Exercise 5, § 8

Let $G$ be a Hausdorff topological group, $H$ a closed subgroup of $G$ and $\pi$ the canonical mapping of $G$ onto $G/H$. Suppose that $H$ is a finite-dimensional real Lie group. There exist a neighbourhood $U$ of $\pi(e)$ in $G/H$ and a continuous mapping $\sigma$ of $U$ into $G$ such that $\pi \circ \sigma = \mathrm{Id}_U$. (Let $\rho$ be an analytic linear representation of $H$ on $\mathbf{GL}(n, \mathbf{R})$, which is locally a homeomorphism (§ 6, Corollary to Theorem 1). Let $f$ be a continuous function $\geqslant 0$ on $G$, equal to 1 at $e$ and zero outside a sufficiently small neighbourhood of $e$. Let $ds$ be a left Haar measure of $H$. For $x \in G$, we write

Then $g(xt) = g(x)\rho(t)$ for $x \in G$ and $t \in H$. If $V$ is sufficiently small,

for $x$ sufficiently close to $e$. Finally use the fact that the theorem to be proved is true locally for $\mathbf{GL}(n, \mathbf{R})$ and $\rho(H)$.