Exercise 1, § 2

Let $G$ be a locally compact group, $H$ a closed subgroup of $G$. For $\xi \in H$, set $\chi(\xi) = \Delta_H(\xi)\Delta_G(\xi)^{-1}$. Regard $H$ as operating on the right in $G$ by translations. Show that there exists on $\mathscr{X}^\chi(G)$ a nonzero positive linear form $I$ and, up to a constant factor, only one, that is invariant under left translation. (Make use of Prop. 3 with $X = G$, while taking $\mu$ to be a left Haar measure of $G$.)