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  "body": "\n## Statement\n\nLet $A \\subseteq \\R^n$ be a Borel set and let $s \\ge 0$.\nLet $\\mathcal{H}^s$ be the $s$-dimensional [[Hausdorff measure|Q1TGZK8RN6]], and let $B(x, r)$ be the closed ball of centre $x$ and radius $r$.\nThen $\\mathcal{H}^s(A) \u003e 0$ if and only if there is a nonzero [[Radon measure|HZNEQQBCW4]] $\\mu$ on $\\R^n$ with these two properties:\n\n1. The [[support|FQTXV6BQZG]] of $\\mu$ is a [[compact set|0R6EWFKRCF]] contained in $A$.\n2. $\\mu(B(x, r)) \\le r^s$ for every $x \\in \\R^n$ and every $r \u003e 0$.\n\n## Remarks\n\nThe direction from $\\mu$ to $\\mathcal{H}^s(A) \u003e 0$ is the mass distribution principle.\nIf $A \\subseteq \\bigcup_i U_i$, each nonempty $U_i$ lies in a closed ball of radius $\\operatorname{diam} U_i$ centred at a point of $U_i$.\nSo $0 \u003c \\mu(A) \\le \\sum_i \\mu(U_i) \\le \\sum_i (\\operatorname{diam} U_i)^s$, and the infimum over covers is at least $\\mu(A)$.\nFor the other direction, with $A$ compact, one builds $\\mu$ on dyadic cubes from the finest scale upwards, cutting the mass wherever the bound fails, and takes a weak limit.\nThe Borel case reduces to the compact case, since a Borel set with $\\mathcal{H}^s(A) \u003e 0$ contains a compact set with positive $\\mathcal{H}^s$ measure.\n\nAs a consequence, $\\dim_H A = \\sup \\set{ s \\ge 0 : \\text{a measure as above exists} }$ for every Borel set $A$, where $\\dim_H$ is the [[Hausdorff dimension|SSNFTHZ6YJ]].\n\nExample: for $A = [0, 1] \\subseteq \\R$ and $s = 1$, Lebesgue measure restricted to $[0, 1]$ works, since an interval of radius $r$ has length $2r$.\nAfter scaling by $\\frac12$, it satisfies $\\mu(B(x, r)) \\le r$.\n\n## Explanations\n\n### L1 human\n\nA set is at least $s$-dimensional in the sense of Hausdorff measure exactly when one can spread a positive amount of mass on it that is never too concentrated.\nHere \"not too concentrated\" means that a ball of radius $r$ carries at most $r^s$ of the mass.\nThis turns a question about all covers of the set into the search for one good measure.\n\n### L2 human\n\nLower bounds for Hausdorff dimension are usually proved by constructing a Frostman measure, since upper bounds come from explicit covers.\nThe lemma leads to the energy method: if $A$ carries a nonzero measure with finite $t$-energy, then $\\dim_H A \\ge t$.\nIt is a standard tool in projection theorems and in the study of Kakeya sets.\nIt also links Hausdorff dimension to capacity in potential theory.\n",
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