invariant measure on homogenous space

Question

Let $G$ be a locally compact Hausdorff topological group, and $\Gamma$ be a discrete subgroup of $G$. Is it necessarily true that $G/\Gamma$ has a $G$ invariant measure?

Answer 1

This is true if and only if the modular function of $G$ is constantly equal to $1$ on $\Gamma$. For more details, see e.g. Deitmar and Echterhoff's Principles.of Harmonic Analysis, chapter 1.