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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Point-finite Borel-additive families are of bounded class

Author: R. W. Hansell
Journal: Proc. Amer. Math. Soc. 83 (1981), 375-378
MSC: Primary 54H05; Secondary 04A15
MathSciNet review: 624935
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Abstract: We prove the following theorem, which answers a question originally raised by J. Kaniewski and R. Pol:

Theorem. If $ \mathfrak{X}$ is a point-finite family of subsets of a metric space $ X$ such that the union of every subfamily is a Borel set of $ X$, then there exists a fixed countable ordinal $ \alpha $ such that each member of $ \mathfrak{X}$ is a Borel set of class $ \alpha $ in $ X$.

The proof is given in the general setting of abstract measurable spaces. An application is made to the study of measurable selectors for compact-valued mappings and to the Borel measurability of graphs.

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Keywords: Borel-additive family, point-finite family, Borel classifications, measurable selectors
Article copyright: © Copyright 1981 American Mathematical Society