For any $x,y\in I=[0,1]$, define $x\sim y$ iff $x-y\in\mathbb Q$. Then $I$ is a disjoint union of equivalence classes of $\sim$. By the axiom of choice, one can form a set $V$ by choosing one representative from each equivalence class.
In many textbooks, $V$ is shown to be (Lebesgue) non-measurable by a proof of contrapositive. However, while $V$ is not measurable, its Lebesgue inner or outer measures still exist. Is it possible to explicitly calculate the the inner and outer measures for some choice of $V$, so as to offer a direct proof of non-measurability of this set?
(Please excuse me if this is a duplicate. As $V$ doesn't seem to have a standard name, I cannot search for a duplicate effectively.)
Edit. Noah Schweber has pointed out that such a set $V$ is called a Vitali set, and it can be constructed with known inner and outer measures. This fact is mentioned in neither Wikipedia nor the books I read (Royden, Rudin as well as Wilcox and Myers' An Introduction to Lebesgue Integratioin and Fourier Series). However, this was proved in Vitali-type set with given outer measure .
