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

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 ∞(BX ∗ ) by means of the mapping j : cwk(X)→ ∞(BX ∗ ) defined by j (C)(x ∗ ) = sup(x ∗ (C)), then j ◦F is integrable with respect to a norming subset of B ∞(BX ∗ ) ∗ . 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. © 2006 Elsevier Inc. All rights reserved.