Can we find an example of non-mesuarable set which their outer measure could be computed?

Question

We know there is non-measuarable set and we know every set has outer measure, so can anyone give me an example of a non-measuarable and there outer measure could be computed ?

Answer 1

Let us consider the Lebesgue (outer) measure on $[0,1]$ and let $V\subset [0,1]$ be a Vitali set. Then $V$ is not measurable and for any measurable set $E\subset V$, $E$ is a measure-zero set.

Note that $[0,1]\setminus V$ is non-measurable and for every $a\in [0,1]$, $[0,a]\setminus V$ has outer-measure $a$. As a result, there exists $a_0\in[0,1)$, such that $$\{a\in [0,1] : [0,a]\setminus V \text{ is measurable }\}=[0,a_0].$$ Consequently, for every $a\in(a_0,1]$, $[0,a]\setminus V$ is non-measurable with outer-measure $a$.

Remark: I guess $a_0=0$, but I have no idea how to prove or disprove it.