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).

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 ∞ (B X * ) by means of the mapping j : cwk(X) → ∞ (B X * ) defined by j (C)(x * ) = sup(x * (C)), then j • F is integrable with respect to a norming subset of B ∞ (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 .

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Abstract. We study compactness and related topological properties in the space L 1 (m) of a Banach space valued measure m when the natural topologies associated to the convergence of the vector valued integrals are considered. The resulting topological spaces are shown to be angelic and the relationship of compactness and equi-integrability is explored. A natural norming subset of the dual unit ball of L 1 (m) appears in our discussion and we study when it is a boundary. The (almost) complete continuity of the integration operator is analyzed in relation with the positive Schur property of L 1 (m). The strong weakly compact generation of L 1 (m) is discussed as well.

Let (Ω, Σ, µ) be a complete probability space and u : X → Y an absolutely summing operator between Banach spaces. We prove that for each Dunford integrable (i.e., scalarly integrable) function f : Ω → X the composition u • f is scalarly equivalent to a Bochner integrable function. Such a composition is shown to be Bochner integrable in several cases, for instance, when f is properly measurable, Birkhoff integrable or McShane integrable, as well as when X is a subspace of an Asplund generated space or a subspace of a weakly Lindelöf space of the form C(K). We also study the continuity of the composition operator f → u • f . Some other applications are given.  2005 Elsevier Inc. All rights reserved.