= Solution
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>:
$$
\boxed{\{U_T^nf:n\in\mathbb Z\}\text{ is totally bounded in }L^2(\mu).}
$$
Equivalently, for every $\varepsilon>0$ it has a finite $\varepsilon$-net in $L^2$. For a noninvertible system use the forward orbit $n\geq0$.
A <factor map> $\varphi:X\to Y$ satisfies $\varphi_*\mu=\nu$ and $\varphi T=S\varphi$ <almost everywhere>. Disintegrate $\mu=\int_Y\mu_y\,d\nu(y)$ over this map and write
$$
\|h\|_y=\left(\int_X|h|^2\,d\mu_y\right)^{1/2}
$$
for the <conditional L2 norm>. A <relatively almost periodic observable> is an $f\in L^2(\mu)$ such that, for every $\varepsilon>0$, there are finitely many $g_1,\ldots,g_r\in L^2(\mu)$ with
$$
\boxed{\min_{1\leq j\leq r}\|U_T^nf-g_j\|_y<\varepsilon
\quad\text{for all }n\in\mathbb Z,\text{ for }\nu\text{-almost every }y.}
$$
The 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 $(Sy)_j=y_{j+1}$. Set
$$
X=Y\times\{0,1\},\qquad
T(y,i)=(Sy,i+1\pmod2),\qquad \varphi(y,i)=y,
$$
using 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 $U_T^nf(y,i)=y_n$. Distinct coordinates are independent and have mean zero and variance one, hence
$$
\boxed{\|U_T^nf-U_T^mf\|_2^2
=\mathbb E[(y_n-y_m)^2]=2\qquad(n\ne m).}
$$
Its orbit has no finite sufficiently small net. Thus \b[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 $U_T^nf$ 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 \b[$f$ 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 $a(y)+(-1)^ib(y)$, 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 $X'$ 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> give
$$
T_*\mu_y=\mu_{Sy},\qquad
\boxed{\|U_T^nh\|_y=\|h\|_{S^ny}\quad(n\in\mathbb Z)}
$$
on common full-measure sets. For the first equality, $y\mapsto T_*\mu_{S^{-1}y}$ is another disintegration over the same fibers: it is supported on $\varphi^{-1}\{y\}$ 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 $f\,\mathbf1_C\circ\varphi$ is also <relatively almost periodic>. Indeed, on a fiber,
$$
U_T^n\bigl(f\,\mathbf1_C\circ\varphi\bigr)
=(U_T^nf)\mathbf1_C(S^ny).
$$
When that indicator is one, use a center for $U_T^nf$; when it is zero, use the additional center $0$. Secondly, <relative almost periodicity> is closed under convergence in the <uniform conditional L2 norm>
$$
\|h\|_{2,\infty\mid Y}
=\mathop{\mathrm{ess\,sup}}_{y\in Y}\|h\|_y.
$$
If 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 $\varepsilon/2$-net for one sufficiently close $f_j$ is consequently an $\varepsilon$-net for $f$.
Now take $f=\mathbf1_A$. The <compact extension> supplies <relatively almost periodic observables> $f_j$ with
$$
\|f_j-f\|_2^2\leq2^{-j}.
$$
Put $e_j(y)=\|f_j-f\|_y$. The <Tonelli theorem> gives
$$
\int_Y\sum_{j=1}^\infty e_j(y)^2\,d\nu(y)
=\sum_{j=1}^\infty\|f_j-f\|_2^2<\infty.
$$
Hence $e_j(y)\to0$ for $\nu$-almost every $y$. Since $\int_Y\mu_y(A)d\nu=\mu(A)>0$, choose $\eta>0$ for which
$$
E=\{y:\mu_y(A)\geq\eta\}
$$
has positive measure. The <Egorov theorem> gives a measurable $C\subseteq E$ of positive measure on which $e_j\to0$ uniformly. Define
$$
\boxed{B=A\cap\varphi^{-1}C.}
$$
Then
$$
\boxed{\mu(B)=\int_C\mu_y(A)\,d\nu(y)\geq\eta\nu(C)>0.}
$$
For $h_j=f_j\,\mathbf1_C\circ\varphi$, the localization fact makes every $h_j$ <relatively almost periodic>, while
$$
\|h_j-\mathbf1_B\|_{2,\infty\mid Y}
=\mathop{\mathrm{ess\,sup}}_{y\in C}e_j(y)\longrightarrow0.
$$
The conditional-norm closure fact therefore proves \b[$\mathbf1_B$ 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>.
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