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Concerning the Vitali Covering Theorem, what is the significance of closed balls in the hypothesis? In particular wouldn't open balls also work?

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There are so many theorems going under this name or variants, and so many proofs, that it's hard to say this for sure; but in most of the theorems of this sort, and most of the proofs of those theorems, that I've seen, one uses the fact that a point that hasn't been covered by a finite collection of balls is at a positive distance from them. Of course, this requires closed-ness. – L Spice May 26 '11 at 13:08
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Assuming that you are considering the Vitali covering theorem for Radon measures in $\mathbb{R}^n$ the answer is that open balls don't work. I'll leave the task of finding a counterexample (for example already in $\mathbb{R}$) to you, as I remember it being a homework assignment on our real analysis course.

(Notice that open balls work trivially for any measure that gives zero measure to the sphere.)

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