Borel functions of bounded class

Fremlin, D. H.; Hansell, R. W.; Junnila, H. J. K.

doi:10.1090/S0002-9947-1983-0694392-0

Borel functions of bounded class
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by D. H. Fremlin, R. W. Hansell and H. J. K. Junnila PDF

Trans. Amer. Math. Soc. 277 (1983), 835-849 Request permission

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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