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Let $A\subset\mathbb{R}$ be a set with empty interior. Does it then follow that $A$ is Lebesgue measurable?

Equivalently, can you provide a subset of $\mathbb{R}$ with empty interior that is not Lebesgue measurable?

I'm having trouble wrapping my head around the relationship of sets with empty interior and Lebesgue measurability.

I understand that a set with measure zero must have empty interior, and that empty interior does not in general imply measure zero. To prove the latter claim, fat Cantor sets are usually invoked and shown to have positive Lebesgue measure, but they still have a Lebesgue measure.

om harmoni
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    There are empty interior sets with positive measure, and every set of positive measure has a non-measurable subset. Therefore, because * holds, it follows that ** holds. Here * is a certain hereditary property and ** is what you want. – Dave L. Renfro Oct 11 '24 at 00:55
  • If a set with empty interior has positive measure, it contains a non-measurable subset. Any subset of a set with empty interior itself has empty interior. Therefore, the non-measurable set in question has empty interior, which is an example I'm asking for in the second statement of my original question.

    Is this right?

     – om harmoni Oct 11 '24 at 01:13
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    Is this right? --- Yes, this is correct, assuming you can use or prove each part. However, I see that @quasi has an argument that relies on results a bit easier to prove. For example, proving that the removal of countably many points from a non-measurable set results in a non-measurable set is much easier than proving that any set with positive measure has a non-measurable subset. Regarding sets with empty interior as compared to nowhere dense sets, see this MSE answer. – Dave L. Renfro Oct 11 '24 at 03:35
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    Related: https://math.stackexchange.com/questions/4228885/non-lebesgue-measurable-set-with-empty-interior – Henry Oct 11 '24 at 12:01

1 Answers1

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Let $B$ be a non-measurable subset of $\mathbb{R}$, and let $A=B{\setminus}\mathbb{Q}$.

Then $A$ is non-measurable and has empty interior.

quasi
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