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Suppose $S$ is a set of real of numbers with strictly positive Lebesgue measure. Does there exist a subset $N$ of $S$ such that $N$ is non-Lebesgue measurable? My second question is, suppose $N$ is a non-Lebesgue measurable set of reals. Does there exist an $\epsilon > 0$, such that for all $\delta$ with $0 < \delta \leq \epsilon$, there exists a subset $M$ of $N$ such that $M$ is Lebesgue measurable with the measure of $M$ equal to $\delta$?

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    These are good questions, but they have been asked here before: your first question has a positive answer here, and your second question has a negative answer here. (I've used the former as the duplicate target since we can only pick one.) – Noah Schweber Nov 29 '22 at 17:34
  • Besides the MSE questions/answers @Noah Schweber cited, this answer gives links to more precise results and states a conjecture that might be suitable for an undergraduate or Masters project, especially if examples can be proved to not exist for all the possibilities described. – Dave L. Renfro Nov 29 '22 at 18:02

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