Show that there exists $E\subset [0,1]$ such that $$m^*(E)=m^*([0,1]\setminus E)=1$$

I have thought to use the Theorem: $A\subset R$ with the property that $A\cap B$ is unmeasurable for every Legesgue measurable set $B$ with $m(B)>0$

But I have trouble to considering why the outer measure of a vitali set can be any number.


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