Exercise 8, § 2

Let $\mathfrak{G}$ be a locally compact group, $G$ and $H$ two closed subgroups of $\mathfrak{G}$. Assume that every $s \in \mathfrak{G}$ can be written in one and only one way in the form $s = \xi x = y \eta$ $(\xi, \eta \in G,\ x, y \in H)$, and that $\xi, x, y, \eta$ depend continuously on $s$. Every $\xi \in G$ defines a homeomorphism $\widehat{\xi}$ of H onto H by $\xi x \in \widehat{\xi}(x)G\ (x \in H)$. Every $x \in H$ defines a homeomorphism $\widehat{x}$ of G onto G by $\xi x \in H \widehat{x}(\xi)\ (\xi \in G)$. Let $\mu, \alpha, \beta$ be left Haar measures on $\mathfrak{G}, G, H$. Show that

(Let $f \in \mathscr{K}(G)$. Express the fact that $\int f(u) d\mu(u)$, calculated with the help of (23), either for $X = G,\ Y = H$, or for $X = H,\ Y = G$, is invariant when $f$ is subjected to left translation by an element of G or an element of H.)