Convex functions on dual Orlicz spaces

Abstract

In the dual $$L_{\varPhi ^*}$$ of a $$\varDelta _2$$ -Orlicz space $$L_\varPhi $$ , 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 $$\tau (L_{\varPhi ^*},L_\varPhi )$$ if and only if on each order interval $$[-\zeta ,\zeta ]=\{\xi : -\zeta \le \xi \le \zeta \}$$ ( $$\zeta \in L_{\varPhi ^*}$$ ), it is lower semicontinuous (resp. continuous) for the topology of convergence in probability. For this purpose, we provide the following Komlos type result: every norm bounded sequence $$(\xi _n)_n$$ in $$L_{\varPhi ^*}$$ admits a sequence of forward convex combinations $${{\bar{\xi }}}_n\in \text {conv}(\xi _n,\xi _{n+1},\ldots )$$ such that $$\sup _n|{\bar{\xi }}_n|\in L_{\varPhi ^*}$$ and $${\bar{\xi }}_n$$ converges a.s.