Say that a class of equivalence relations ${\cal C}$ has the finite union property if every equivalence relation that is the union of finitely many members of ${\cal C}$ must itself be a member of ...

Let $(X, \mathscr{B})$ be a standard Borel space, $R \subset X \times X$ an equivalence relation $\in \mathscr{B} \times \mathscr{B}$. Assume each equivalence class ...