2
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.