Exercise 2, § 1

Let $G$ be a locally compact group and $\Gamma$ the compact space obtained by adjoining to $G$ a point at infinity $\omega$. One extends the law of composition of $G$ to $\Gamma$ by setting $x \omega = \omega x = \omega$ for every $x \in \Gamma$. To every measure $\mu$ on $\Gamma$ there corresponds, on the one hand a bounded measure $\mu_1$ on $G$, on the other a complex number $\mu(\{\omega\})$. Show that if $*$ (resp. $\widehat{*}$) denotes the convolution defined by the multiplication in $G$ (resp. $\Gamma$), then $(\mu \widehat{*} \nu)_1 = \mu_1 * \nu_1$ and

for any two measures $\mu$ and $\nu$ on $\Gamma$.