If the irrationals in [0,1] are partitioned into an uncountable number of countable sets via the use of Vitali equivalence classes and each set is then bijected with the set of rationals on the interval [0, 1), then can there be a uniform mapping from the irrationals of [0,1] onto the rationals of [0, 1)? I believe the answer should be no, but an example of this has sat for a day or two and am looking for some feedback:
https://www.physicsforums.com/threads/selecting-a-natural-and-a-real-uniformly-at-random.911544/