Disintegration of a probability measure

Statement

Let $p:X\to Y$ be measurable between standard Borel spaces, let $\mu$ be a Borel probability on $X$ , and let $\nu=p_*\mu$ . There is a probability kernel $y\mapsto\mu_y$ , unique for $\nu$ -almost every $y$ , such that $\mu_y(p^{-1}\{y\})=1$ almost everywhere and

\int_X f\,d\mu=\int_Y\left(\int_X f\,d\mu_y\right)d\nu(y)

for every bounded Borel $f$ . It is called the disintegration of $\mu$ over $p$ .

Conditional expectation

The function $x\mapsto\int f\,d\mu_{p(x)}$ is a version of conditional expectation onto the sigma-algebra generated by $p$ . Work on Borel models before completing measures; the component measures can be singular with respect to $\mu$ , so they need not extend to every set in its completion simultaneously.

References

Terence Tao, “254A, Lecture 9: Ergodicity”, Theorem 4 and Exercise 13.