On robust representation of conditional risk measures on a $L^\infty$-type module

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.