On the convexity and compactness of the integral of a Banach space valued correspondence

Abstract

We characterize the class of finite measure spaces ( T , T , μ ) which guarantee that for a correspondence ϕ from ( T , T , μ ) to a general Banach space the Bochner integral of ϕ is convex. In addition, it is shown that if ϕ has weakly compact values and is integrably bounded, then, for this class of measure spaces, the Bochner integral of ϕ is weakly compact, too. Analogous results are provided with regard to the Gelfand integral of correspondences taking values in the dual of a separable Banach space, with “weakly compact” replaced by “weak *-compact.” The crucial condition on the measure space ( T , T , μ ) concerns its measure algebra and is consistent with having T = [ 0 , 1 ] and μ to be an extension of Lebesgue measure.