Theorem 3

— Let T be a locally compact space with a countable base, $\mu$ a positive measure on T, and R a $\mu$-measurable equivalence relation in T. Then, there exists a $\mu$-measurable subset S of T that intersects each class with respect to R at one and only one point (a 'measurable section' for R).

We may clearly suppose that the measure $\mu$ is bounded and that $\mu(T) \leq 1$ (Ch. V, §5, No. 6, Prop. 11). We are going to define a sequence $(S_n)$ of Borel subsets (GT, IX, §6, No. 3) such that each equivalence class with respect to R intersects the union $S'$ of the $S_n$ in at most one point, that for every $n$ the saturation $T_n$ of the union of the $S_p$ with index $p \leq n$ is $\mu$-measurable, and that $\mu(T - T_n) \leq 1/2^n$. The saturation $T'$ of $S'$ will therefore be $\mu$-measurable and $N = T - T'$ will have measure zero. If $S''$ is any section of N for the relation $R_N$, $S = S' \cup S''$ will meet the requirements, since $S'$, being a Borel set, is $\mu$-measurable (Ch. IV, §5, No. 4, Cor. 3 of Th. 2), and $S''$ has measure zero.

By Prop. 2, $R\{x, y\}$ is equivalent to the relation $p(x) = p(y)$, where $p$ is a $\mu$-measurable mapping of T into a locally compact space F. Suppose the $S_k$ defined for $k \leq n$. Since $T - T_n$ is $\mu$-measurable and of measure $\leq 1/2^n$, there exists a compact subset K of $T - T_n$ such that $\mu(T - (T_n \cup K)) \leq 1/2^{n+1}$ and such that the restriction of $p$ to K is continuous. Since the induced relation $R_K$ is closed and K is metrizable, we know that there exists a Borel subset $S_{n+1}$ of K such that, in K, each point is equivalent (mod R) to one and only one point of $S_{n+1}$ (GT, IX, §6, No. 8, Th.4). Therefore $p(S_{n+1}) = p(K)$, which is a compact set in F; the saturation of $S_{n+1}$ for R is the inverse image $\overline{p^{-1}(p(K))}$, which is therefore μ-measurable (Ch. IV, §5, No. 5, Prop. 7); it is clear that this set contains K, therefore the union T_{n+1} of T_n and $\overline{p}^{-1}(p(K))$ is μ-measurable, saturated for R, and is such that $\mu(T - T_{n+1}) \leq 1/2^{n+1}$, which completes the proof.