Lemma 3

— Let G be a locally compact group, $\mu$ a left Haar measure on G, $\nu$ and $\nu'$ two nonzero quasi-invariant measures on G. If, for every $s \in G$, the densities of $\gamma(s)\nu$ with respect to $\nu$ and of $\gamma(s)\nu'$ with respect to $\nu'$ are equal locally $\mu$-almost everywhere, then $\nu$ and $\nu'$ are proportional.

Write $\nu = \rho \cdot \mu$, $\nu' = \rho' \cdot \mu$, where $\rho, \rho'$ are locally $\mu$-integrable functions on $G$ and are everywhere nonzero ($§ 1$, No. 9, Prop. 11). For every $s \in G$,

and the hypothesis implies that $\rho^{-1} \cdot \gamma(s)\rho = {\rho'}^{-1} \cdot \gamma(s)\rho'$ locally $\mu$-almost everywhere. Set $\sigma = \rho'/\rho$, which is a $\mu$-measurable function on $G$. For every $s \in G$, $\gamma(s)\sigma = \sigma$ locally $\mu$-almost everywhere. Therefore $\sigma$ is equal to a constant locally $\mu$-almost everywhere, by Cor. 2 of Prop. 6 applied with $X = H = G$.

Let $G$ be a locally compact group, $H$ a closed subgroup of $G$. Consider the homogeneous space $G/H$ of left cosets with respect to $H$, on which $G$ operates continuously on the left. We are going to show that there exists one and only one class of nonzero quasi-invariant measures on $G/H$.

Note that $H$ operates on $G$ continuously and properly by right translations; and the quotient space, which is none other than $G/H$, is paracompact (GT, III, $§ 4$, No. 6, Prop. 13). We may therefore apply the results of Nos. 1 to 4, with $X = G$. We thus have mappings $f \mapsto f^b$ of $\mathscr{K}(G)$ onto $\mathscr{K}(G/H)$, and $\lambda \mapsto \lambda^\#$ of $\mathscr{M}(G/H)$ into $\mathscr{M}(G)$ (once a left Haar measure $\beta$ on $H$ has been fixed). The fact that $G$ operates on the left in $G/H$ gives rise to a supplementary property:

Indeed, for every $x \in G$,

whence formula (15), which implies formula (16).