It is, but you're right to question it because it's a very nontrivial property that makes irrational rotations a bit of a special case. The fact that it's uniq

It is, but you're right to question it because it's a very nontrivial property that makes irrational rotations a bit of a special case. The fact that it's uniquely ergodic (which roughly means that the uniform Lebesgue measure is the only measure that's invariant under irrational rotation) is required to ensure it works for all starting points. Most ergodic systems only ensure that it works for almost all starting points i.e. all starting points except for a measure 0 set.