Lemma 2

Let f be a function on X, with values in a topological space.

a) If f is a numerical function $\geq 0$ then, for $x \in X$,

b) For f to be $\beta_{\dot{x}}$-measurable, it is necessary and sufficient that the function $\xi \mapsto f(x\xi)$ on H be $\beta$-measurable.

c) Suppose that f is a function on X, with values in a Banach space or in $\overline{\mathbf{R}}$; then, for f to be $\beta_{\dot{x}}$-integrable (resp. essentially $\beta_{\dot{x}}$-integrable), it is necessary and sufficient that the function $\xi \mapsto f(x\xi)$ on H be $\beta$-integrable (resp. essentially $\beta$-integrable), in which case $\int_X f(y)\, d\beta_{\dot{x}}(y) = \int_H f(x\xi)\, d\beta(\xi)$.

This follows from Ch. V, §4, Prop. 2, Prop. 3 and Th. 2.

Since $f^\flat \in \mathscr{K}(X/H)$ for $f \in \mathscr{K}(X)$, formula (8) proves that the mapping $u \mapsto \beta_u$ of $X/H$ into $\mathscr{M}(X)$ is vaguely continuous, that the family $(\beta_u)$ is $\lambda$-adequate$^1$ for any positive measure $\lambda$ on $X/H$, and that

which furnishes a new interpretation of $\lambda^\#$.

$^1$ In the sense of the first edition of Ch. V, hence a fortiori in the sense of the second (cf. the footnote to the Example of Ch. VI, §1, No. 1).