Exercise 13, § 2

Let $G$ be a compact group, $\mu$ a Haar measure on $G$, $u$ an endomorphism of $G$ such that $u(G)$ is an open subgroup of $G$ and the kernel $\overline{u}(e)$ (denoted $G_u$) a finite subgroup of $G$.

a) Show that there exist a real number $h(u) > 0$ and an open neighborhood U of e in G such that, for every open set $V \subset U$, $u(V)$ is open in G and $\mu(u(V)) = h(u)\mu(V)$ (cf. §1, No. 7, Cor. of Prop. 9).

b) Show that $h(u) = \mathrm{Card}(G/u(G))/\mathrm{Card}(G_u)$ (calculate $\mu(u(G))$ in two ways, using a) and Prop. 10 of No. 7).