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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A selection theorem for multifunctions


Author: H. Sarbadhikari
Journal: Proc. Amer. Math. Soc. 71 (1978), 285-288
MSC: Primary 54C65; Secondary 04A20, 54A25, 54C60
MathSciNet review: 500507
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Abstract: In this paper the following theorem is proved. X is any set, H is a family of subsets of X which is $ \lambda $-additive, $ \lambda $-multiplicative and satisfies the $ \lambda $-WRP for some cardinal $ \lambda > {\aleph _0}$. Suppose Y is a regular Hausdorff space of topological weight $ \leqslant \lambda $ such that given any family of open sets, there is a subfamily of cardinality $ < \lambda $ with the same union. Let $ F:X \to {\mathbf{C}}(Y)$, where $ {\mathbf{C}}(Y)$ is the family of nonempty compact subsets of Y, satisfy $ \{ x:F(x) \cap C \ne \emptyset \} \in {\mathbf{H}}$ for any closed subset C of Y. Then F admits a $ {({\mathbf{H}} \cap {{\mathbf{H}}^c})_\lambda }$ measurable selector.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1978-0500507-0
PII: S 0002-9939(1978)0500507-0
Keywords: $ \tau $-additive family of sets, $ \tau $-multiplicative family of sets, $ \tau $-weak reduction principle, H-measurable function, selector
Article copyright: © Copyright 1978 American Mathematical Society