The purpose of this paper is to prove some convergence theorems of closed and convex set valued sub-and super-martingales in the Kuratowski-Mosco sense. For getting submartingale convergence theorem, we give sufficient conditions for Kudo-Aumann integral and Hiai-Umegaki conditional expectation to be closed both for compact convex set valued random variables and for closed convex set valued random variables. We also give an example of a bounded closed convex set valued random variable whose Kudo-Aumann integral is not closed.