A convergence theorem for convex set valued supermartingales ∗

Functions whose values are convex subsets provide a natural setting for the study of goal uncertainty in decision making. In fact under reasonable assumptions the successive estimates of the convex valued conditional expectations of the utility function form a supermartingale. It is of course important to determine if such estimates converge

It is shown that if {F_n, H_n} is a supermartingale where F_n has as values convex subsets of a Banach space X with separable dual and provided that F_n is “uniformly integrable” then F_n converges in some appropriate mode made precise in the work