Corollary

*Let G, G' be two locally compact groups, V (resp. V') an open neighborhood of the neutral element of G (resp. G'), and $\varphi$ a local isomorphism of G' with G (GT, III, §1, No. 3, Def. 2) defined on V', such that $\varphi(V') = V$. Let $\alpha'$ be a left Haar measure on G', and $\alpha'_V$, its restriction to $V'$. Then $\varphi(\alpha'_{V'})$ is the restriction to $V$ of a unique left Haar measure $\alpha$ on $G$.

Let $V_1$ be an open neighborhood of $e$ in $G$ such that $V_1 V_1^{-1} \subset V$. Let $\mu$ be the restriction of $\varphi(\alpha'_{V'})$ to $V_1$. Let $U$ be an open subset of $V_1$ and let $s \in G$ be such that $sU \subset V_1$. Then $s \in V_1 V_1^{-1} \subset V$, therefore $s = \varphi(s')$ for some $s' \in V'$. Let $x \in U$. Then $x = \varphi(x')$ for some $x' \in V'$, therefore $sx = \varphi(s')\varphi(x') = \varphi(s'x')$ since $sx \in sU \subset V$. Since the left translations in $G'$ preserve $\alpha'$, one sees that $V_1$ and $\mu$ satisfy the conditions of Prop. 9. Let $\alpha$ be the left Haar measure on $G$ inducing $\mu$ on $V_1$. For every $t \in V$, there exists an open neighborhood $W$ of $e$ in $V_1$ such that $tW \subset V$. Then the restriction of $\varphi(\alpha'_{V'})$ to $tW$ may be deduced by translation from the restriction of $\mu$ to $W$, hence is the restriction of $\alpha$ to $tW$. Therefore $\varphi(\alpha'_{V'})$ is the restriction of $\alpha$ to $V$.

One says that $\alpha$ is deduced from $\alpha'$ by means of the local isomorphism $\varphi$.