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

AbstractKuratowski 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 (n…