The aim of this paper is to study Birkhoff integrability for multi-valued maps F : Ω → cwk(X), where (Ω, Σ, µ) is a complete finite measure space, X is a Banach space and cwk(X) is the family of all non-empty convex weakly compact subsets of X. It is shown that the Birkhoff integral of F can be computed as the limit for the Hausdorff distance in cwk(X) of a net of Riemann sums n µ(A n )F (t n ). We link Birkhoff integrability with Debreu integrability, a notion introduced to replace sums associated to correspondences when studying certain models in Mathematical Economics. We show that each Debreu integrable multi-valued function is Birkhoff integrable and that each Birkhoff integrable multi-valued function is Pettis integrable. The three previous notions coincide for finite dimensional Banach spaces and they are different even for bounded multi-valued functions when X is infinite dimensional and X * is assumed to be separable. We show that when F takes values in the family of all non-empty convex norm compact sets of a separable Banach space X, then F is Pettis integrable if, and only if, F is Birkhoff integrable; in particular, these Pettis integrable F 's can be seen as single-valued Pettis integrable functions with values in some other adequate Banach space. Incidentally, to handle some of the constructions needed we prove that if X is an Asplund Banach space, then cwk(X) is separable for the Hausdorff distance if, and only if, X is finite dimensional.  2004 Elsevier Inc. All rights reserved.

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 A F 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 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 sat-✩ B. Cascales and J. Rodríguez were supported by MEC and FEDER (project MTM2005-08379) and Fundación Séneca (project 00690/PI/04). J. Rodríguez was also supported by the "Juan de la Cierva" Programme (MEC and FSE).isfying stronger measurability properties but without the classical requirement of the range Banach space being separable are also obtained.

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We prove that McShane and Pettis integrability are equivalent for functions taking values in a subspace of a Hilbert generated Banach space. This generalizes simultaneously all previous results on such equivalence. On the other hand, for any super-reflexive generated Banach space having density character greater than or equal to the continuum, we show that Birkhoff integrability lies strictly between Bochner and McShane integrability. Finally, we give a ZFC example of a scalarly null Banach space-valued function (defined on a Radon probability space) which is not McShane integrable. * The second-named author was supported by MEC and FEDER (project MTM2005-08379).