  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.
