A Vitali set is non-measurable, direct proof, without using countable additivity
Solution 1:
Let $n$ be such that $m^*(V)>1/n$, choose $n$ distinct rationals $q_1,\dots,q_n\in\mathbb{Q}/\mathbb{Z}$, and write $V_k=\bigcup_{i=1}^k V\oplus q_i$. Then if $V\oplus q_k$ is measurable, taking $A=V_k$ we find that $$m^*(V_k)=m^*(V\oplus q_k)+m^*(V_{k-1}).$$ Thus if $V$ were measurable, we could conclude by translation-invariance of $m^*$ and induction that $m^*(V_k)=km^*(V)$ for each $k$. In particular, $m^*(V_n)=nm^*(V)>1$, which is a contradiction.