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

Asked Nov 07 '17 at 08:38

Active Nov 08 '17 at 07:09

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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.

edited Nov 08 '17 at 07:09

asked Nov 07 '17 at 08:38

Sustcer_Shuai

1

You may benefit from the following along with the linked and related questions: https://math.stackexchange.com/q/157532/ – Jonas Meyer Nov 08 '17 at 03:10
@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 08 '17 at 05:41
See the mathoverflow questions Non Lebesgue measurable subsets with “large” outer measure AND Is there a maximum to the amount of disjoint non-measurable subsets of the unit interval with full outer measure?. – Dave L. Renfro Nov 08 '17 at 13:08

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