Let $A\subset \mathbb{R}$ be a non-measurable set with infinite Lebesgue outer measure, i.e., $m^*(A)=\infty$. Does there exist a ${\bf closed}$ subset $F\subset A$ with (infinite Lebesgue (outer) measure) $m^*(F)=\infty$?
I think the answer is "positive", but what would be an easy answer? Thanks in advance.
No, as motivated by here. Take a Vitali set in $[0, 1]$ and any closed subset (which is measurable) has measure $0$. Now take a countable union of disjoint translates of this set for your counterexample.