User:Boris Tsirelson/sandbox2

Negative results

$\newcommand{\M}{\mathscr M}$ As was noted, the normal form of an object $M\in\M$ is a "selected representative" from the equivalence class $[M]$, usually possessing some nice properties. The set of all these "representatives" intersect each equivalence class exactly once; such set is called a transversal (for the given equivalence relation). Existence of a transversal is ensured by the axiom of choice for arbitrary equivalence relation on arbitrary set. However, a transversal in general is far from being nice. For example, consider the equivalence relation "$x-y$ is rational" for real numbers $x,y$. Its transversal (so-called Vitali set) cannot be Lebesgue measurable!

Typically, the set $\M$, endowed with its natural σ-algebra, is a standard Borel space, and the set $\{(x,y)\in\M\times\M:x\sim y\}$ is a Borel subset of $\M\times\M$; this case is well-known as a "Borel equivalence relation". Still, existence of a Borel transversal is not guaranteed (for an example, use the Vitali set again).