Ergodic components of a rational circle rotation (${\mu }_x=\frac{1}{q}{\sum }_{j=0}^{q-1}{\delta }_{x+jp/q}$)

For rotation by $p/q$ with coprime integers and Haar probability on the circle, components are the equal measures on the $q$-point orbits. The quotient map is $x\mapsto qx(\mathrm{mod}1)$. Integration over the quotient recovers Haar measure, and the rotation permutes each orbit transitively, making its probability measure ergodic.