Exercise 2, § 4

Let H be the additive group $\mathbf{R}$ equipped with the discrete topology. Let G be the locally compact group $\mathbf{R} \times \mathbf{R} \times H$. Let $\alpha$ be a Haar measure on G, $\beta$ a Haar measure on $\mathbf{R} \times H$, and $\mu = \varepsilon_0 \otimes \beta \in \mathscr{M}(G)$. Construct a function $f \geq 0$ on G, locally $\alpha$-negligible (hence such that $\mu$ and $f$ are convolvable) but such that no translate of $f$ is $\mu$-measurable. (Imitate the construction of Ch. V, §3, Exer. 4.)