Convex functions on dual Orlicz spaces

Abstract

In the dual $L_{Φ^*}$ of a $Δ_2$-Orlicz space $L_Φ$, 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 $τ(L_{Φ^*},L_Φ)$ if and only if on each order interval $[-ζ,ζ]=\{ξ: -ζ\leq ξ\leqζ\}$ ($ζ\in L_{Φ^*}$), it is lower semicontinuous (resp. continuous) for the topology of convergence in probability. For this purpose, we provide the following Komlós type result: every norm bounded sequence $(ξ_n)_n$ in $L_{Φ^*}$ admits a sequence of forward convex combinations $\barξ_n\in\mathrm{conv}(ξ_n,ξ_{n+1},...)$ such that $\sup_n|\barξ_n|\in L_{Φ^*}$ and $\barξ_n$ converges a.s.