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

Kuratowski and Ryll-Nardzewski’s theorem about the existence of measurable selectors for multifunctions 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 AF dμ 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 ω1. 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 ∗ ∈ BX ∗} ⊂ RΩ. Results about norm-Borel measurable selectors for multi-functions sat-isfying stronger measurability properties but without the classical requirement of the range Banach space being separable are also obtained