Remarks on the set-valued integrals of Debreu and Aumann

Some recently obtained sufficient conditions for the weak compactness of subsets of L¹(m, X) are used to show that for functions whose values are compact, convex subsets of a Banach space the Debreu integral, when it exists, is the same as the Aumann integral. Here no assumption is made concerning the reflexivity of X. This result extends to functions whose values are weakly compact, convex subsets of Banach space.