Multimeasures with Values in Conjugate Banach Spaces and the Weak Radon-Nikodým 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-Nikod\'{y}m property via functions (cf. the author, {\it The weak Radon-Nikod\'{y}m property of Banach spaces}, Studia Math. 64 (1979) 151--174, or {\it Pettis integral}, in: {\it Handbook of Measure Theory I}, Elsevier, Amsterdam (2002) 532--586). The main tool is a lifting of a multifunction, that is Effros measurable with respect to the weak$^*$ open subsets of $X^*$.