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Can every bounded Lebesgue measurable set be approximated from the inside by countably many disjoint closed rectangles?

A citation or proof would be nice.

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What about $([0,1]\times[0,1])\setminus(\mathbb Q\times\mathbb Q)$? – Jonas Meyer Dec 15 '12 at 18:16
Ok. What conditions would I need to have this property? Jordan measurability? – dcs24 Dec 15 '12 at 18:22
Jordan measurability will surely do and I think that it is an if and only if. That is, if a bounded Lebesgue measurable set can be approximated from the inside with rectangles, then it is Jordan measurable. Indeed, if this is the case, then by approximating the complementary of the closure of the given set with open rectangles, and working a bit, I think that one can prove that the boundary has null measure. – Giuseppe Negro Dec 15 '12 at 18:31

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