Before revision: The purpose of this paper is to establish a robust representation theorem for conditional risk measures by using a module-based convex analysis, where risk measures are defined on a L^\infty-type module. We define and study a Fatou property for this kind of risk measures, which is a generalization of the already known Fatou property for static risk measures. In order to prove this robust representation theorem we provide a modular version of Krein-Smulian theorem .
After revision: The theory of theory of locally L^0-convex modules was introduced as the analytic basis for conditional L^0-convex risk measures. In this paper we first give some preliminaries of this theory and discuss about two kinds of countable concatenation properties. Second we extend to this framework some results from classical convex analysis, namely we provide randomized versions of Mazur lemma and Krein-\v{S .