Quasi-invariant Radon measure on G/H

Let $G$ be a locally compact Hausdorff group and $H\le G$ a closed subgroup. Write $X=G\mathrm{/}H$ and let $g_{\ast }\mu (E)=\mu (g^{-1}E)$ for the pushforward under the left action. A nonzero Radon measure $\mu$ on $X$ is quasi-invariant if $g_{\ast }\mu$ and $\mu$ are equivalent for every $g\in G$, meaning they have the same null Borel sets. A representative is strongly quasi-invariant when the Radon–Nikodym densities $d(g_{\ast }\mu )\mathrm{/}d\mu$ can be chosen jointly continuous and positive as a function of $(g,xH)$.

The first condition depends only on the measure class. The stronger condition names a regular representative and a continuous density cocycle; it is the version constructed from a rho-function below.