 Disintegration theorem for a probability measure

For a probability measure on a standard Borel space and a measurable map $\pi$ to another standard Borel space, there is an almost-everywhere unique measurable probability kernel $y\mapsto {\mu }_y$ supported on ${\pi }^{-1}\{y\}$. It integrates to $\mu$, and $\int fd{\mu }_{\pi (x)}$ realizes conditional expectation given $\pi$. These measures are the conditional measures of a factor.