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

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.