Exercise 12, § 3

Let G be a locally compact group that is not unimodular.

a) Show that there exists a bounded positive measure $\mu$ on G such that $\Delta_G \cdot \mu$ is not bounded (take $\mu$ to be discrete).

b) Let $\mu'$ be a left Haar measure on G. Then $\mu$ and $\mu'$ are convolvable (Prop. 5). Show that $\mu'$ and $\mu$ are not convolvable.