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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 ${\Xi }_{\mathrm{erg}}(F)$ for every $F\in \Omega$.

Restrict the input class to the circle rotationsFrom 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 forbecause 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:

The indicator of the rational numbers in the real numbers is discontinuous everywhere because both the rationals and irrationals are dense. The discontinuity theorem for Baire class one functions therefore shows that it is not Baire class one.