Remark

Prop. 14 applies in particular when G is the topological semi-direct product of X by Y (GT, III, §2, No. 10). In this case, $Z = \{e\}$ and $\Omega = G$. Since $yx = xi_x(y)$ for $x \in X,\ y \in Y$, the measure $\mu_G$ is also, up to a constant factor, the image of $(\operatorname{mod}\ i_x)\mu_X \otimes \mu_Y$ under the mapping $(x, y) \mapsto yx$ of $X \times Y$ into G.