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  "body": "\n## Statement\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]].\nFor a [[closed ball|05DSV40EB9]] $B = \\bar B(x, r)$ with $r \u003e 0$ and $c \u003e 0$, let $cB = \\bar B(x, cr)$.\n\n1. Finite version: for closed balls $B_1, \\dots, B_k$ of positive radius in $X$, there is a subfamily $B_{j_1}, \\dots, B_{j_m}$ of pairwise disjoint balls with $B_1 \\cup \\dots \\cup B_k \\subseteq 3B_{j_1} \\cup \\dots \\cup 3B_{j_m}$.\n2. Infinite version: let $\\mathcal{F}$ be a family of closed balls of positive radius in $X$ with bounded radii. There is a subfamily $\\mathcal{G} \\subseteq \\mathcal{F}$ of pairwise disjoint balls with this property: each $B \\in \\mathcal{F}$ meets some $B' \\in \\mathcal{G}$ of radius at least half that of $B$. In particular $\\bigcup_{B \\in \\mathcal{F}} B \\subseteq \\bigcup_{B' \\in \\mathcal{G}} 5B'$.\n\nIf $X$ is [[separable|0Z2P41SVDP]], for example $X = \\R^n$, the subfamily $\\mathcal{G}$ in part 2 is countable.\n\n## Remarks\n\nThe finite version is greedy: pick the largest ball, discard every ball that meets it, and repeat.\nA discarded ball meets a chosen ball of at least its radius, so it lies in the triple of that ball.\nThe infinite version sorts the balls by radius into dyadic classes $R/2^{k+1} \u003c r \\le R/2^k$ and chooses a maximal disjoint subfamily in each class in turn, using Zorn's lemma.\nIf $B$ meets $B'$ and the radius of $B$ is at most twice that of $B'$, then $B \\subseteq 5B'$.\n\nExample: in $\\R^n$ the lemma gives $\\lambda\\big(\\bigcup_i B_i\\big) \\le 3^n \\sum_m \\lambda(B_{j_m})$, which is the key step in the weak type estimate for the Hardy–Littlewood maximal function.\n\nThe bound on the radii is needed in part 2.\nThe balls $\\bar B(0, k)$ for $k = 1, 2, \\dots$ in $\\R$ pairwise meet, so a disjoint subfamily has one ball, and no fixed enlargement of it covers $\\R$.\n\n## Explanations\n\n### L1 human\n\nGiven a bunch of overlapping balls, one can keep a few of them that do not overlap at all.\nAfter blowing the kept balls up by a fixed factor, they cover everything the original balls covered.\nSo the total size of a union of balls is controlled by the size of a disjoint subfamily.\n\n### L2 human\n\nThe lemma turns overlapping covers into disjoint ones at the price of a constant factor.\nIt gives the weak $(1,1)$ bound for the maximal function and so the Lebesgue differentiation theorem.\nThe Vitali covering theorem built on it says that a fine cover of a set by balls contains a disjoint subfamily that covers almost all of the set.\n",
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