Convex functions on dual Orlicz spaces

Abstract

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

12 pages; added a new characterisation of the ${\Delta }_2$-Orlicz spaces as well as a few minor changes

References in corpus (1)

• On the C-property and $w^{\ast }$-representations of risk measures

Cited by in corpus (3)

• Robust utility maximization in markets with transaction costs
• Surplus-invariant risk measures
• Fatou Property, representations, and extensions of law-invariant risk measures on general Orlicz spaces