Compact extension of a measure-preserving system (${\overline{\mathrm{AP}(X\mid Y)}}^{L^2}=L^2(X)$)

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 $\mu (B)>0$.