Corollary 1

— Suppose that $\pi$ is $\nu$-measurable.

a) If $N \subset T$ is locally $\mu$-negligible, then $\overline{\pi}^{-1}(N)$ is locally $\nu$-negligible. b) If $f$ is a $\mu$-measurable mapping of $T$ into a topological space $G$, then $f \circ \pi$ is $\nu$-measurable.

We take up again the notations $\Lambda_h$, $\nu_h$, $g_h$ of the end of the preceding proof: $\nu_h$ being a bounded measure for every $h \in \mathscr{H}$, $\pi$ is $\nu_h$-proper, $g_h$ is locally $\mu$-integrable, and $\pi(\nu_h) = g_h \cdot \mu$, a measure with base $\mu$. It follows that $N$ is locally negligible (resp. that $f$ is measurable) for the measure $\pi(\nu_h)$ (§ 5, No. 3, Cor. 1 of Prop. 3 and Prop. 4). Consequently $\overline{\pi}^{-1}(N)$ is locally negligible (resp. $f \circ \pi$ is measurable) for the measure $\nu_h$ (Cor. 2 of

Prop. 2, resp. Prop. 3). Finally, one notes that the measures $\nu_h$ form an increasing directed family of positive measures whose supremum is $\nu$, and one applies Cor. 1 (resp. Cor. 2) of Prop. 11 of §1, No. 4.