On the extension of the Namioka-Klee theorem and on the Fatou property for Risk Measures

This paper has been motivated by general considerations on the topic of Risk Measures, which essentially are convex monotone maps defined on spaces of random variables, possibly with the so-called Fatou property.
We show first that the celebrated Namioka-Klee theorem for linear, positive functionals holds also for convex monotone maps π on Frechet lattices.
It is well-known among the specialists that the Fatou property for risk measures on the space of bounded random variables enables a simplified dual representation, via probability measures only. The Fatou property in a general framework of lattices is nothing but the lower order semicontinuity property for π. Our second goal is thus to prove that a similar simplified dual representation holds also for order lower semicontinuous, convex and monotone functionals π defined on more general spaces (locally convex Frechet lattices). To this end, we identify a link between the topology and the order structure - the C-property - that enables the simplified representation. One main application of these results leads to the study of convex risk measures defined on Orlicz spaces and of their dual representation.