There are restrictions on the choice function C.

First, given r, we don't choosesomeand then take the intersection . We must choose so that .

More importantly, after we have chosen for some , the choice for all such that is fixed: we must have . This is because, in Wikipedia description, the choice function is fromequivalence classesto [0,1]: all reals that differ by a rational number produce a single element in the resulting Vitali set V.