Lemma 7

Let K be a compact subset of X/H', and L a compact subset of X. Then $\bigcup_{x \in L} \omega_x^{-1}(K)$ is relatively compact in H/H'.

Let $K_1$ be a compact subset of X such that $\pi'(K_1) = K$. Let $K_2$ be the set of $\xi \in H$ such that $L\xi$ intersects $K_1$. Then $K_2$ is compact (GT, III, §4, No. 5, Th. 1). Let $\xi \in H$ be such that $p(\xi) \in \bigcup_{x \in L} \omega_x^{-1}(K)$. Thus, there exists an $x \in L$ such that $\omega_x(p(\xi)) \in K$, in other words such that $\pi'(x\xi) \in K$. Since $\pi'(K_1) = K$, there exists $\xi' \in H'$ such that $x\xi\xi' \in K_1$. Then $\xi\xi' \in K_2$, therefore $p(\xi) = p(\xi\xi') \in p(K_2)$. We have thus shown that $\bigcup_{x \in L} \omega_x^{-1}(K) \subset p(K_2)$.

This lemma shows first of all that the mapping $\omega_x$ is proper. One can therefore form the measure $\omega_x(\beta/\beta')$ on $X/H'$, which is concentrated on $\omega_x(H/H') = \pi'(\psi_x(H)) = \pi'(xH)$. If $f \in \mathscr{K}(X/H')$, Lemma 7 and §1, No. 1, Lemma 1 show that the function $x \mapsto \langle f, \omega_x(\beta/\beta') \rangle$ is continuous on X; moreover, $\langle f, \omega_x(\beta/\beta') \rangle$ is zero when Supp $f$ does not intersect $\pi'(xH)$, in other words when $\pi(x)$ does not belong to the canonical image of Supp $f$ in $X/H$.

Moreover, if $x \in H$ then

The mapping $x \mapsto \omega_x(\beta/\beta')$ of X into $\mathscr{M}(X/H')$ therefore defines by passage to the quotient a mapping $u \mapsto (\beta/\beta')_u$ of $X/H$ into $\mathscr{M}(X/H')$. The foregoing shows that, for every $f \in \mathscr{K}(X/H')$, the mapping $u \mapsto \langle f, (\beta/\beta')_u \rangle$ is continuous with compact support. Consequently the mapping $u \mapsto (\beta/\beta')_u$ is a vaguely continuous and $(\mu/\beta)$*-adequate family of measures on* $X/H'$, with $X/H$ as index set.

Let $x \in X$, and $u = \pi(x) \in X/H$. Let $f$ be a function on $X/H'$, with values in a Banach space or in $\overline{\mathbf{R}}$. By Ch. V, §4, Th. 2, for $f$ to be $(\beta/\beta')_u$-integrable, it is necessary and sufficient that the function $p(\xi) \mapsto f(\omega_x(p(\xi))) = f(\pi'(x\xi))$ on $H/H'$ be $(\beta/\beta')$-integrable, in which case

One has analogous properties for measurability, the upper integral and the essential integral.