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I'm working on the Vitali Covering Lemma. I'd like to see a demonstration of the statement in the title.

I'm looking for a demonstration of the fact that an arbitrary union of sets (each with non-empty interior) is Lebesgue measurable in $R^n$ using the Vitali Covering Lemma.

Even just a paper or links to other works are acceptable, and it doesn't matter if you're not directly answering. I've browsed past questions without finding any useful results. Hope somebody can help.

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    This does not answer your question, but a special case of it: http://mathoverflow.net/questions/43721/is-arbitrary-union-of-closed-balls-in-rn-lebesgue-measurable.2012-12-19
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    Perhaps first use VCL to show the arbitrary union of closed balls is measurable. At least that one is true...2012-12-19

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