Multimeasures with Values in Conjugate Banach Spaces and the Weak Radon-Nikodym Property

I prove that for a Banach space X the conjugate space X* has the WRNP if and only if for every complete probability space (Omega, Sigma, mu), every mu-continuous multimeasure of sigma-finite variation that takes as its values closed (closed bounded, weak*-compact) and convex subsets of X* can be represented as a Pettis integral of a multifunction with closed bounded (closed bounded, weak* compact) and convex values. This generalizes the known characterization of conjugate Banach spaces with the weak Radon-Nikodym property via functions (cf. [18] or [21]). The main tool is a lifting of a multifunction (see section 3), that is Effros measurable with respect to the weak* open subsets of X*.