Let $V$ be a Vitali set on $\mathbb{R}$ and suppose that $V$ is not bounded, ie the representatives $v$ of the cosets of $\mathbb{Q}$ are chosen in a such a way that $|v|\geq M$ for all $M\in\mathbb{R}$. If such a Vitali set exists, can it contain a set with positive measure?
I know that all measurable subsets of the standard Vitali set in $[0,1]$ have measure 0, but what if $V$ cannot be contained in a bounded set?
The answer is no.
Say that a set $X\subseteq\mathbb{R}$ is pre-Vitali if $X$ contains at most one real in each $\mathbb{Q}$-equivalence class. Then any bounded pre-Vitali set contains no measurable set of positive measure, since every bounded pre-Vitali set is contained in a bounded Vitali set.
Now suppose $V$ is an unbounded Vitali set. For each $n$, the set $V_n=V\cap [-n, n]$ is a pre-Vitali set. Suppose $M$ is measurable with positive measure; we'll show $M\not\subseteq V$.
The key point is the following: for some $n$, $M_n=M\cap [-n, n]$ also has positive measure. Why? Well, the union of countably many measure-zero sets has measure zero, so if $M_n$ had measure zero for each $n$, $M$ itself would have measure zero; but we assumed $M$ had positive measure.
OK, so some $M_n$ has positive measure. Well, each $M_n$ is measurable, so $M_n\not\subseteq V_n$. But this means $M\not\subseteq V$, so we're done.