Factor map between measure-preserving systems (${\phi }_{\ast }\mu =\nu ,\phi T=S\phi$)

A measurable measure-preserving map intertwining the transformations of two probability measure-preserving systems. The equalities may be imposed on invariant full-measure domains. It represents the target's observable information inside the source. The conditional measures of a factor describe the information remaining on each fiber.

An extension for which the relatively almost periodic observables are dense in global $L^2$. Density does not assert that every $L^2$ function already meets the uniform finite-net condition on fibers. Localization yields the positive-measure almost periodic indicator in a compact extension.

For a skew-product extension with finitely many labeled fibers, the orbit of a bounded observable on each fiber lies in a fixed bounded subset of a finite-dimensional vector space. A finite net there supplies global centers constant in the base coordinate. Bounded observables are dense in $L^2$, so this is a compact extension. Global almost periodicity can still fail because the base can be weakly mixing.

Approximate $1_A$ by relative almost periodic functions with summable squared global errors. Tonelli theorem makes their conditional errors tend to zero almost everywhere. Choose a positive base set on which the conditional mass of $A$ is bounded below, then use the Egorov theorem to make those errors uniform on a smaller positive set $C$. Localization of relative almost periodicity to base sets and conditional-norm closure prove $1_B$ is relatively almost periodic and $\mu (B)>0$.

For every positive $\epsilon$, require finitely many global $L^2$ centers $g_j$ such that the displayed inequality holds for every integer $n$ on almost every fiber of the factor map. A center can depend on the fiber and time; its finite list is fixed. This is weaker than almost periodic observable in global norm. For noninvertible systems use nonnegative times.

Multiplying a relatively almost periodic observable by a base-set indicator preserves relative almost periodicity: on each fiber an orbit value is either unchanged or zero, so add zero to the finite list of centers. Conditional norm covariance under a factor map also proves closure under convergence in the uniform conditional L2 norm.