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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
DOI: https://doi.org/10.1090/S0002-9939-1978-0500507-0
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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Keywords: <IMG WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img3.gif" ALT="$\tau$">-additive family of sets, <IMG WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\tau$">-multiplicative family of sets, <IMG WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$\tau$">-weak reduction principle, <B>H</B>-measurable function, selector
Article copyright: © Copyright 1978 American Mathematical Society