Weak topologies for modules over rings of bounded random variables

Abstract

In order to establish a functional analytic basis for representation theorems for conditional and multi-period risk measures, we study locally convex modules over the ring λ=L∞(G). Their topology is determined by λ-seminorms. As expected, central mathematical tools of the analysis are Hahn–Banach type and separation theorems which however have to be treated more carefully in the module case. Once a dual λ-module is introduced, one can establish a module version of the Bipolar theorem. We also prove the Krein–Šmulian as well as the Alaoglu–Bourbaki theorem for λ-modules. For Banach λ-modules their reflexivity is characterized by a compactness criterium in a (very) weak-* topology.