Exercise 3, § 5

Let $G$ be a locally compact group, $\chi$ a continuous representation of $G$ in $\mathbf{R}_+^*$. Generalize Props. 4, 5 and 6 to the set $\Gamma_\chi$ of measures $\alpha \in \Gamma$ such that the restriction of $\chi$ to $H_\alpha$ is the modulus of $\alpha$. (Replace the measure $\mu$ by $\chi \cdot \mu$ and consider the quotient measures $(\chi \cdot \mu)/\check{\alpha}$.)