Exercise 2, § 5

Let $\nu$ be Lebesgue measure on $\mathbf{R}$, g a continuous mapping of $\mathbf{R}$ into $[0, 1]$ with support contained in $]-1, 2[$ and equal to 1 on $[0, 1]$, and let $\mu$ be the measure $g \cdot \nu$ on $\mathbf{R}$. The set $H_0 \subset [0, 1]$ defined in Exer. 8 of §4 is not $\mu$-measurable, but the set $H = H_0 - 2$ is $\mu$-negligible.

a) If one sets $f(x) = x - 2$, then $f$ is continuous and $\varphi_H$ is $\mu$-measurable, but the composed function $\varphi_H \circ f$ is not $\mu$-measurable.

b) If one sets $h(x) = x + 2$, the image $h(H)$ of the $\mu$-measurable set H under the continuous function $h$ is not $\mu$-measurable.