, 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(LΦ∗,LΦ)
if and only if on each order interval
[−ζ,ζ]={ξ:−ζ≤ξ≤ζ}
(
ζ∈LΦ∗
), 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