Abstract: | Let (T,
, P) be a probability space,
a P-complete sub-δ-algebra of
and X a Banach space. Let multifunction t → Γ(t), t T, have a
(X)-measurable graph and closed convex subsets of X for values. If x(t) ε Γ(t) P-a.e. and y(·) ε Ep
x(·), then y(t) ε Γ(t) P-a.e. Conversely, x(t) ε 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 =
, 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. |