Corollary

Let $T$ and $T'$ be two locally compact spaces, $\mu$ a positive measure on $T$, $\pi$ a bijective mapping of $T$ onto $T'$, and $\pi^{-1}$ the inverse mapping. Suppose that $\pi$ is $\mu$-proper, and let $\mu' = \pi(\mu)$. Then $\pi^{-1}$ is $\mu'$-proper and $\pi^{-1}(\pi(\mu)) = \mu$.