The Pettis integral for multi-valued functions via single-valued ones☆

Abstract We study the Pettis integral for multi-functions F : Ω → cwk ( X ) defined on a complete probability space ( Ω , Σ , μ ) with values into the family cwk ( X ) of all convex weakly compact non-empty subsets of a separable Banach space X. From the notion of Pettis integrability for such an F studied in the literature one readily infers that if we embed cwk ( X ) into l ∞ ( B X ∗ ) by means of the mapping j : cwk ( X ) → l ∞ ( B X ∗ ) defined by j ( C ) ( x ∗ ) = sup ( x ∗ ( C ) ) , then j ○ F is integrable with respect to a norming subset of B l ∞ ( B X ∗ ) ∗ . A natural question arises: When is j ○ F Pettis integrable? In this paper we answer this question by proving that the Pettis integrability of any cwk ( X ) -valued function F is equivalent to the Pettis integrability of j ○ F if and only if X has the Schur property that is shown to be equivalent to the fact that cwk ( X ) is separable when endowed with the Hausdorff distance. We complete the paper with some sufficient conditions (involving stability in Talagrand's sense) that ensure the Pettis integrability of j ○ F for a given Pettis integrable cwk ( X ) -valued function F.