Proposition 9

Let G be a locally compact group, V an open subset of G, and $\mu$ a nonzero positive measure on V having the following property: if U is an open subset of V and if $s \in G$ is such that $sU \subset V$, then the image of the measure $\mu_U$ induced by $\mu$ on U, under the homeomorphism $x \mapsto sx$ of U onto sU, is $\mu_{sU}$. Then, there exists one and only one left Haar measure $\alpha$ on G that induces $\mu$ on V.

For every $s \in G$, let $\mu_s$ be the image of $\mu$ under the homeomorphism $x \mapsto sx$ of V onto sV. The restriction of $\mu_s$ to $V \cap sV$ is the image of $\mu_{s^{-1}V \cap V}$ under the restriction of $x \mapsto sx$ to $s^{-1}V \cap V$; by hypothesis, this image is $\mu_{V \cap sV}$. By translation, one concludes from this that $\mu_s$ and $\mu_t$ have the same restriction to $sV \cap tV$ for any $s, t$. Therefore, by Prop. 1 of Ch. III, §2, No. 1, there exists a measure $\alpha$ on G that induces $\mu_s$ on sV for every $s$. It is clear that $\alpha$ is the unique left Haar measure on G inducing $\mu$ on V.