Corollary 1

— a) Let f be a function on X/H', with values in a Banach space or in $\overline{\mathbf{R}}$, integrable for $\mu/\beta'$. There exists a $(\mu/\beta)$-negligible subset N of X/H having the following property: if $x \in X$ is such that $\pi(x) \notin N$, then the function $f \circ \omega_x$ on H/H', that is, the function $\dot{\xi} \mapsto f(\pi'(x\xi))$, is integrable for $\beta/\beta'$. The integral $\int_{H/H'} f(\pi'(x\xi)) d(\beta/\beta')(\dot{\xi})$ depends only on $\dot{x} = \pi(x)$, and is a $(\mu/\beta)$-integrable function of $\dot{x}$; and

b) Let $f$ be a function $\geq 0$ on $X/H'$, measurable for $\mu/\beta'$ and zero outside a countable union of $(\mu/\beta')$-integrable sets. Then

is $(\mu/\beta)$-measurable, and

c) Let $f$ be a function on $X/H'$ with values in a Banach space or in $\overline{\mathbf{R}}$, measurable for $\mu/\beta'$ and zero outside a countable union of $(\mu/\beta')$-integrable sets. Then, for $f$ to be $(\mu/\beta')$-integrable, it is sufficient that