Closedness of convex sets in Orlicz spaces with applications to dual representation of risk measures

Abstract

Let $(Φ,Ψ)$ be a conjugate pair of Orlicz functions. A set in the Orlicz space $L^Φ$ is said to be order closed if it is closed with respect to dominated convergence of sequences of functions. A well known problem arising from the theory of risk measures in financial mathematics asks whether order closedness of a convex set in $L^Φ$ characterizes closedness with respect to the topology $σ(L^Φ,L^Ψ)$. (See [26, p.3585].) In this paper, we show that for a norm bounded convex set in $L^Φ$, order closedness and $σ(L^Φ,L^Ψ)$-closedness are indeed equivalent. In general, however, coincidence of order closedness and $σ(L^Φ,L^Ψ)$-closedness of convex sets in $L^Φ$ is equivalent to the validity of the Krein-Smulian Theorem for the topology $σ(L^Φ,L^Ψ)$; that is, a convex set is $σ(L^Φ,L^Ψ)$-closed if and only if it is closed with respect to the bounded-$σ(L^Φ,L^Ψ)$ topology. As a result, we show that order closedness and $σ(L^Φ,L^Ψ)$-closedness of convex sets in $L^Φ$ are equivalent if and only if either $Φ$ or $Ψ$ satisfies the $Δ_2$-condition. Using this, we prove the surprising result that: \emph{If (and only if) $Φ$ and $Ψ$ both fail the $Δ_2$-condition, then there exists a coherent risk measure on $L^Φ$ that has the Fatou property but fails the Fenchel-Moreau dual representation with respect to the dual pair $(L^Φ, L^Ψ)$}. A similar analysis is carried out for the dual pair of Orlicz hearts $(H^Φ,H^Ψ)$.