Proposition 9

Let $G$ be a locally compact abelian group, generated by a compact neighborhood of $e$. For a subset $A$ of $N_c$ to be relatively compact in $N_c$, it is necessary and sufficient that it satisfy the following two conditions:

(i) There exists an open neighborhood $U$ of $e$ in $G$ such that $H_\alpha \cap U = \{e\}$ for all $\alpha \in A$.

(ii) There exists a constant $k$ such that $\mu_\alpha(G/H_\alpha) \leq k$ for all $\alpha \in A$.

If $A \subset N_c$ is relatively compact in $N_c$, it is a fortiori so in $N$, and the necessity of the conditions (i) and (ii) therefore follows from No. 3, Props. 6 and 7 (without assuming $G$ to be abelian). Conversely, suppose that $A \subset N_c$ satisfies these conditions; if $\overline{A}$ is the closure of $A$ in $N$, then $\overline{A}$ is compact by virtue of No. 3, Prop. 7; moreover, since $\alpha \mapsto \| \mu_\alpha \|$ is lower semi-continuous on $\Gamma^0$ for the vague topology (No. 2, Prop. 4), the condition (ii) implies that one also has $\| \mu_\alpha \| \leq k$ for all $\alpha \in \overline{A}$. Now, since $G$ is abelian, $\mu_\alpha = \mu/\alpha$ is a Haar measure on the group $G/H_\alpha$, and $G/H_\alpha$ is therefore compact for every $\alpha \in \overline{A}$ (Ch. VII, §1, No. 2, Prop. 2). This means that $\overline{A} \subset N_c$, thus $A$ is relatively compact in $N_c$.