Remark

— 1) If $\mu$ is bounded (in particular, if $\mu$ has compact support) then every $\mu$-measurable mapping of $T$ into $X$ is $\mu$-proper (Ch. IV, § 5, No. 3, Th. 1 and No. 6, Th. 5).

2) If $\pi$ is $\mu$-measurable and if, for every compact subset $K$ of $X$, $\overline{\pi}^{-1}(K)$ is relatively compact, then $\pi$ is $\mu$-proper (Ch. IV, § 5, No. 5, Prop. 7 and No. 6, Th. 5); in particular, every proper continuous mapping of $T$ into $X$ (GT, I, § 10, No. 2, Th. 1) is $\mu$-proper for every positive measure $\mu$ on $T$. More particularly, this is true of every homeomorphism $\pi$ of $T$ onto X; the measure $\nu = \pi(\mu)$ is then none other than the measure on X that is the transport of $\mu$ by $\pi$ (Ch. III, §1, No. 3).

3) Suppose that the topology of X admits a countable base; then every mapping $\pi$ of T into X that satisfies condition b) of Prop. 1 is $\mu$-measurable, hence is $\mu$-proper. It suffices to apply Th. 4 of Ch. IV, §5, No. 5, on observing that X is then metrizable (GT, IX, §2, No. 9, Cor. of Prop. 16) and that, for any metric compatible with the topology of X, every closed ball is a countable union of compact sets.