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 $\mathsf{ZF}+\mathsf{DC}+{\mathsf{AD}}_{\mathbb{R}}+V=L(\mathcal{P}(\mathbb{R}))$, 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.