# The Vitali Covering Theorem and use of closed balls

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

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.