Closedness of convex sets in Orlicz spaces with applications to dual representation of risk measures
Volume 249 / 2019
Abstract
We study various types of closedness of convex sets in an Orlicz space $L^\varPhi$ and its heart $H^\varPhi$ and their relations to a natural version of the Krein–Šmulian property. Let $L^\varPsi$ be the conjugate Orlicz space and $H^\varPsi$ be the heart of $L^\varPsi$. Precisely, we show that the following statements are equivalent:
(i) Every order closed convex set in $L^\varPhi$ is $\sigma(L^\varPhi,L^\varPsi)$-closed.
(ii) Every boundedly a.s. closed convex set in $H^\varPhi$ is $\sigma(H^\varPhi,H^\varPsi)$-closed.
(iii) Every $\sigma(L^\varPhi,L^\varPsi)$-sequentially closed convex set in $L^\varPhi$ is $\sigma(L^\varPhi,L^\varPsi)$-closed.
(iv) Every $\sigma(H^\varPhi,H^\varPsi)$-sequentially closed convex set in $H^\varPhi$ is $\sigma(H^\varPhi,H^\varPsi)$-closed.
(v) $\sigma(L^\varPhi,L^\varPsi)$ (respectively, $\sigma(H^\varPhi,H^\varPsi)$) has the Krein–Šmulian property.
(vi) Either $\varPhi$ or its conjugate $\varPsi$ satisfies the $\Delta_2$-condition.
The implication (i)$\Rightarrow$(vi) solves an open question raised by Owari (2014) and has applications in the dual representation theory of risk measures.
