Let $G$ be a locally compact Hausdorff group. A discrete subgroup $\Gamma\le G$ is called lattice, if $G/\Gamma$ admits a non-trivial finite invariant Radon measure (or regular Borel measure, if you wish). On Wikipedia it is stated that this definition is equivalent to $\Gamma$ having a fundamental domain of finite volume, i.e. some measurable $V\subseteq G$ such that $V\Gamma=G$ and $V\gamma_1\cap V\gamma_2=\emptyset$ for $\gamma_1\ne\gamma_2$ and $m(V)<\infty$, where $m$ is the Haar measure on G.
The statement that every lattice admits a measurable fundamental domain is somewhat standard in the literature, however I only ever saw it being proven for Abelian groups or groups with some other extra assumption such as $\sigma$-compactness.
How do I prove that every lattice on a locally compact Hausdorff group admits a fundamental domain?