Proposition 9

— We conserve the hypotheses and notations of Prop. 8. Let $\lambda$ be a positive measure on $X/H$.

a) The pair $(\pi, h)$ is $\lambda^\#$-adapted, and $\int_X h(x) \varepsilon_{\pi(x)} d\lambda^\#(x) = \lambda$.

b) The mapping $\pi$ is proper for the measure $h \cdot \lambda^\#$, and $\pi(h \cdot \lambda^\#) = \lambda$.

c) Let $k$ be a function on $X/H$, with values in a Banach space or in $\overline{\mathbf{R}}$. For $k$ to be measurable (resp. locally integrable, essentially integrable, integrable) for $\lambda$, it is necessary and sufficient that $h \cdot (k \circ \pi)$ be so for $\lambda^\#$; and, if $k$ is essentially integrable for $\lambda$, then

Let $f \in \mathscr{K}(X/H)$. Then $h \cdot (f \circ \pi) \in \mathscr{K}(X)$ and

whence a). The assertion b) is proved similarly. The assertions of c) concerning measurability, essential integrability and formula (14) may then be obtained by applying the results of Ch. V (§ 4, Prop. 3, § 5, Th. 1, § 4, Th. 2). If $k$ is $\lambda$-integrable, then $h \cdot (k \circ \pi)$ is $\lambda^\#$-integrable (Ch. V, § 3, No. 3, Th. 1). If $h \cdot (k \circ \pi)$ is $\lambda^\#$-integrable, Prop. 5 proves that $(h \cdot (k \circ \pi))^b = h^b \cdot k = k$ is $\lambda$-integrable. If $k$ is locally $\lambda$-integrable, then $h \cdot (k \circ \pi)$ is locally $\lambda^\#$-integrable (Prop. 6). Finally, suppose $h \cdot (k \circ \pi)$ locally $\lambda^\#$-integrable; for every $f \in \mathscr{H}(X/H)$, $h \cdot (k \circ \pi) \cdot (f \circ \pi)$ has compact support, and

where $M = \sup |f|$; therefore $h \cdot ((kf) \circ \pi)$ is $\lambda^\#$-integrable, consequently $kf$ is $\lambda$-integrable, by what has already been proved; this proves that $k$ is indeed locally $\lambda$-integrable.