Abstract

I prove that for a Banach space XX the conjugate space X∗X^* has the WRNP if and only if for every complete probability space (Ω,Σ,μ)(\Omega,\Sigma,\mu), every μ\mu-continuous multimeasure of σ\sigma-finite variation that takes as its values closed (closed bounded, weak∗^*-compact) and convex subsets of X∗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 Radon-Nikodým property via functions (cf. the author, The weak Radon-Nikodým property of Banach spaces, Studia Math. 64 (1979) 151–174, or Pettis integral, in: Handbook of Measure Theory I, Elsevier, Amsterdam (2002) 532–586). The main tool is a lifting of a multifunction, that is Effros measurable with respect to the weak∗^* open subsets of X∗X^*.

Author information

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Keywords

  • Multimeasures
  • multifunctions
  • weak Radon-Nikodým property
  • Pettis integral
  • lifting

Mathematics Subject Classification

28B20; 28B05, 46G10, 54C60

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Suggested citation

K. Musiał. “Multimeasures with Values in Conjugate Banach Spaces and the Weak Radon-Nikodým Property.” Journal of Convex Analysis 28 (2021), No. 3, 879–902. https://doi.org/10.68381/jca28051

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