Definition 1

With hypotheses and notations as in Prop. 4, $\lambda$ is called the quotient of $\mu$ by $\beta$ and is denoted $\frac{\mu}{\beta}$ or $\mu/\beta$.

The mapping $\lambda \mapsto \lambda^\#$ of $\mathscr{M}(X/H)$ into $\mathscr{M}(X)$ is none other than the transpose of the mapping $f \mapsto f^b$ of $\mathscr{K}(X)$ into $\mathscr{K}(X/H)$. Let $\mathfrak{F}$ be a filter on $\mathscr{M}(X/H)$; to say that $\lim_{\lambda,\mathfrak{F}} \lambda^\#(f) = 0$ for all $f \in \mathscr{K}(X)$ is equivalent to saying that $\lim_{\lambda,\mathfrak{F}} \lambda(f') = 0$ for all $f' \in \mathscr{K}(X/H)$; therefore the mapping $\lambda \mapsto \lambda^\#$ is, for the vague topologies, an isomorphism of $\mathscr{M}(X/H)$ onto a linear subspace of $\mathscr{M}(X)$. This subspace is vaguely closed, since it is the set of $\mu \in \mathscr{M}(X)$ such that $\delta(\xi)\mu = \Delta_H(\xi)\mu$ for all $\xi \in H$. It is clear that the conditions $\lambda \geqslant 0$ and $\lambda^\# \geqslant 0$ are equivalent.

The formula (6) may, by analogy with the usual notation for double integrals, be written

This involves an abuse of notation, the integral $\int_H f(x\xi) d\beta(\xi)$ being regarded as a function of $\dot{x}$ and not of $x$; this manner of writing will be used frequently in what follows provided no confusion can arise.