Proposition 13

— *Suppose in addition that Z is compact. Let $\mu_G, \mu_X, \mu_Y$ be left Haar measures on G, X, Y, and $\Lambda$ the restriction of $\Delta_G$ to Y. Then the restriction $\mu$ of $\mu_G$ to $\Omega$ is, up to a constant factor, the image of $\mu_X \otimes (\Lambda^{-1} \cdot \mu_Y)$ under the mapping $(x, y) \mapsto xy^{-1}$ of $X \times Y$ onto $\Omega$ (a mapping that is proper).

For $x \in X, y \in Y$,

Identifying $\Omega$ with the homogeneous space $(X \times Y)/Z_0$ and choosing a suitable Haar measure on $Z_0$, one sees that $\mu^\#$ is the product of the left Haar measure of $X \times Y$, namely $\mu_X \otimes \mu_Y$, by the function $(x, y) \mapsto \Delta_G(y)^{-1}$ (No. 6, Lemma 6). On the other hand $\mu$ is, up to a constant factor, the image of $\mu^\#$ under the canonical mapping of $X \times Y$ onto $\Omega$ (No. 3, Remark 2).