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  "body": "\n## Statement\n\nFor each integer $n \\ge 1$ there is an integer $N_n$, depending only on $n$, with the following property.\nLet $\\mathcal{F}$ be a family of [[closed balls|05DSV40EB9]] of positive radius in [[Euclidean space|0XPR1E85W0]] $\\R^n$ with $\\sup \\set{\\operatorname{diam} B : B \\in \\mathcal{F}} \u003c \\infty$.\nLet $A$ be the set of centres of the balls of $\\mathcal{F}$.\nThen there are subfamilies $\\mathcal{G}_1, \\dots, \\mathcal{G}_{N_n} \\subseteq \\mathcal{F}$ such that:\n\n1. Each $\\mathcal{G}_i$ is a countable family of pairwise disjoint balls.\n2. $A \\subseteq \\bigcup_{i=1}^{N_n} \\bigcup_{B \\in \\mathcal{G}_i} B$.\n\n## Remarks\n\nThe proof chooses balls greedily, each time a ball of nearly maximal radius centred at a point not yet covered.\nA geometric argument in $\\R^n$ shows that each chosen ball meets only a bounded number of earlier chosen balls of comparable or larger size, with a bound depending on $n$.\nThe chosen balls are then sorted into $N_n$ disjoint subfamilies.\n\nUnlike the Vitali covering lemma, no enlargement of the balls is needed and no measure enters the statement.\nSo its consequences hold for every Radon measure on $\\R^n$, doubling or not.\nA consequence concerns a locally finite Borel measure $\\mu$ on $\\R^n$ and a set $A$.\nIf every point of $A$ is the centre of balls of $\\mathcal{F}$ of arbitrarily small radius, then some countable disjoint subfamily of $\\mathcal{F}$ covers $\\mu$-almost all of $A$.\n\nThe fact that the balls are centred at points of $A$ is essential.\nThe theorem does not extend to general metric spaces.\n\n## Explanations\n\n### L1 human\n\nEvery point of a set is the centre of a ball.\nOne can choose some of these balls so that they still cover all the centres, and so that no point is in too many of the chosen balls.\nThe bound on the overlap depends only on the dimension.\n\n### L2 human\n\nThe theorem is the covering tool for measures that are not doubling.\nIt gives the differentiation theorem $\\lim_{r \\to 0} \\nu(B(x,r)) / \\mu(B(x,r)) = \\frac{d\\nu}{d\\mu}(x)$ for Radon measures $\\mu, \\nu$ on $\\R^n$ with $\\nu \\ll \\mu$, for $\\mu$-almost every $x$.\nIt is a basic tool of geometric measure theory.\n",
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