Remark 2

A proper continuous mapping (GT, I, §10, No. 1, Def. 1) of T into X is $\mu$-proper for every measure $\mu$. For, let $x \in X$; since $\overline{\pi}(x)$ is compact (loc. cit., No. 2, Th. 1), the set $\overline{\pi}(x)$ has an open neighborhood H such that $\mu^\bullet(H) < +\infty$. Set $V = X - \pi(T - H)$; since $\pi$ is closed, V is open in X, contains x, and satisfies $\overline{\pi}(V) \subset H$, whence $\mu^\bullet(\overline{\pi}(V)) \leq \mu^\bullet(H) < +\infty$.