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For an invertible probability measure-preserving system, an almost periodic observable is an $f\in L^2(\mu )$ whose Koopman operator orbit is totally bounded:Equivalently, for every $\epsilon >0$ it has a finite $\epsilon$-net in $L^2$. For a noninvertible system use the forward orbit $n\ge 0$.

A factor map $\phi :X\to Y$ satisfies ${\phi }_{\ast }\mu =\nu$ and $\phi T=S\phi$ almost everywhere. Disintegrate $\mu ={\int }_Y{\mu }_{y}d\nu (y)$ over this map and writefor the conditional L2 norm. A relatively almost periodic observable is an $f\in L^2(\mu )$ such that, for every $\epsilon >0$, there are finitely many $g_1,\dots ,g_r\in L^2(\mu )$ withThe center may depend on both $n$ and $y$; the finite list itself does not. Since $\mathbb{Z}$ is countable, the exceptional null sets can be combined. A compact extension of a measure-preserving system is a factor map for which these relatively almost periodic observables are dense in $L^2(\mu )$. This density definition should not be replaced by a claim that every $L^2$ observable already satisfies the uniform fiberwise finite-net condition.

For a proper factor map example, let $Y=\{-1,1{\}}^{\mathbb{Z}}$ carry independent fair coordinates and the two-sided Bernoulli shift $S$, with $(Sy{)}_j=y_{j+1}$. Setusing the product of the Bernoulli measure and the equal two-point measure. These are invertible measure-preserving systems, and the projection preserves measure and intertwines the transformations, so it is a two-to-one factor map.

Take $f(y,i)=y_0$. Then $U_T^{n}f(y,i)=y_n$. Distinct coordinates are independent and have mean zero and variance one, henceIts orbit has no finite sufficiently small net. Thus the Bernoulli coordinate observable is not almost periodic in global $L^2$. In contrast, the conditional measure over $y$ is the equal measure on $(y,0),(y,1)$, and $U_T^{n}f$ is constant on that fiber with value $y_n\in \{-1,1\}$. The two centers $g_1=1,g_2=-1$ give zero error on every fiber for every $n$. Therefore $f$ is a relatively almost periodic observable for this map.

The example is also a finite-fiber compact extension. For a bounded observable on $X$, each orbit value on a fiber is a pair in a fixed bounded subset of ${\mathbb{C}}^2$, which has a finite net. Its centers can be chosen as functions constant in $y$, with prescribed values on the two labels. Bounded observables are dense in $L^2$, so the extension is compact. The Bernoulli shift is mixing: cylinder events involving disjoint coordinate sets are independent for all sufficiently large shifts, and approximation by cylinder events extends this to arbitrary events. Thus $S$ and $S^2$ are ergodic. If an invariant function on $X$ is written $a(y)+(-1{)}^{i}b(y)$, invariance gives $a\circ S=a$ and $b\circ S=-b$. The first function is constant; the second is $S^2$-invariant and hence constant, and then its sign equation forces zero. This proves that the example's source is also ergodic.

For the positive-subset assertion, interpret the printed $X^{\prime }$ as $X$, or as an invariant full-measure domain of the factor map; the prime is otherwise undefined and does not change the measure-theoretic argument.

First establish conditional norm covariance under a factor map. Invariance of the measures and uniqueness in the disintegration theorem for a probability measure giveon common full-measure sets. For the first equality, $y\mapsto T_{\ast }{\mu }_{S^{-1}y}$ is another disintegration over the same fibers: it is supported on ${\phi }^{-1}\{y\}$ and integrates to $\mu$. Uniqueness identifies it with ${\mu }_y$, and iteration gives the norm formula.

Two elementary facts explain localization of relative almost periodicity to base sets. If $f$ is a relatively almost periodic observable and $C\subseteq Y$ is measurable, then $f{1}_C\circ \phi$ is also relatively almost periodic. Indeed, on a fiber,When that indicator is one, use a center for $U_T^{n}f$; when it is zero, use the additional center $0$. Secondly, relative almost periodicity is closed under convergence in the uniform conditional L2 normIf this norm of $f-f_j$ tends to zero, covariance bounds the corresponding error between every pair of orbit values by the same number. A finite $\epsilon /2$-net for one sufficiently close $f_j$ is consequently an $\epsilon$-net for $f$.

Now take $f=1_A$. The compact extension supplies relatively almost periodic observables $f_j$ withPut $e_j(y)=\Vert f_j-f{\Vert }_y$. The Tonelli theorem givesHence $e_j(y)\to 0$ for $\nu$-almost every $y$. Since ${\int }_Y{\mu }_y(A)d\nu =\mu (A)>0$, choose $\eta >0$ for whichhas positive measure. The Egorov theorem gives a measurable $C\subseteq E$ of positive measure on which $e_j\to 0$ uniformly. DefineThenFor $h_j=f_j{1}_C\circ \phi$, the localization fact makes every $h_j$ relatively almost periodic, whileThe conditional-norm closure fact therefore proves $1_B$ is a relatively almost periodic observable, with $B\subseteq A$ and positive measure. This is the positive-measure almost periodic indicator in a compact extension. No ergodic component or uncountable intersection of exceptional sets is needed in this construction; it in fact works under the same disintegration and compactness hypotheses without using the assumed ergodicity.

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.