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

I prove that for a Banach space

the conjugate space

X^* X ∗

has the WRNP if and only if for every complete probability space

(\Omega,\Sigma,\mu) ( Ω , Σ , μ )

, every

\mu μ

-continuous multimeasure of

\sigma σ

-finite variation that takes as its values closed (closed bounded, weak

^* ∗

-compact) and convex subsets of

X^* 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-Nikodým property via functions (cf. the author, The weak Radon-Nikodým property of Banach spaces, Studia Math. 64 (1979) 151–174, or Pettis integral, in: 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^* X ∗

.