Borel canonization of analytic sets with Borel sections

Abstract:

Kanovei, Sabok and Zapletal asked whether every proper $\sigma$-ideal satisfies the following property: given $E$ an analytic equivalence relation with Borel classes, there exists a set $B$ which is Borel and $I$-positive such that $E\restriction _{B}$ is Borel. We propose a related problem – does every proper $\sigma$-ideal satisfy: given $A$ an analytic subset of the plane with Borel sections, there exists a set $B$ which is Borel and $I$-positive such that $A\cap (B\times \omega ^{\omega })$ is Borel. We answer positively when a measurable cardinal exists, and negatively in $L$, where no proper $\sigma$ ideal has that property. We show that a positive answer for all ccc $\sigma$-ideals implies that $\omega _{1}$ is inaccessible to the reals and Mahlo in $L$.