Proposition 3

Let $\mu$ be a measure on $X$. The following conditions are equivalent:

a) There exists a relatively bounded linear form I on $\mathscr{K}^\chi(X)$ such that $I(f^\chi) = \mu(f)$ for all $f \in \mathscr{K}(X)$. b) $\delta(\xi)\mu = \chi(\xi)^{-1} \Delta_H(\xi)\mu$ for all $\xi \in H$. c) For all $f, g$ in $\mathscr{K}(X)$,

(2)

d) If $f \in \mathscr{K}(X)$ is such that $f^\chi = 0$, then $\mu(f) = 0$.

a) $\Rightarrow$ b): If $\mu(f) = I(f^\chi)$ then, taking into account Prop. 1 d),

whence $\delta(\xi)\mu = \chi(\xi)^{-1} \Delta_H(\xi)\mu$.

b) $\Rightarrow$ c): Suppose hypothesis b) is satisfied. Note that the functions $(x, \xi) \mapsto f(x)g(x\xi)$ and $(x, \xi) \mapsto f(x\xi)g(x)$ on $X \times H$ are continuous with compact support (because $H$ operates properly in $X$); this established, Th. 2 of Ch. III, §4, No. 1 permits us to write:

which proves c).

c) $\Rightarrow$ d): If c) is verified and if $f^\chi = 0$, then $\mu(f \cdot g^1) = 0$ for all $g \in \mathscr{K}(X)$, thus $\mu(f) = 0$ on choosing $g \in \mathscr{K}(X)$ such that $g^1 = 1$ on $\mathrm{Supp}\, f$ (which is possible by Prop. 2 applied with $\chi = 1$).

d) $\Rightarrow$ a): If condition d) is satisfied, there exists a linear form I on $\mathscr{K}^\chi(X)$ such that $\mu(f) = I(f^\chi)$ for $f \in \mathscr{K}(X)$, and this form is relatively bounded by virtue of Prop. 2.