Exercise 7, § 3

Let T be a Polish locally compact space, $\mu$ a bounded positive measure on T, and R a $\mu$-measurable equivalence relation in T. Let $\Omega$ be the subset of the space $\mathscr{M}(T)$ of measures on T, formed by the measures $\lambda \geq 0$, of total mass $\leq 1$ and nonzero; $\Omega$ is locally compact when it is equipped with the vague topology (Ch. III, §1, No. 9, Cor. 2 of Prop. 15). Show that there exists one and only one positive measure $\rho$ on $\Omega$ such that: $1^\circ \ \mu = \int_{\Omega} \lambda \, d\rho(\lambda) ; \ 2^\circ \ \rho$ is concentrated on a subset $B_0$ of $\Omega$ whose elements are measures of total mass 1, carried by the classes mod R, two distinct elements of $B_0$ being carried by distinct classes. (Consider a disintegration $b \mapsto \lambda_b$ of $\mu$ by the relation R, such that all of the $\lambda_b$ have total mass 1, and the image of B under the mapping $b \mapsto \lambda_b$ of B into $\Omega$; make use of Th. 4.)