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I'm stuck with this question:

"Find a bounded Lebesgue measurable subset of $\mathbb {R}^n$ with positive Lebesgue measure that is neither a Borel set, nor Jordan measurable."

Can someone help me?

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  • $\begingroup$ Do you know how to do the two parts separately - get a positive-measure set which isn't Borel, and get a positive-measure set which isn't Jordan measurable? $\endgroup$ – Noah Schweber Sep 20 '17 at 18:49
  • $\begingroup$ Positive-measure set which is not Jordan I know. But I only know a set with 0-mesure that is not Borel $\endgroup$ – Matheus Manzatto Sep 20 '17 at 18:53
  • $\begingroup$ OK, do you see a way to put those two examples together to get what you want? (HINT: don't do anything complicated, just ... "put them together.") $\endgroup$ – Noah Schweber Sep 20 '17 at 19:02
  • $\begingroup$ "Positive-measure set which is not Jordan I know. But I only know a set with 0-mesure that is not Borel" Put a copy of one (scale and translate if necessary) in the interval $[0,1]$ and a copy of the other (scale and translate if necessary) in the interval $[2,3].$ Then use the union of these sets for your example. (If neither set has positive Lebesgue measure, include also in the union the interval $[4,5].)$ Make sure you explain why the union is not Borel and also not Jordan measurable. $\endgroup$ – Dave L. Renfro Sep 20 '17 at 19:03
  • $\begingroup$ I don't think it's possible to "find" a non-Borel set in a strict sense of "find"; the existence of non-Borel sets relies on the axiom of choice, and that implies that any characterization of a non-Borel set will be non-constructive. So presumably the question allows "find" to include non-constructive characterization. $\endgroup$ – Acccumulation Sep 20 '17 at 19:04

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