Remark 2

If G is a Lie group, we shall see later that the function $\rho$ of Th. 2 can be chosen to be infinitely differentiable.*

Under the conditions of Th. 2, certain results of Nos. 3 and 4 may be specialized as follows (on taking into account Ch. V, §4, Th. 2 and Prop. 2 for passing from properties relative to $\mu$ to properties relative to $\rho \cdot \mu$):

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

b) Let $f$ be a $\mu$-measurable function $\geqslant 0$ on G, zero outside a countable union of $\mu$-integrable sets. Then, the function

on $G/H$ is $\lambda$-measurable and

c) Let $f$ be a $\rho \cdot \mu$-integrable function on G, with values in a Banach space or in $\overline{\mathbf{R}}$. Then, the set of $\dot{x} \in G/H$ such that $\xi \mapsto f(x\xi)$ is not $\beta$-integrable is $\lambda$-negligible; the function $\dot{x} \mapsto \int_{H} f(x\xi) d\beta(\xi)$ is $\lambda$-integrable, and

d) There exists a continuous function $h \geqslant 0$ on G, whose support has compact intersection with the saturation KH of every compact subset K of G, and such that $\int_{H} h(x\xi) d\beta(\xi) = 1$ for every $x \in G$. For a function g on G/H to be measurable (resp. locally integrable, essentially integrable, integrable) for $\lambda$, it is necessary and sufficient that $h \cdot (g \circ \pi)$ be so for $\rho \cdot \mu = \lambda^{\sharp}$; and, when g is essentially integrable for $\lambda$, one has