BOREL CANONIZATION OF ANALYTIC SETS WITH BOREL SECTIONS

摘要

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 (sic) 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 boolean AND (B x 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.