I have been looking for a direct answer for this question a little while. Ichecked a lot of topics and none seemed to answer that in really direct way.
Why does the measurability definition exclude Vitali sets? I`d like to see a proof(It could be a reference) that Vitali sets are not measurable(in all that desirable way...) using the definition(*) of a measurable set.
Also, I`d like some intuition on it. Deeply, I want to understand why the definition of measurable sets works well in the task of saying wether a set behaves nicely or not in the sense of a desirable measure.
What I`ve thought so far: it all lies on the fact that Vitali sets are based on picking representatives which are arbitrary. Am I in the right direction on the interpreatation?
Edit:(*) by definition of measurable set I meant:
A set E is said to be measurable if for each set A we have $$m^∗A=m^∗(A∩E)+m^∗(A∩Ec)$$
where $m^*$ denotes the exterior measure.