Exercise 13, § 1

Let $G$ be a locally compact group operating continuously on the left in a locally compact Polish space $T$, $\beta$ a left Haar measure on $G$, and $\nu$ a positive measure on $T$ quasi-invariant under $G$. a) There exists a function $(s, x) \mapsto \chi(s, x) > 0$ on $G \times T$, locally $(\beta \otimes \nu)$-integrable, such that for every $s \in G$, the function $x \mapsto \chi(s^{-1}, x)$ is locally $\nu$-integrable and satisfies $\gamma(s)\nu = \chi(s^{-1}, \cdot) \cdot \nu$. (Show that $(s, x) \mapsto (s, sx)$ transforms $\beta \otimes \nu$ into an equivalent measure, by making use of criterion 3) of Th. 2 of Ch. V, §5, No. 5; let $\chi(s^{-1}, x)d\beta(s)\, d\nu(x)$ be this equivalent measure. For $\varphi \in \mathscr{H}(G)$ and $\psi \in \mathscr{H}(T)$,

whence $\int \psi(sx)\, d\nu(x) = \int \psi(x)\chi(s^{-1}, x)\, d\nu(x)$ except on a locally $\beta$-negligible set $N(\psi)$. Next make use of Lemma 1 of Ch. VI, §3, No. 1; then modify $\chi$ on a $(\beta \otimes \nu)$-negligible set.)

b) Show that for any $s, t$ in $G$, one has

except on a $\nu$-negligible set of values of $x$ (make use of the relation $\gamma(st)\nu = \gamma(s)(\gamma(t)\nu)$).

c) Show that the function $\chi$ of a) is determined up to a locally $(\beta \otimes \nu)$-negligible subset of $G \times T$.