Convex functions on dual Orlicz spaces

In the dual $L_{{\mathrm{\Phi }}^{\ast }}$ of a ${\mathrm{\Delta }}_2$-Orlicz space $L_{\mathrm{\Phi }}$, 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 $\tau (L_{{\mathrm{\Phi }}^{\ast }},L_{\mathrm{\Phi }})$ if and only if on each order interval $[-\zeta ,\zeta ]=\{\xi :-\zeta \le \xi \le \zeta \}$ ($\zeta \in L_{{\mathrm{\Phi }}^{\ast }}$), 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 $L_{{\mathrm{\Phi }}^{\ast }}$ admits a sequence of forward convex combinations ${\bar{\xi }}_n\in \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}({\xi }_n,{\xi }_{n+1},\mathrm{.}\mathrm{.}\mathrm{.})$ such that ${\sup }_n\mathrm{\mid }{\bar{\xi }}_n\mathrm{\mid }\in L_{{\mathrm{\Phi }}^{\ast }}$ and ${\bar{\xi }}_n$ converges a.s.