Exercise 7, § 6

Let $\mu$ be Lebesgue measure on $\mathbf{R}$, I the interval $[0,1]$ of $\mathbf{R}$. If one sets $\theta(x) = |x|$, $\theta$ is proper for the measure $\varphi_I \cdot \mu$ and one has $\theta(\varphi_I \cdot \mu) = \varphi_I \cdot \mu$. Give an example of a set negligible for $\varphi_I \cdot \mu$, whose image under $\theta$ is not measurable for $\varphi_I \cdot \mu$ (cf. Ch. IV, §4, Exer. 8).