Proposition 7

Assume $X/H$ paracompact. Let $\chi$ be a continuous representation of $H$ in $\mathbf{R}_+^*$.

a) There exists on $X$ a continuous function $r$, with values $> 0$, such that $r(x\xi) = \chi(\xi)r(x)$ for all $x \in X$ and $\xi \in H$.

b) The mapping $g \mapsto g/r$ is an isomorphism of the vector space $\mathscr{K}^\chi(X)$ onto the vector space $\mathscr{K}^1(X)$.

Let us apply Prop. 1 of No. 1 on taking $f$ to be a function $\geqslant 0$ that is not identically zero on any orbit (this is possible by Lemma 1 of Appendix 1); then $r = f^\chi$ satisfies the properties of a). The assertion b) is obvious.