Convexity and Conditional Expectations

Abstract

If a $n$-dimensional function is with probability one in a convex set, the same holds true for the conditional expectation (with respect to any sub-$\sigma$-field). An extreme point of this convex set can be assumed by the conditional expectation only if it is assumed by the original function and if this function is partially measurable with respect to the conditioning sub-$\sigma$-field. These results are used to prove Jensen's inequality for conditional expectations of $n$-dimensional functions, and to give a condition for strict inequality.