Proposition 11

Let $\pi$ be a local homeomorphism of a locally compact space $X$ into a locally compact space $T$, and let $\mu$ be a positive measure on $T$. Let $n$ be the numerical function that associates to every $t \in T$ the number of elements of $\overline{\pi}^{-1}(t)$ if this number is finite, and $+\infty$ in the contrary case. For $\pi$ to be $\overline{\pi}^{-1}(\mu)$-proper, it is necessary and sufficient that $n$ be locally $\mu$-integrable, in which case