Convergence of unbounded multivalued supermartingales in the Mosco and slice topologies

Abstract

Our starting point is the Mosco-convergence result due to Hess ([He'89]) for integrable multivalued supermartingales whose values may be unbounded, but are majorized by a w-ball-compact-valued function. It is shown that the convergence takes place also in the slice topology. In the case when both the underlying space X and its dual X have the Radon-Nikodym property a weaker compactness assumption guarantees convergence of the multi-valued supermartingales in the slice topology. This result implies convergence in the Mosco topology and gives an analogue of Hess' result in the case when X and X have the RNP. Finally the results are restated in terms of normal integrands.