a non-Borel equivalence relation whose every equivalence class is Borel Let $(X, \mathcal{A})$ be a standard Borel space where $\mathcal{A}$ is the sigma algebra of Borel sets of $X$. An equivalence relation $E$ on $X$ is Borel if it is a Borel subset of $X^2$ (with respect to the product topology). Say $E$ is measurable (my temrinology) if every equivalence class of $E$ is a Borel set of $X$. I am curious if there is an equivalence relation on $X$ that is measurable but not Borel in this sense (my suspicion is yes but I have yet been able to prove it). Also I suspect that the converse is false as well -- there should be Borel equivalence relations that are non-measurable (given that the projections of a Borel subset is not always Borel). But again that's mere conjecture. Any hint would be much appreciated.

Consider equivalence relations on $\mathbb R$ for which all equivalence classes are $2$-element sets (which are of course Borel sets). There are $2^\mathfrak c$ such equivalence relations, and they are not all Borel, seeing as there are only $\mathfrak c$ Borel subsets of $\mathbb R^2$.
