INTEGRATION OF CORRESPONDENCES ON LOEB SPACES

We study the Bochner and Gel'fand integration of Banach space valued c orrespondences on a general Loeb space. Though it is well known that t he Lyapunov type result on the compactness and convexity of the integr al of a correspondence and the Fatou type result on the preservation o f upper semicontinuity by integration are in general not valid in the setting of an infinite dimensional space, we show that exact versions of these two results hold in the case we study. We also note that our results on a hyperfinite Loeb space capture the nature of the correspo nding asymptotic results for the large finite case; but the unit Lebes gue interval fails to provide such a framework.