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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]], and let $s \u003e 0$.\nFor $U \\subseteq X$, let $\\operatorname{diam} U = \\sup \\set{d(x, y) : x, y \\in U}$, with $\\operatorname{diam} \\emptyset = 0$.\nThe *$s$-dimensional Hausdorff content* of a set $E \\subseteq X$ is\n$$\\mathcal{H}^s_\\infty(E) = \\inf \\set{\\sum_{i=1}^{\\infty} (\\operatorname{diam} U_i)^s : E \\subseteq \\bigcup_{i=1}^{\\infty} U_i}.$$\nThe infimum is over all countable covers $(U_i)$ of $E$ by subsets of $X$.\n\n## Remarks\n\nThe map $\\mathcal{H}^s_\\infty$ is an [[outer measure|BJ314H17HC]] on $X$.\nIt has the same null sets as the [[Hausdorff measure|Q1TGZK8RN6]] $\\mathcal{H}^s$, so $\\dim_H E = \\inf \\set{s \u003e 0 : \\mathcal{H}^s_\\infty(E) = 0}$, with the convention $\\inf \\emptyset = \\infty$.\nExample: in $\\R$, $\\mathcal{H}^{1}_\\infty([0,1]) = 1$.\nThe cover by $[0,1]$ itself gives at most $1$, and any countable cover has total diameter at least the Lebesgue measure $1$.\nNon-example of additivity: in $\\R$ with $s = 1/2$, each of $[0,1]$ and $[1,2]$ has content $1$.\nIndeed, a cover that uses a set of diameter at least $1$ has total at least $1$.\nIn a cover with all diameters below $1$, one has $(\\operatorname{diam} U_i)^{1/2} \\ge \\operatorname{diam} U_i$, and the diameters add up to at least $1$.\nBut the single set $[0,2]$ covers their union, so the union has content at most $\\sqrt 2 \u003c 2$.\n",
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