International Journal of Mathematics and Mathematical Sciences
Volume 12 (1989), Issue 3, Pages 477-486
doi:10.1155/S016117128900061X

On the continuity of the vector valued and set valued conditional expectations

Abstract

In this paper we study the dependence of the vector valued conditional expectation (for both single valued and set valued random variables), on the σ–field and random variable that determine it. So we prove that it is continuous for the L1(X) convergence of the sub–σ–fields and of the random variables. We also present a sufficient condition for the L1(X)–convergence of the sub–σ–fields. Then we extend the work to the set valued conditional expectation using the Kuratowski–Mosco (K–M) convergence and the convergence in the Δ–metric. We also prove a property of the set valued conditional expectation.