Bimeasurable functions

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.