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On convex functions on the duals of $\Delta_2$-Orlicz spaces. (arXiv:1611.06218v1 [math.FA])

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In the dual $L^{\Phi^*}$ of a $\Delta_2$-Orlicz space $L^\Phi$, 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.


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