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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]], and let $\\operatorname{diam} U$ be the [[diameter|0KW3PN547K]] of a set $U \\subseteq X$.\nLet $s \\ge 0$, with the convention $(\\operatorname{diam} U)^0 = 1$ for every $U$.\nFor $A \\subseteq X$ and $\\delta \u003e 0$, let\n$$\\mathcal{H}^s_\\delta(A) = \\inf \\set{ \\sum_{i=1}^\\infty (\\operatorname{diam} U_i)^s : A \\subseteq \\bigcup_{i=1}^\\infty U_i, \\ \\operatorname{diam} U_i \\le \\delta },$$\nwhere the [[infimum|0QF1MZ9FRG]] runs over all countable covers of $A$ by sets $U_i \\subseteq X$ of diameter at most $\\delta$.\nThe *$s$-dimensional Hausdorff measure* of $A$ is\n$$\\mathcal{H}^s(A) = \\lim_{\\delta \\to 0} \\mathcal{H}^s_\\delta(A) = \\sup_{\\delta \u003e 0} \\mathcal{H}^s_\\delta(A).$$\n\n## Remarks\n\nThe limit exists in $[0, \\infty]$ because $\\mathcal{H}^s_\\delta(A)$ grows as $\\delta$ decreases.\nThe map $\\mathcal{H}^s$ is a [[metric outer measure|3T6WG315GC]], so all Borel sets are measurable and its restriction to them is a measure.\nSome authors multiply by a constant so that $\\mathcal{H}^n$ equals [[Lebesgue measure|2Z2QVWZ262]] on $\\R^n$; without it, $\\mathcal{H}^n$ is a constant multiple of Lebesgue measure.\n\nExample: $\\mathcal{H}^0$ is counting measure.\nOn $\\R$, $\\mathcal{H}^1$ is Lebesgue measure: covers by intervals realize the infimum, and the length of an interval is its diameter.\n\nNon-example: the quantity without the limit $\\delta \\to 0$, the Hausdorff content $\\mathcal{H}^s_\\infty$ with no bound on diameters, is not a measure on Borel sets.\nIn $\\R^2$ with $s = 1$, the segments $A = [0, 1] \\times \\set{0}$ and $B = [0, 1] \\times \\set{1}$ each have content $1$.\nBut $A \\cup B$ lies in the unit square, of diameter $\\sqrt 2$, so its content is at most $\\sqrt 2 \u003c 2$.\n\n## Explanations\n\n### L1 human\n\nCover the set by small pieces, add the $s$-th powers of their sizes, and take the best cover.\nLetting the pieces become arbitrarily small gives the $s$-dimensional size of the set.\n",
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