Prove that a set $E \subset R$ has non-measurable subsets if and only if $m∗(E)>0$

Question

Prove that a set $E \subset R$ has non-measurable subsets if and only if $m*(E)>0$

I dont know how to define what is a non measurable set, i read a little about vitali set, but don't understand how to use this to solve the problem.

Answer 1

Suppose that $E$ has a non-measurable subset and then suppose for a contradiction that $m^{*}(E)=0$. It follows that $m_{*}(E)=0$ where $m_{*}$ is inner measure and consequently, $E$ is measurable with $m(E)=0$. Finally, Lebesgue measure is complete so $F\subset E\implies F$ is measurable and $m(F)=0$ and, in particular, this contradicts the fact that $E$ is supposed to have a non-measurable subset.

If $m^{*}(E)>0$, then you can use Vitali set construction to build a non-measurable set.