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 $\displaystyle v_r$ = r + $\displaystyle q_r$ for some rational $\displaystyle q_r$. (This requires a choice function C).

Then $\displaystyle V_C$ = {v_r : r in R} $\displaystyle \cap$ [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 $\displaystyle q_r$ 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 $\displaystyle q_r$ = 0 for all r, then $\displaystyle V_C$ = [0,1]. So, shouldn't we put on some other conditions in the definition to ensure non-measurability?all