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

K. Musial

Journal of convex analysis, Tome 31 (2024) no. 1, pp. 179-193

Résumé

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.

Détail

Classification : 28B20, 28B05, 46G10, 46B22
Mots-clés : Measurable multimeasures, rich multimeasures, Radon-Nikodym property, integral representations, lifting