Conditional L2 norm ($\Vert f{\Vert }_y={\left(\int |f{|}^{2}d{\mu }_y\right)}^{1/2}$)

Disintegrate over a factor map and take the $L^2$ norm using the conditional probability on each fiber. Its square integrates to the global squared norm. The uniform conditional L2 norm takes the essential supremum of these fiber norms.

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.

The essential supremum over the base of the conditional L2 norm. It may be infinite for a globally square-integrable function. For probability fibers, $\Vert f{\Vert }_2\le \Vert f{\Vert }_{2,\infty \mid Y}$; the uniform norm controls all fibers outside a null set rather than only their average.