
Journal of Convex Analysis 28 (2021), No. 3, 879902 Copyright Heldermann Verlag 2021 Multimeasures with Values in Conjugate Banach Spaces and the Weak RadonNikodým Property Kazimierz Musial Institute of Mathematics, Wroclaw University, 50384 Wroclaw, Poland kazimierz.musial@math.uni.wroc.pl [Abstractpdf] 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 RadonNikod\'{y}m property via functions (cf. the author, {\it The weak RadonNikod\'{y}m property of Banach spaces}, Studia Math. 64 (1979) 151174, or {\it Pettis integral}, in: {\it Handbook of Measure Theory I}, Elsevier, Amsterdam (2002) 532586). The main tool is a lifting of a multifunction, that is Effros measurable with respect to the weak$^*$ open subsets of $X^*$. Keywords: Multimeasures, multifunctions, weak RadonNikod\'{y}m property, Pettis integral, lifting. MSC: 28B20; 28B05, 46G10, 54C60. [ Fulltextpdf (185 KB)] for subscribers only. 