Measurable selectors and set-valued Pettis integral in non-separable Banach spaces

Abstract

Kuratowski and Ryll-Nardzewski's theorem about the existence of measurable selectors for multi-functions is one of the keystones for the study of set-valued integration; one of the drawbacks of this result is that separability is always required for the range space. In this paper we study Pettis integrability for multi-functions and we obtain a Kuratowski and Ryll-Nardzewski's type selection theorem without the requirement of separability for the range space. Being more precise, we show that any Pettis integrable multi-function F : Ω → cwk ( X ) defined in a complete finite measure space ( Ω , Σ , μ ) with values in the family cwk ( X ) of all non-empty convex weakly compact subsets of a general (non-necessarily separable) Banach space X always admits Pettis integrable selectors and that, moreover, for each A ∈ Σ the Pettis integral ∫ A F d μ coincides with the closure of the set of integrals over A of all Pettis integrable selectors of F. As a consequence we prove that if X is reflexive then every scalarly measurable multi-function F : Ω → cwk ( X ) admits scalarly measurable selectors; the latter is also proved when ( X ∗ , w ∗ ) is angelic and has density character at most ω 1 . In each of these two situations the Pettis integrability of a multi-function F : Ω → cwk ( X ) is equivalent to the uniform integrability of the family { sup x ∗ ( F ( ⋅ ) ) : x ∗ ∈ B X ∗ } ⊂ R Ω . Results about norm-Borel measurable selectors for multi-functions satisfying stronger measurability properties but without the classical requirement of the range Banach space being separable are also obtained.