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

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.

• You may benefit from the following along with the linked and related questions: math.stackexchange.com/q/157532 – Jonas Meyer Nov 8 '17 at 3:10
• @JonasMeyer Thanks, but I don't understand why the outer measure of vitali set is an arbitrary number belongs to $[0,1]$ – Sustcer_Shuai Nov 8 '17 at 5:41
• – Dave L. Renfro Nov 8 '17 at 13:08