Integration of Correspondences on Loeb Spaces

Sun, Yeneng

doi:10.1090/S0002-9947-97-01825-4

Integration of Correspondences on Loeb Spaces

Author: Yeneng Sun
Journal: Trans. Amer. Math. Soc. 349 (1997), 129-153
MSC (1991): Primary 03H05, 28B20; Secondary 46G10, 90A14
DOI: https://doi.org/10.1090/S0002-9947-97-01825-4
MathSciNet review: 1401529
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Abstract: We study the Bochner and Gel $^{\prime }$ fand integration of Banach space valued correspondences on a general Loeb space. Though it is well known that the Lyapunov type result on the compactness and convexity of the integral of a correspondence and the Fatou type result on the preservation of 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 corresponding asymptotic results for the large finite case; but the unit Lebesgue interval fails to provide such a framework.

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