Countable-test proof of ergodicity of conditional components (${\mathbb{E}}_{{\mu }_y}[1_A\mid {\mathcal{I}}_y]={\mu }_y(A)$)

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.