On the continuity of the vector valued and set valued conditional expectations (Q1121581)

Summary: We study the dependence of the vector valued conditional expectation (for both single valued and set valued random variables), on the \(\sigma\)- field and the random variables that determine it. So we prove that it is continuous for the \(L^ 1(X)\) convergence of the sub-\(\sigma\)-fields and of the random variables. We also present a sufficient condition for the \(L^ 1(X)\)-convergence of the sub-\(\sigma\)-fields. Then we extend the work to the set valued conditional expectation using the Kuratowski-Mosco (K-M) convergence and the convergence in the \(\Delta\)-metric. We also prove a property of the set valued conditional expectation.