Extensions of Borel Measurable Maps and Ranges of Borel Bimeasurable Maps
We prove an abstract version of the Kuratowski extension theorem for Borel measurable maps of a given class. It enables us to deduce and improve its nonseparable version due to Hansell. We also study the ranges of not necessarily injective Borel bimeasurable maps $f$ and show that some control on the relative classes of preimages and images of Borel sets under $f$ enables one to get a bound on the absolute class of the range of $f$. This seems to be of some interest even within separable spaces.