 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.