Exercise 8, § 4

Let G be a locally compact group countable at infinity, operating continuously on the left in a Polish locally compact space T. Let $\nu$ be a positive measure on T that is quasi-invariant under G. Let R be a $\nu$-measurable equivalence relation on T, compatible with G. There then exists (Ch. VI, §3, No. 4, Prop. 2) a Polish locally compact space B, and a $\nu$-measurable mapping $p$ of T into B, such that $R\{x, y\}$ is equivalent to $p(x)) = p(y)$. Let $\nu'$ be a pseudo-image measure of $\nu$ under $p$, and let $b \mapsto \lambda_b$ ($b \in B$) be a disintegration of $\nu$ by R. Show that the $\lambda_b$ are, for almost every $b \in B$, quasi-invariant under G.

(Let $\chi$ be a function on $G \times T$ satisfying the conditions of Exer. 13 of Ch. VII, §1. Show that for every $s \in G$, there exists a $\nu'$-negligible subset N(s) of B such that $\chi(s^{-1}, \cdot)$ is locally $\lambda_b$-integrable for $b \notin N(s)$; for this, observe that for every $\psi \in \mathscr{K}(T)$, the function $x \mapsto \psi(x)\chi(s^{-1}, x)$ is $\lambda_b$-integrable except for $b$ belonging to a $\nu'$-negligible set $N(s, \psi)$, and make use of Lemma 1 of Ch. VI, §3, No. 1. Set $\lambda_{b,s}' = 0$ if $b \in N(s)$, and $\lambda_{b,s}' = \chi(s^{-1}, \cdot) \cdot \lambda_b$ if $b \notin N(s)$. Show that the mapping $b \mapsto \lambda_{b,s}'$ is $\nu'$-adequate (use Lemma 3 of Ch. VI, §3, No. 1) and that $\gamma(s)\beta = \int \lambda_{b,s}'\, d\nu'(b)$. Show that on the other hand $\gamma(s)\beta = \int \gamma(s)\lambda_b\, d\nu'(b)$ and deduce from this that, for every $s \in G$, one has $\gamma(s)\lambda_b = \chi(s^{-1}, \cdot) \cdot \lambda_b$ for almost every $b$, therefore for almost every $b$ one has $\gamma(s)\lambda_b = \chi(s^{-1}, \cdot) \cdot \lambda_b$ for almost every $s$. Then use Cor. 2 of Prop. 17 of §4 to infer that, for almost every $b$, $\gamma(s)\lambda_b$ is equivalent to $\lambda_b$ for all $s \in G$.) Show that if $\nu$ is relatively invariant under G with multiplier $\chi$, then the $\lambda_b$ are, for almost every $b$, relatively invariant with multiplier $\chi$.