Non-measurable set: divide points of a unit circle into equivalence classes: consider two points equal if an angle between them is rational. As there are strictly more points than rational rotations, there will be countably many equivalence classes. Using axiom of choice, choose a representative from each class. Rotated over rational angle, this set will become disjoint with initial. Hence, a disjoint union of all such rotations gives the whole circle (as the circle is partitioned by equivalence relation). But the circle is measurable, hence there must be a way to assign a measure to each rotated copy to make the series convergent. As all the sets are congruent, their measures must be equal (as the Lebesgue measure is both translation- and rotation-invariant). Nevertheless, regardless of the measures assigned, series will either converge to zero or diverge to infinity. Hence, the circle must either have zero or infinite length, which is impossible. Hence, such a set must be non-measurable.