When an Equivalence Relation with All Borel Classes will be Borel Somewhere?

Abstract

In $\mathsf{ZFC}$, if there is a measurable cardinal with infinitely many Woodin cardinals below it, then for every equivalence relation $E \in L(\mathbb{R})$ on $\mathbb{R}$ with all $\mathbf{\Delta}_1^1$ classes and every $\sigma$-ideal $I$ on $\mathbb{R}$ so that the associated forcing $\mathbb{P}_I$ of $I^+$ $\mathbf{\Delta}_1^1$ subsets is proper, there exists some $I^+$ $\mathbf{\Delta}_1^1$ set $C$ so that $E \upharpoonright C$ is a $\mathbf{\Delta}_1^1$ equivalence relation. In $\mathsf{ZF} + \mathsf{DC} + \mathsf{AD}_\mathbb{R} + V = L(\mathscr{P}(\mathbb{R}))$, for every equivalence relation $E$ on $\mathbb{R}$ with all $\mathbf{\Delta}_1^1$ classes and every $\sigma$-ideal $I$ on $\mathbb{R}$ so that the associated forcing $\mathbb{P}_I$ is proper, there is some $I^+$ $\mathbf{\Delta}_1^1$ set $C$ so that $E \upharpoonright C$ is a $\mathbf{\Delta}_1^1$ equivalence relation.