Theorem of Kuratowski-Suslin for measurable mappings

Abstract:

The purpose of this paper is to describe these Borel mappings on a separable complete metric space X which transform every measurable set (with respect to some measure $\mu$ on X) onto a measurable one. It is shown that a one-to-one Borel mapping f on X fulfills the above property if and only if the measure $\mu$ is absolutely continuous with respect to the measure ${\mu _f}$ (an image of $\mu$ under the mapping f). Our results are a generalization of the classical results of Suslin and Kuratowski.