Proposition 3

— Let T be a Polish locally compact space, $\mu$ a positive measure on T, and R an equivalence relation in T. The following properties are equivalent:

a) $R$ is $\mu$-measurable.

b) There exists a sequence of mappings $p_n : T \to F_n$ into Hausdorff topological spaces, such that each $p_n$ is $\mu$-measurable and such that the relation $R\{x, y\}$ is equivalent to «for all $n$, $p_n(x) = p_n(y)$».

c) There exists a sequence $(A_n)$ of $\mu$-measurable sets, saturated for R, such that for every $x \in T$ the class of x with respect to R is the intersection of those $A_n$ that contain x.

With notations as in the statement of b), set $p(x) = (p_n(x))$; the property b) means that the mapping $p$ of T into the product space $\prod F_n$ is measurable (Ch. IV, §5, No. 3, Th. 1) and that the relation $R\{x, y\}$ is equivalent to $p(x) = p(y)$; thus b) implies a).

Next let us show that c) implies b). Suppose c) verified; then the characteristic functions $\varphi_{A_n}$ are $\mu$-measurable, and the hypothesis c) means that the relation $R\{x, y\}$ is equivalent to «for every $n$, $\varphi_{A_n}(x) = \varphi_{A_n}(y)$».

Finally, let us show that a) implies c). By Prop. 2, there exist a locally compact space B with a countable base, and a $\mu$-measurable mapping $p$ of T into B, such that the relation $R\{x, y\}$ is equivalent to $p(x) = p(y)$. Let $(U_n)$ be a countable base for the topology of B. The sets $A_n = \overline{p^{-1}(U_n)}$ are $\mu$-measurable (Ch. IV, §5, No. 5, Prop. 7) and saturated for R; and if $x, y$ are points of T such that $p(x) \neq p(y)$, there exists an index $n$ such that $p(x) \in U_n$ and $p(y) \notin U_n$, which means that $x \in A_n$ and $y \notin A_n$.