Lebesgue property for convex risk measures on Orlicz spaces

We present a robust representation theorem for monetary convex risk measures rho : X -> R such that lim(n) rho(X-n) = rho(X) whenever (X-n) almost surely converges to X, vertical bar X-n vertical bar <= Z is an element of X, for all n is an element of N and X is an arbitrary Orlicz space. The separable (L-1, L-infinity) case of Jouini et al. (Adv Math Econ 9:49-71, 2006), as well as the non-separable version of Delbaen [7], are contained as a particular case here. We answer a natural question posed by Biagini and Fritelli [2]. Our approach is based on the study for unbounded sets, as the epigraph of a given penalty function associated with rho, of the celebrated weak compactness Theorem due to James (Isr J Math 13:289-300, 1972).