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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Borel functions of bounded class

Authors: D. H. Fremlin, R. W. Hansell and H. J. K. Junnila
Journal: Trans. Amer. Math. Soc. 277 (1983), 835-849
MSC: Primary 54H05; Secondary 03E15
MathSciNet review: 694392
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Abstract: Let $ X$ and $ Y$ be metric spaces and $ f:X \to Y$ a Borel measurable function. Does $ f$ have to be of bounded class, i.e. are the sets $ {f^{ - 1}}[ H ]$, for open $ H \subseteq Y$, of bounded Baire class in $ X?$ This is an old problem of A. H. Stone. Positive answers have been given under a variety of extra hypotheses and special axioms. Here we show that (i) unless something similar to a measurable cardinal exists, then $ f$ is of bounded class and (ii) if $ f$ is actually a Borel isomorphism, then $ f\,({\text{and}}\ {f^{ - 1}})$ are of bounded class.

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Article copyright: © Copyright 1983 American Mathematical Society