Exercise 6, § 2

Let $G$ be a locally compact group, $G_1, G_2, G_3$ closed subgroups such that $G_3 \subset G_1 \cap G_2$. Assume that $G, G_1, G_2, G_3$ are unimodular, and that $G_1/G_3$ has finite total measure (for every measure invariant under $G_1$). Let $\lambda, \mu, \nu$ be invariant measures on $G/G_1, G/G_2, G_2/G_3$, and $\varphi$ the canonical mapping of $G$ onto $G/G_1$. Let $f \in \mathscr{K}(G/G_1)$. Show that

$(\dot{x} = xG_2,\ \dot{\xi} = \xi G_3 )$, where $a$ is a constant independent of $f$.