Abstract. Motivated by the randomized version of the classical Bolzano–Weierstrass theorem, in this paper we ?rst introduce the notion of a random sequentially compact set in a random normed module and develop the related theory systematically. From these developments, we prove the corresponding Schauder ?xed point theorem: let E be a random normed module and G a random sequentially compact L0-convex set of E, then every σ-stable continuous mapping from G to G has a ?xed point, which uni?es all the previous random generalizations of the Schauder ?xed point theorem. As one of the applications of the theorem, we prove the existence of Nash equilibrium points in the context of conditional information. It should be pointed out that the main challenge in this paper lies in overcoming noncompactness since a random sequentially compact set is generally noncompact.
