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Convergence and approximation results for measurable multifunctions

Authors: R. Lucchetti, N. S. Papageorgiou and F. Patrone
Journal: Proc. Amer. Math. Soc. 100 (1987), 551-556
MSC: Primary 28A20; Secondary 54C60, 60G99
MathSciNet review: 891162
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Abstract: In this note we consider measurable multifunctions taking values in a separable Banach space. We show that if $ F(\omega ) \subseteq {\text{s}} - \lim {F_n}(\omega )$, then any Castaing representation of the $ F( \cdot )$ can be obtained as the strong limit of Castaing representations of the $ {F_n}$. We also prove that any weakly measurable multifunction is the Kuratowski-Mosco limit of a sequence of countably simple multifunctions. Then we show that in reflexive Banach spaces this approximation property is equivalent to weak measurability. Finally we discuss the problem of measurability of the inferior and $ {\text{w}}$-superior limits of a sequence of measurable multifunctions.

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Keywords: Kuratowski-Mosco convergence, measurable multifunction, Castaing representation, Aumann's selection theorem, complete $ \sigma $-field, countably simple multifunction, Trojanski's renorming theorem
Article copyright: © Copyright 1987 American Mathematical Society

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