Based on the recently developed theory of σ-stable sets and stable compactness, we first establish the random Markov-Kakutani fixed point theorem in a random locally convex module: let (E,P) be a random locally convex module and G be a nonempty stably compact L⁰-convex subset of E, then every commutative family of T_c(P_cc)-continuous L⁰-affine mappings from G to G has a common fixed point, where P_cc is the σ-stable hull of P and T_c(P_cc) is the locally L⁰-convex topology induced by P_cc. Second, we prove that the random Markov-Kakutani fixed point theorem implies the algebraic form of the known random Hahn-Banach theorem. Finally, we establish a more general strict separation theorem in a random locally convex module, which provides not only a more general geometric form of the random Hahn-Banach theorem but also another proof for the random Markov-Kakutani fixed point theorem. Therefore, as a byproduct, the work of this paper also shows that the algebraic and geometric forms of the random Hahn-Banach theorem are equivalent. It should be pointed out that the main challenge in this paper lies in overcoming noncompactness since a stably compact set is generally noncompact.

This work was supported by the National Natural Science Foundation of China (Grant No.12371141) and the Natural Science Foundation of Hunan Province of China (Grant No.2023JJ30642).