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Proceedings of the American Mathematical Society

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

 

 

Bimeasurable functions


Author: R. Daniel Mauldin
Journal: Proc. Amer. Math. Soc. 83 (1981), 369-370
MSC: Primary 54C65; Secondary 04A15, 28C15, 54H05
DOI: https://doi.org/10.1090/S0002-9939-1981-0624933-4
MathSciNet review: 624933
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Abstract: Let $ X$ and $ Y$ be complete, separable metric spaces, $ B$ a Borel subset of $ X$ and $ f$ a Borel measurable map of $ B$ into $ Y$. The map $ f$ is said to be bimeasurable if the image of every Borel subset of $ B$ is a Borel subset of $ Y$. R. Purves proved that $ f$ is bimeasurable if and only if $ y:{f^{ - 1}}(y)$ is uncountable is countable. The purpose of this note is to demonstrate how Purves' theorem can be obtained directly from some results concerning parametrization of sets which have been developed in the past few years.


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DOI: https://doi.org/10.1090/S0002-9939-1981-0624933-4
Keywords: Borel isomorphism, bimeasurable function
Article copyright: © Copyright 1981 American Mathematical Society