Prove that there exists a bounded set $A\subset\mathbb{R}$ such that $m^*(F)\le m^*(A)-1$ for every closed set $F\subset A$.
Here $m^*$ is the outer mesure.
I used the bounded and non measurable Vitali set $V$, and defined $A=tV$, where $t>0$, but I could only prove that $$m^*(A\setminus F)\ge1$$ However, since $A$ is non measurable either, I have that $m^*(A)\le m^*(F)+m^*(A\setminus F)$, so I don't know how to finish the proof.