Ergodic component (${\mu }_y$)

A conditional probability of an invariant probability measure given its invariant sigma-algebra. In a standard Borel probability system, almost every such probability is invariant and ergodic. Their integral recovers the original measure. The countable-test proof of ergodicity of conditional components explains why invariance alone is not the whole conclusion.

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.