$-closed if and only if either $\mathcal{X}$ or the norm dual $\mathcal{X}^*$ of $\mathcal{X}$ is order continuous. In addition, we also give a characterization of $\mathcal{X}$ for which the order closedness of a convex set in $\mathcal{X}$ is equivalent to the closedness with respect to the topology $\sigma(\mathcal{X},\mathcal{X}_{uo}^\sim)$ where $\mathcal{X}_{uo}^\sim$ is the unbounded order continuous dual of $\mathcal{X}$.