Disintegration theorem for a probability measure ($\mu =\int {\mu }_{y}d({\pi }_{\ast }\mu )(y)$)

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.

The fiber-supported measurable probability kernel supplied by the disintegration theorem for a probability measure for a factor map. For integrable $f$, $\mathbb{E}[f\mid {\phi }^{-1}{\mathcal{B}}_Y](x)=\int fd{\mu }_{\phi (x)}$ almost everywhere. The same construction applies to general measurable maps and countably generated sub-sigma-algebras modulo null sets.