Closedness of convex sets in Orlicz spaces with applications to dual representation of risk measures
Tom 249 / 2019
Streszczenie
We study various types of closedness of convex sets in an Orlicz space 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.
