Multifunctions of faces for conditional expectations of selectors and Jensen's inequality

Let (T, , P) be a probability space, a P-complete sub-[delta]-algebra of and X a Banach space. Let multifunction t --> [Gamma](t), t [set membership, variant] T, have a [circle times operator] (X)-measurable graph and closed convex subsets of X for values. If x(t) [epsilon] [Gamma](t) P-a.e. and y(·) [epsilon] Ep x(·), then y(t) [epsilon] [Gamma](t) P-a.e. Conversely, x(t) [epsilon] F([Gamma](t), y(t)) P-a.e., where F([Gamma](t), y(t)) is the face of point y(t) in [Gamma](t). If X = , then the same holds true if [Gamma](t) is Borel and convex, only. These results imply, in particular, extensions of Jensen's inequality for conditional expectations of random convex functions and provide a complete characterization of the cases when the equality holds in the extended Jensen inequality.