Measurability and Perfect Set Theorems for Equivalence Relations with Small Classes

Abstract

We ask whether ${\mathbf{\Delta }}_2^1$ or ${\mathbf{\Sigma }}_2^1$ equivalence relations with $I$-small classes for $I$ a $\sigma$-ideal must have perfectly many classes. We show that for a wide class of ccc $\sigma$-ideals, a positive answer for ${\mathbf{\Delta }}_2^1$ equivalence relations is equivalent to the $I$-measurability of ${\mathbf{\Delta }}_2^1$ sets. However, the analogous statement for ${\mathbf{\Sigma }}_2^1$ equivalence relations is false: ${\mathbf{\Sigma }}_2^1$ equivalence relations with meager classes have a perfect set of pairwise inequivalent elements if and only if ${\mathbf{\Delta }}_2^1$ sets have the Baire property.