Theorem 4

*Let $X$ be a locally compact space countable at infinity, $H$ a discrete group operating continuously and properly on the right in $X$, π the canonical mapping of X onto X/H, μ a positive measure on X invariant under H, β the normalized Haar measure of H, and λ = μ/β. Let F be a μ-measurable fundamental domain.

a) The pair $(\pi, n^{-1}\varphi_F)$ is μ-adapted, and

b) The mapping π is proper for $n^{-1}\varphi_F \cdot \mu$, and $\pi(n^{-1}\varphi_F \cdot \mu) = \lambda$.

c) Let k be a function on X/H. For k to be λ-measurable (resp. λ-integrable), it is necessary and sufficient that $n^{-1}\varphi_F(k \circ \pi)$ be μ-measurable (resp. μ-integrable); and, if k is λ-integrable then

We have $\mu = \lambda^\sharp$. Let $f \in \mathscr{H}_+(X/H)$. Then $n^{-1}\varphi_F(f \circ \pi)$ is μ-measurable and $\geq 0$, and by Prop. 5 b) of No. 3 we have

and $\int_H^* n(x\xi)^{-1} \varphi_F(x\xi) d\beta(\xi) = n(x)^{-1} \sum_{\xi \in H} \varphi_F(x\xi) = 1$ by Lemma 8. Thus $n^{-1}\varphi_F \cdot (f \circ \pi)$ is μ-integrable and

This proves a). The assertion b) is proved similarly. The assertion c) may be deduced from a) and from Ch. V, §4, Prop. 3 and Th. 2.