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.