Birkhoff integral for multi-valued functions

The aim of this paper is to study Birkhoff integrability for multi-valued maps <f>F :Ω→cwk(X)</f>, where <f>(Ω,Σ,μ)</f> is a complete finite measure space, <f>X</f> is a Banach space and <f>cwk(X)</f> is the family of all non-empty convex weakly compact subsets of <f>X</f>. It is shown that the Birkhoff integral of <f>F</f> can be computed as the limit for the Hausdorff distance in <f>cwk(X)</f> of a net of Riemann sums <f>∑nμ(An)F(tn)</f>. 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 <f>X</f> is infinite dimensional and <f>X*</f> is assumed to be separable. We show that when <f>F</f> takes values in the family of all non-empty convex norm compact sets of a separable Banach space <f>X</f>, then <f>F</f> is Pettis integrable if, and only if, <f>F</f> is Birkhoff integrable; in particular, these Pettis integrable <f>F</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 <f>X</f> is an Asplund Banach space, then <f>cwk(X)</f> is separable for the Hausdorff distance if, and only if, <f>X</f> is finite dimensional. [Copyright &y& Elsevier]