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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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.) 

