In my textbook the Vitali set is shown as a classic example of non-measurable sets. The proof is done by showing that you can derive an impossible measure of this set if it is measurable. I also searched the internet and there are other proofs more or less the same. One proof can be found here "Is outer measure a measure?".
However, no material I read directly shows how the Vitali set (denoted as $V$) violates the definition of measurable sets, that is for any set $A$ the equation $|A|_e=|A\bigcap V|_e+|A\bigcap V^C|_e$ holds, where $|A|_e$ denotes outer measure. Since the Vitali set is non-measurable, there must exist some set $A$ that does not satisfy the equation. Can anyone help find an example of such an $A$?