Exercise 6, § 3

Let T be a Polish locally compact space, $\mu$ a positive measure on T, R a $\mu$-measurable equivalence relation in T, $p$ a $\mu$-measurable mapping of T into a Polish locally compact space B such that $p(x) = p(y)$ is equivalent to $R\{x,y\}$, $\nu$ a pseudo-image measure of $\mu$ under $p$, and $b \mapsto \lambda_b$ a disintegration of $\mu$ by R. For every $b \in B$ such that $\lambda_b \neq 0$, let $C(b)$ be the class mod R that carries $\lambda_b$; show that $\varphi_{C(b)} \cdot \mu$ is proportional to $\lambda_b$. Give an example where $\varphi_{C(b)} \cdot \mu = 0$ for every $b \in B$.