Exercise 8, § 3

Let T be the interval $[0,2]$ in $\mathbf{R}$, $\mu$ the Lebesgue measure on T. Let A be a non-Borel set contained in the Cantor set $K \subset [0,1]$ (cf. GT, IV, §2, No. 4, Example and IX, §6, Exer. 6). Let S be the equivalence relation in T whose equivalence classes are the sets $\{x\}$ for $x \notin A \cup (A+1)$ and the sets $\{x, x+1\}$ for $x \in A$. Show that S is $\mu$-measurable but that there does not exist any Borel section for S.