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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]], let $s \\ge 0$, and let $A \\subseteq X$.\nFor $\\delta \u003e 0$, a *$\\delta$-packing* of $A$ is a countable family of pairwise disjoint closed balls $\\bar B(x_i, r_i)$ with centres $x_i \\in A$ and radii $0 \u003c r_i \\le \\delta$.\nThe construction has three steps.\n\n1. $P^s_\\delta(A) = \\sup \\sum_i (2 r_i)^s$, the [[supremum|0R3H5PANRQ]] over all $\\delta$-packings of $A$, and $P^s_0(A) = \\lim_{\\delta \\to 0} P^s_\\delta(A)$.\n2. The *$s$-dimensional packing measure* of $A$ is\n$$\\mathcal{P}^s(A) = \\inf \\set{ \\sum_{i=1}^\\infty P^s_0(A_i) : A \\subseteq \\bigcup_{i=1}^\\infty A_i }.$$\n3. The *packing dimension* of $A$ is $\\dim_P A = \\inf \\set{ s \\ge 0 : \\mathcal{P}^s(A) = 0 }$.\n\n## Remarks\n\nThe limit in step 1 exists since $P^s_\\delta(A)$ decreases as $\\delta$ decreases.\nThe map $\\mathcal{P}^s$ is a [[metric outer measure|3T6WG315GC]], so it is a measure on the Borel sets.\nSome authors use the diameters of the balls instead of $2 r_i$; in $\\R^n$ the two agree.\n\nExample: on $\\R$ with $s = 1$, a $\\delta$-packing of $[a, b]$ by intervals has total length at most $b - a + 2\\delta$, and packings with total length close to $b - a$ exist.\nSo $P^1_0([a, b]) = b - a$, and in fact $\\mathcal{P}^1$ is Lebesgue measure on $\\R$.\n\nStep 2 is needed: $P^s_0$ alone is not countably subadditive.\nThe set $\\Q \\cap [0, 1]$ has $P^1_0 = 1$, because centres in a dense set can approximate any packing of $[0, 1]$.\nBut it is a countable union of points, each with $P^1_0 = 0$, so $\\mathcal{P}^1(\\Q \\cap [0, 1]) = 0$.\n\nFor $A \\subseteq \\R^n$, $\\dim_H A \\le \\dim_P A$, where $\\dim_H$ is defined by [[Hausdorff measure|Q1TGZK8RN6]].\n\n## Explanations\n\n### L1 human\n\nPacking measure fills a set with disjoint small balls centred in it and adds the $s$-th powers of their diameters.\nA second step, through countable covers, turns this into a true measure.\n",
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