Before 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 .
After revision: The theory of theory of locally L^0-convex modules some results from classical convex analysis , namely we provide randomized versions of Mazur lemma and Krein-\v{S .