0
$\begingroup$

Find all Lebesgue measurable subsets $A \subset\mathbb{R}$ such that all $B\subset A$ is measurable.

I argued that if the measure is positive then $A$ is an interval so we can construct the Vitali set and thus it'd have non-measurable subsets. So $A$ must have measure $0$. Is it correct to say that such $A$ is the power set of $\mathbb{Q}$? Or we need to worry about algebraic irrationals as well( since they're countable)?

$\endgroup$
1
$\begingroup$

You are on the right track but what about the irrationals? The have measure >0 but certainly do not form an interval. As for your question: Any singleton set is measurable (since it has outer measure zero). Now suppose this set were $\{\pi\}$. Does your conclusion still hold?

$\endgroup$
  • $\begingroup$ You are right. I was totally misguided. so the answer to the question is All $A$ with measure $0$. $\endgroup$ – Spock Nov 26 '13 at 14:54

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.