Borel Canonization of Analytic Sets with Borel Sections

TL;DR

This paper investigates conditions under which analytic sets with Borel sections can be restricted to Borel and positive sets to become Borel, revealing consistency and independence results related to large cardinals and constructibility.

Contribution

It establishes positive results assuming a measurable cardinal and negative results in L, providing new insights into Borel canonization of analytic sets and equivalence relations.

Findings

Positive under measurable cardinal assumptions

Negative in the constructible universe L

Counterexamples for certain equivalence relations and ideals

Abstract

Given an analytic equivalence relation, we tend to wonder whether it is Borel. When it is non Borel, there is always the hope it will be Borel on a "large" set -- nonmeager or of positive measure. That has led Kanovei, Sabok and Zapletal to ask 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. Assuming $\omega_{1}$ is inaccessible to the reals but not Mahlo in $L$, we construct a ccc $\sigma$ ideal $I$ not having this property -- in fact, forcing with $I$ adds a non Borel section to a certain analytic set with Borel sections, and a non Borel class to a certain analytic equivalence relation with Borel classes. Various counterexamples are given for the case of a $\mathbf{\Delta_{2}^{1}}$ equivalence relation as well as for the case of an improper ideal.