On the C-property and $w^*$-representations of risk measures

We identify a large class of Orlicz spaces $X$ for which the topology $σ(X,X_n^\sim)$ fails the C-property introduced in [7]. We also establish a variant of the C-property and use it to prove a $w^*$-representation theorem for proper convex increasing functionals on dual Banach lattices that satisfy a suitable version of Delbaen's Fatou property. Our results apply, in particular, to risk measures on all Orlicz spaces over $[0,1]$ which is not $L_1[0,1]$.

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