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

Let (T, A, P) be a probability space, B a P-complete sub-δ-algebra of A and X a Banach space. Let multifunction t → Γ(t), t ∈ T, have a B ⊗ B(X)-measurable graph and closed convex subsets of X for values. If x(t) ε{lunate} Γ(t) P-a.e. and y(·) ε{lunate} E_p^B x(·), then y(t) ε{lunate} Γ(t) P-a.e. Conversely, x(t) ε{lunate} F(Γ(t), y(t)) P-a.e., where F(Γ(t), y(t)) is the face of point y(t) in Γ(t). If X = Rⁿ, then the same holds true if Γ(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.