consider the torus map $(x,y) \mapsto (x+y, y)$. For every $y$, this gives a rotation on the circle $S^1 \times y$. That is, Lebesgue measure on any "horizontal" circle is preserved. For $y$ rational it is not ergodig but for $y$ rational it is. Well, that's it. Lebesgue on a circle with rational rotation is weak* approximated by Lebesgue on circles with irrational rotation.

consider the torus map $(x,y) \mapsto (x+y, y)$. For every $y$, this gives a rotation on the circle $S^1 \times y$. That is, Lebesgue measure on any "horizontal" circle is preserved. For $y$ rational it is not ergodic, but for $y$ irrational it is ergodic. Well, that's it. Lebesgue on a circle with rational rotation is weak* approximated by Lebesgue on circles with irrational rotation.