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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
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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Additional Information

Yeneng Sun
Affiliation: Department of Mathematics, National University of Singapore, Singapore 119260
Address at time of publication: Cowles Foundation, Yale University, New Haven, Connecticut 06520
Email: gs53@econ.yale.edu

DOI: https://doi.org/10.1090/S0002-9947-97-01825-4
Keywords: Correspondences, Loeb spaces, Bochner integral, Gel$^{\prime }$fand integral, convexity, semicontinuity, weak compactness, weak$^{*}$ compactness
Received by editor(s): February 23, 1995
Additional Notes: The main results were presented at the Fifth Asian Logic Conference held in Singapore in June 1993. The author is grateful to Professors Robert Anderson, Donald Burkholder, Chi Tat Chong, Ward Henson, Zhuxin Hu, Jerome Keisler, Ali Khan, Peter Loeb, Walter Trockel, and Jerry Uhl for helpful conversations and encouragement. The research is partially supported by the National University of Singapore, grant no. RP3920641.
Article copyright: © Copyright 1997 American Mathematical Society