Theorem 1

Let $G$ be a locally compact group, $H$ a closed subgroup of $G$.

a) Any two nonzero quasi-invariant measures on $G/H$ are equivalent; the subsets of $G/H$ locally negligible for these measures are those whose inverse image in $G$ is locally negligible for a Haar measure.

b) Let $\lambda, \lambda'$ be two nonzero quasi-invariant measures on $G/H$. If, for every $s \in G$, the densities of $\gamma_{G/H}(s)\lambda$ with respect to $\lambda$ and of $\gamma_{G/H}(s)\lambda'$ with respect to $\lambda'$ are equal almost everywhere for $\lambda$ (or $\lambda'$), then $\lambda$ and $\lambda'$ are proportional.

The assertion a) follows at once from Lemma 4. Let $\lambda$ and $\lambda'$ be two nonzero quasi-invariant measures satisfying the condition of b). Then, for every $s \in G$, the densities of $\gamma_G(s)\lambda^\#$ with respect to $\lambda^\#$ and of $\gamma_G(s){\lambda'}^\#$ with respect to ${\lambda'}^\#$ are equal locally $\mu$-almost everywhere, therefore (Lemma 3) $\lambda^\#$ and ${\lambda'}^\#$ are proportional, hence $\lambda$ and $\lambda'$ are proportional.

On the other hand, Lemma 4 reduces the search for nonzero quasi-invariant measures on $G/H$ to that for the measures on $G$ equivalent to

Haar measure and of the form $\lambda^\#$. On this subject we have the following lemma: