Borel sets with $\sigma$-compact sections for nonseparable spaces
Volume 199 / 2008
Fundamenta Mathematicae 199 (2008), 139-154
MSC: Primary 54H05; Secondary 54C65, 28A05.
DOI: 10.4064/fm199-2-4
Abstract
We prove that every (extended) Borel subset $E$ of $X\times Y$, where $X$ is complete metric and $Y$ is Polish, can be covered by countably many extended Borel sets with compact sections if the sections $E_x=\{y\in Y:(x,y)\in E\}$, $x\in X$, are $\sigma$-compact. This is a nonseparable version of a theorem of Saint Raymond. As a by-product, we get a proof of Saint Raymond's result which does not use transfinite induction.
