Lemma 2

*Let T, T', T'' be three locally compact spaces, $\mu$ a measure on T, $\pi$ a $\mu$-measurable mapping of T into T', $\pi'$ a continuous mapping of T' into T'', and $\pi'' = \pi' \circ \pi$. If $\pi''$ is $\mu$-proper, then $\pi$ is $\mu$-proper, $\pi'$ is $\pi(\mu)$-proper, and $\pi''(\mu) = \pi'(\pi(\mu))$.

Let K' be a compact subset of T'. Then $K'' = \pi'(K')$ is compact, therefore $\pi''(K'')$ is essentially $\mu$-integrable, therefore $\pi^{-1}(K') \subset \pi''(K'')$ is essentially $\mu$-integrable, thus $\pi$ is $\mu$-proper. Then $\pi'$ is $\pi(\mu)$-proper and $\pi''(\mu) = \pi'(\pi(\mu))$ by Ch. V, §6, No. 3, Prop. 4.