Conditional norm covariance under a factor map ($T_{\ast }{\mu }_y={\mu }_{Sy},\Vert U_T^{n}f{\Vert }_y=\Vert f{\Vert }_{S^{n}y}$)

For invertible probability systems linked by a factor map, invariance and uniqueness of disintegration identify the pushforward of each conditional measure with the conditional measure on the transformed fiber. The displayed norm identity follows for all integer times on a common full-measure set. It transports uniform conditional error bounds along whole orbits.