Existence of invariant measures on cosets Let $G$ a locally compact group. It is know it admits a left Haar measure (e.g. as a consequence of Brouwer fixed point theorem).
Let $L$ be a subgroup of $G$. The space $G/L$ of right cosets of $G$ modulo $L$ seems to have a right $G$-invariant Borel measure.
Is there an easy proof of this fact, or a clear reference? I wonder in particular if certain properties are required for $L$.

You need $H$ to be closed in $G$. Even that is not enough, though, as there is a rather technical condition involving the modular functions of $G$ and $H$.
In particular, you want $\Delta_G \restriction H = \Delta_H$.
See here for more information online, or Theorem 2.51 in Folland's "A Course in Abstract Harmonic Analysis" (page 62 of my edition). That entire section (2.6, "Homogenous Spaces") will probably be of interest.

I hope this helps ^_^
