Let $G$ be a locally compact group with closed subgroup $H$. Let $m$ be a probability Radon measure on $G/H$ such that for every $g\in G$ the measures $g_*m$ and $m$ are Radon-Nykodym equivalent, which implies that we have $g_*m=\mu(g)m$ for some $\mu:G\to C(G/H,\mathbb{R}^\times_{>0})\cong C(G/H,\mathbb{R})$. The space $C(G/H,\mathbb{R})$ is a $\mathbb{C}[G]$-module and the map $\mu$ is a cocycle for the cohomology group $H^1\big (G,C(G/H,\mathbb{R}\big )$. I would like to understand the induced cohomology class. I know it is non-trivial if $G/H$ does not have a $G$-invariant measure and that it does not depend on the choice of $m$.
My questions are: Does it generate the cohomology? If not, does it play a significant role otherwise? How large is the cohomology? Can a natural basis be given?
I think, the answers are known, but as this is not my field, I don't know where to start looking.
