
Journal of Convex Analysis 31 (2024), No. 1, 179193 Copyright Heldermann Verlag 2024 Lifting Approach to Integral Representations of Rich Multimeasures with Values in Banach Spaces I Kazimierz Musial Institute of Mathematics, Wroclaw University, Wroclaw, Poland kazimierz.musial@math.uni.wroc.pl Let (Ω, Σ, μ) be a complete probability space and M be a μcontinuous multimeasure of σfinite variation with values in the family of nonempty closed convex subsets of a Banach space X. I prove that if M is rich in countably additive selections possessing strongly measurable and Pettis integrable densities, then there exists an Effros measurable multifunction that is a Pettis integrable density of M. The above assumptions are in particular satisfied in case of X with RNP. In particular, if X has RNP and Γ is a multifunction that is Pettis integrable in the family of nonempty closed convex and bounded subsets of X, then there exists an Effros measurable multifunction that is scalarly equivalent to Γ. The paper is a continuation of a previous paper of the author [Multimeasures with values in conjugate Banach spaces and the Weak RadonNikodym Property, J. Convex Analysis 28/3 (2021) 879902], where it has been proven that if X* has WRNP, then a multimeasure as above but with values in X* can be represented as a Pettis integral of a multifunction with closed bounded and convex values that is Effros measurable with respect to weak* open sets. Keywords: Measurable multimeasures, rich multimeasures, RadonNikodym property, integral representations, lifting. MSC: 28B20; 28B05, 46G10, 46B22. [ Fulltextpdf (151 KB)] for subscribers only. 