Exercise 5, § 5

Let $G$ be the discrete group that is the direct sum of an infinite sequence $(G_n)$ of subgroups with two elements, and let $p_n$ be the projection of $G$ onto $G_n$. Set $H_n = \overline{p_n}(e)$, and let $\alpha_n$ be the normalized Haar measure of $H_n$. If $\alpha$ is the normalized Haar measure of $G$, show that in $\Gamma_c^0$ the sequence $(\alpha_n)$ converges to $\alpha$ but that $\| \mu_{\alpha_n} \| = 2$ and $\| \mu_\alpha \| = 1$.