Proposition 5

Let $T$ and $X$ be two locally compact spaces, $\mu$ a positive measure on $T$, $\pi$ a $\mu$-proper mapping of $T$ into $X$, $g$ a finite numerical function $\geqslant 0$, defined on $X$ and such that $g \circ \pi$ is locally integrable for $\mu$. In order that $g$ be locally integrable for $\pi(\mu)$, it is necessary and sufficient that $\pi$ be proper for the measure $(g \circ \pi) \cdot \mu$, in which case

Set $\nu = \pi(\mu)$. For $g$ to be locally $\nu$-integrable, it is necessary and sufficient that $gf$ be $\nu$-integrable for every function $f \in \mathscr{H}(X)$; since $gf$ has compact support, it comes to the same to say that $gf$ is essentially $\nu$-integrable, and this is equivalent to saying that $(g \circ \pi)(f \circ \pi)$ is essentially $\mu$-integrable (Th. 1). But, by Th. 1 of §5, No. 3, this signifies that $f \circ \pi$ is essentially integrable for $\rho = (g \circ \pi) \cdot \mu$, and, by definition, this says that $\pi$ is $\rho$-proper (since $\pi$ is obviously $\rho$-measurable). Moreover,

(No. 2, Th. 1 and §5, Th. 1), which proves the relation (4).