Set-valued measure and set-valued weak Radon-Nikodym derivative of a set-valued measure (Q2831712)

\textit{K. Musial} [Stud. Math. 64, 151--173 (1979; Zbl 0405.46015)] defined the notions of the weak Radon-Nikodym (wRN) derivative and of Banach spaces with the weak Radon-Nikodym property, within the framework of Pettis integration for vector functions. The extension to multimeasures was given by \textit{A. Coste} [C. R. Acad. Sci., Paris, Sér. A 280, 1515--1518 (1975; Zbl 0313.28007)]. Completing the results obtained by Coste, the author presents some further properties of these wRN derivatives of multimeasures.NEWLINENEWLINELet \(X\) be a separable Banach space with dual \(X^*\) and \((\Omega,\Sigma,\mu)\) a complete positive measure space. A wRN derivative of a multimeasure \(M:\Sigma\to 2^X\setminus\{\emptyset\}\) is a Pettis integrable multifunction \(F:\Omega\to CB(X)\) (the family of all nonempty closed convex subsets of \(X\)) such that \(M(A)=\int_AF \, d\mu \) (the Pettis integral of \(F\)), for all \(A\in \Sigma\). The multimeasure \(M\) is called \(\mu\)-continuous if \(\mu(A)=0\) implies \(M(A)=\{0\}\). Coste [loc. cit.] proved that, if the Banach space \(X\) has the wRN property, then any \(\mu\)-continuous multimeasure of \(\sigma\)-finite variation with weakly compact convex values admits a wRN derivative \(F\). The author shows that any measure \(m:\Sigma\to X\), which is a selection of \(M\), is of the form \(m(A)=\int_Af \, d\mu\), for some Pettis integrable selection \(f\) of \(F\). He shows also that this derivative \(F\) is integrably bounded iff \(M\) is of bounded variation.