Proposition 5

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

a) Let $f$ be a $\lambda^\#$-measurable function on $X$, with values in a topological space, constant outside a countable union of $\lambda^\#$-integrable sets. Then, the set of $\dot{x} \in X/H$ such that the function $\xi \mapsto f(x\xi)$ is not $\beta$-measurable is locally $\lambda$-negligible.

b) Let $f$ be a $\lambda^\#$-measurable function $\geq 0$ on $X$, zero outside a countable union of $\lambda^\#$-integrable sets. Then, the function $\dot{x} \mapsto \int^* f(x\xi) d\beta(\xi)$ on $X/H$ is $\lambda$-measurable, and

c) Let $f$ be a $\lambda^\#$-integrable function on $X$, with values in a Banach space or in $\overline{\mathbf{R}}$. Then, the set of $\dot{x} \in X/H$ such that $\xi \mapsto f(x\xi)$ is not $\beta$-integrable is $\lambda$-negligible; the function $f^b$ on $X/H$ defined almost everywhere by the formula

is $\lambda$-integrable, and

and

d) Let $f$ be a $\lambda^\#$-measurable function on $X$, with values in a Banach space or in $\overline{\mathbf{R}}$, and zero outside a countable union of $\lambda^\#$-integrable sets. Then, for $f$ to be $\lambda^\#$-integrable, it is necessary and sufficient that

Taking into account Lemma 2, the assertions a), b) and c) follow from Ch. V, §3, Prop. 4, Prop. 5 and Th. 1 (with the exception of (13), which follows from (12) because it is clear that $|f^b| \leq |f|^b$); d) follows from b).