Convex functions on dual Orlicz spaces

Abstract

In the dual $L_{\Phi^*}$ of a $\Delta_2$-Orlicz space $L_\Phi$, that we call a dual Orlicz space, we show that a proper (resp. finite) convex function is lower semicontinuous (resp. continuous) for the Mackey topology $\tau(L_{\Phi^*},L_\Phi)$ if and only if on each order interval $[-\zeta,\zeta]=\{\xi: -\zeta\leq \xi\leq\zeta\}$ ($\zeta\in L_{\Phi^*}$), it is lower semicontinuous (resp. continuous) for the topology of convergence in probability. For this purpose, we provide the following Koml\'os type result: every norm bounded sequence $(\xi_n)_n$ in $L_{\Phi^*}$ admits a sequence of forward convex combinations $\bar\xi_n\in\mathrm{conv}(\xi_n,\xi_{n+1},...)$ such that $\sup_n|\bar\xi_n|\in L_{\Phi^*}$ and $\bar\xi_n$ converges a.s.