Given a locally compact group $G$, we can consider $G$ acting on its self by conjugation. Is there a quasiinvariant measure on the quotient space of $G$ with repect to multiplication from one side with an integral formula?

Reltaed example: Let $H$, a closed subgroup act on $G$ by multiplication from the right, then there is a formula due to George Mackey $$ \int_G f(g) \rho(g) d g = \int_{G/H} \int_H f(gh) d h d \mu(gH), $$ where $\mu$ is quasiinvariant under the left multiplication by $G$ on $G/H$.