= Solution
Assume for contradiction that a sequence of general algorithms $\Gamma_n$ decides ergodicity from the perfect measurement data, so that $\Gamma_n(F)$ eventually equals $\Xi_{\rm erg}(F)$ for every $F\in\Omega$.
Restrict the input class to the circle rotations
$$
F_a(x)=x+a\pmod{2\pi}.
$$
From <inexact information in the SCI hierarchy> for the real number $a$, one can answer every requested measurement of $F_a(x)$ to the same precision. The supposed algorithms would therefore give a one-limit decision procedure for
$$
a\longmapsto\mathbf1_{\mathbb R\setminus\mathbb Q}(a/\pi),
$$
because part (b)(ii) identifies ergodicity with irrationality.
Every finite-information general algorithm is locally constant on a sufficiently small cylinder of the inexact data. A pointwise limit of a sequence of such functions is a <Baire class one function>. But the rationality indicator is discontinuous at every real number: every interval contains both rational and irrational numbers. The theorem that the discontinuity set of a Baire class one function is meagre, or directly <rationality indicator is not Baire class one>, gives a contradiction.
Hence no one-limit tower of general algorithms can decide ergodicity, even with the perfect measurement device:
$$
\boxed{
\{\Xi_{\rm erg},\Omega,\{0,1\},\Lambda\}^{\Delta_1}
\notin\Delta_2^G.}
$$
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