# Lebesgue measure of algebraic irrational numbers in $\mathbb{R}$

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)?

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–  user110661 Nov 26 '13 at 12:47

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