Exercise 2, § 2

Let $X$ and $X'$ be two locally compact spaces in which a locally compact group $H$ operates on the right, continuously and properly. Let $\theta$ be a proper continuous mapping of $X$ into $X'$, compatible with the identity mapping of $H$ (GT, III, §2, No. 4), and let $\theta' : X/H \to X'/H$ be the mapping deduced from $\theta$ by passage to quotients. Let $f'$ be a continuous function on $X'$ whose support has compact intersection with the saturation of each compact set. Then $f = f' \circ \theta$ has the same properties in $X$, and

(the mappings $f \mapsto f^b ,\ f' \mapsto {f'}^b$ being relative to the choice of a same Haar measure on $H$).