Borel canonization of analytic sets with Borel sections

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