Convex functions on dual Orlicz spaces

2016/11/18 by Delbaen, Freddy, Owari, Keita
#FOS: Economics and business #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Finance (q-fin.MF) #Probability (math.PR)

paper · doi:10.48550/arxiv.1611.06218

Abstract

In the dual L_Φ^* of a Δ₂-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≤ ξ≤ζ\ (ζ∈ 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 ξ_n\inconv(ξ_n,ξ_n+1,...) such that sup_n|ξ_n|∈ L_Φ^* and ξ_n converges a.s.

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