= Countable-test proof of ergodicity of conditional components
{title2=$\mathbb E_{\mu_y}[\mathbf1_A\mid\mathcal I_y]=\mu_y(A)$}
Apply the <Birkhoff ergodic theorem> to a countable generating algebra, and disintegrate the resulting common full-measure convergence set over the <invariant sigma-algebra>. The orbit limits on a fiber are its conditional masses. Applying the theorem to that invariant component shows its invariant projection is constant on all generating indicators, hence on all $L^2$ functions by density. This proves <ergodicity> without an uncountable intersection of null sets.
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