On the C-property and $w^{\ast }$-representations of risk measures

We identify a large class of Orlicz spaces $X$ for which the topology $\sigma (X,X_n^{\sim })$ fails the C-property introduced in [7]. We also establish a variant of the C-property and use it to prove a $w^{\ast }$-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 $L_1[0,1]$.