Lifting Approach to Integral Representations of Rich Multimeasures with Values in Banach Spaces I

Let (Ω, Σ, μ) be a complete probability space and M be a μ-continuous multimeasure of σ-finite variation with values in the family of non-empty 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 non-empty 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 Radon-Nikodym Property, J. Convex Analysis 28/3 (2021) 879–902], 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.