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
University of California, 1015 Department of Mathematics, Davis 95616, California, USA
Received 12 May 1988; Revised 29 January 1989
Copyright © 1989 Nikolaos S. Papageorgiou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.