Exercise 5, § 3

Let T be the union of Cantor’s triadic set $K \subset [0,1]$ and the interval $I = [1,2]$ of $\mathbf{R}$, and let $\mu$ be the measure induced on the compact space T by Lebesgue measure. Let P be a nonmeasurable subset of I having the power of the continuum (Ch. IV, §4, Exer. 8), and let $\psi$ be a bijection of K onto P. Consider the equivalence relation R in T for which every point $x$ not belonging to $K \cup P$ is its own equivalence class, and the class of a point $y \in K$ consists of $y$ and $\psi(y)$. Show that R is $\mu$-measurable, but the saturation of K for R is not $\mu$-measurable.