Fatou's Lemma for unbounded multifunctions with values in a dual space,

A version of Fatou's lemma for multifunctions with unbounded values in infinite dimensions is presented. Motivated by general equilibrium existence question in a model of spacial economy Cornet-Medecin gave a Fatou lemma for Gelfand integrable functions that map into the dual of an infinite-dimensional space. Their result was improved first by Balder. Subsequently, that improvement was again sharpened by Cornet-Martins da Rocha. In their paper it is shown that, unlike the current situation for Bochner integrable functions, an infinite-dimensional result can be formulated for Gelfand integrable functions that does not require a separate and parallel development in finite dimensions. In other words, it can be applied to finite dimensions without any loss of power. The present paper continues this development. It presents a multivalued Fatou-type result that generalizes not only results of Cornet-Martins da Rocha for Gelfand integrable functions, but also the finite-dimensional version of the multivalued Fatou lemmas of Balder-Hess.