A counterexample and an exact version of Fatou's lemma in infinite dimensional spaces (Q1112294)

The following approximate version of Fatou's lemma in infinite dimensional spaces is known. If X is a separable Banach space, T a finite, atomless, complete measure space, \(\phi_ n\) a sequence of corresondences from T to a weakly compact convex non-empty subset K of X, then the weak limit superior of the integral of \(\phi_ n\) is contained in the norm closure of the integral of the weak limit superior of \(\phi_ n\). We show that such result cannot be improved upon, and provide sufficient conditions for an exact version.