This is measure theory, so I am not sure if it should go here, under Set Theory, or Analysis. I will opt for Topology.

I am a little dubious on a point on the proof that a Vitali set is non-measurable. I look at the definition of it being a subset of [0,1] defined as

for all r in R, let = r + for some rational . (This requires a choice function C).

Then = {v_r : r in R} [0,1].

Then there are proofs based on a combination of translation invariance of length and additivity of a measure to show that each V is non-measurable. But can we say that this is true forC? For example, although it is highly unlikely, nonetheless the function that chooses the same value of for all r, although it would be a boring choice function, would give us an interval that clearly had measure one. For example, if C gave us = 0 for all r, then = [0,1]. So, shouldn't we put on some other conditions in the definition to ensure non-measurability?all