Proposition 6

— Let $\lambda$ be a positive measure on $X/H$.

a) Let $N$ be a subset of $X/H$. For $N$ to be locally $\lambda$-negligible, it is necessary and sufficient that $\pi^{-1}(N)$ be locally $\lambda^\#$-negligible.

b) Let g be a function on X/H, with values in a topological space. For g to be $\lambda$-measurable, it is necessary and sufficient that $g \circ \pi$ be $\lambda^\#$-measurable.

c) Let h be a function on X/H, with values in a Banach space or in $\overline{\mathbf{R}}$. For h to be locally $\lambda$-integrable, it is necessary and sufficient that $h \circ \pi$ be locally $\lambda^\#$-integrable, in which case $(h \cdot \lambda)^\# = (h \circ \pi) \cdot \lambda^\#$.

Suppose $h \circ \pi$ is locally $\lambda^\#$-integrable. For every $f \in \mathscr{K}(X)$, $f \cdot (h \circ \pi)$ is $\lambda^\#$-integrable, therefore (Prop. 5) the function $(f \cdot (h \circ \pi))^b = f^b \cdot h$ is $\lambda$-integrable and

Since $f \mapsto f^b$ is a surjective mapping of $\mathscr{K}(X)$ onto $\mathscr{K}(X/H)$, this shows that h is locally $\lambda$-integrable and that

In particular, if $\pi^{-1}(N)$ is locally $\lambda^\#$-negligible, then $\varphi_N \circ \pi$ is locally $\lambda^\#$-negligible, therefore $(\varphi_N \cdot \lambda)^\# = (\varphi_N \circ \pi) \cdot \lambda^\# = 0$, consequently $\varphi_N \cdot \lambda = 0$ and N is locally $\lambda$-negligible. Now suppose that $g \circ \pi$ is $\lambda^\#$-measurable. Let K' be a compact subset of X/H. Let $f \in \mathscr{K}_+(X)$ be such that $f^b = 1$ on K' (No. 1, Prop. 2), and let $K = \operatorname{Supp} f$; one has $\pi(K) \supset K'$. There exists a partition of K formed of a $\lambda^\#$-negligible set M and a sequence $(K_n)$ of compact sets such that $(g \circ \pi)|K_n$ is continuous for every n. Then $g|_{\pi(K_n)}$ is continuous. Let P be the set of points of K' not belonging to $\pi(K_1) \cup \pi(K_2) \cup \ldots$; then $\pi^{-1}(P) \cap K$ is contained in M, hence is $\lambda^\#$-negligible; therefore $f \cdot \varphi_{\pi^{-1}(P)}$ is $\lambda^\#$-negligible; it follows (Prop. 5) that

therefore P is $\lambda$-negligible, and g is $\lambda$-measurable.

If N is locally $\lambda$-negligible, then $\pi^{-1}(N)$ is locally $\lambda^\#$-negligible (Ch. V, §6, No. 6, Cor. 1 of Prop. 10). If g is $\lambda$-measurable, then $g \circ \pi$ is $\lambda^\#$-measurable (ibid.). Finally, suppose h is locally $\lambda$-integrable. Then we already know that $h \circ \pi$ is $\lambda^\#$-measurable. For every $f \in \mathscr{K}_+(X)$ we have, by Prop. 5,

therefore $h \circ \pi$ is locally $\lambda^\#$-integrable.