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  "body": "\n## Statement\n\nLet $n \\ge 3$ and let $\\mu$ be a positive [[Radon measure|HZNEQQBCW4]] on $\\R^n$ with compact [[support|FQTXV6BQZG]] $S$.\nLet\n$$U^\\mu(x) = \\int_{\\R^n} \\abs{x - y}^{2-n} \\, d\\mu(y)$$\nbe its [[Newtonian potential|DCMFQ01QGS]], with values in $[0, \\infty]$.\nLet $M$ be a real number with $U^\\mu(x) \\le M$ for every $x \\in S$.\nThen $U^\\mu(x) \\le M$ for every $x \\in \\R^n$.\n\n## Remarks\n\nProof idea: first assume that $U^\\mu$ restricted to $S$ is continuous.\nThe [[continuity principle|GPJC1NY5RE]] then makes $U^\\mu$ continuous on $\\R^n$.\nOn the open set $\\R^n \\setminus S$ the potential is harmonic, it is at most $M$ on the boundary, which lies in $S$, and it tends to $0$ at infinity.\nThe maximum principle for harmonic functions gives $U^\\mu \\le M$ there.\nThe general case follows by restricting $\\mu$ to compact subsets of $S$ on which $U^\\mu$ is continuous, which exist by Lusin's theorem, and letting these subsets increase.\n\nExample: let $\\mu$ be the uniform measure of total mass $m$ on the unit sphere in $\\R^3$.\nBy the shell theorem, $U^\\mu(x) = m$ for $\\abs{x} \\le 1$ and $U^\\mu(x) = m / \\abs{x}$ for $\\abs{x} \\ge 1$.\nSo $U^\\mu \\le m$ on the support, and indeed $U^\\mu \\le m$ everywhere.\n\nThe positivity of $\\mu$ is needed.\nLet $\\sigma$ be the uniform probability measure on the unit sphere of $\\R^3$ and let $\\mu = -\\sigma$.\nThen $U^\\mu = -1$ on the support, but $U^\\mu(x) = -1/\\abs{x} \u003e -1$ for $\\abs{x} \u003e 1$.\nFor logarithmic potentials in the plane the same statement holds: if $\\int \\log(1/\\abs{z - w}) \\, d\\mu(w) \\le M$ on the support of $\\mu$, then it is at most $M$ on all of $\\C$.\n\n## Explanations\n\n### L1 human\n\nThe potential of a positive charge is largest on the charge itself.\nIf the potential stays below a level $M$ wherever there is charge, it stays below $M$ everywhere in space.\nThis is the potential-theory version of the fact that a harmonic function has no interior maximum.\n\n### L2 human\n\nThe principle is due to Maria and to Frostman.\nIt is the key step that turns an inequality on the support of a measure into an inequality everywhere.\nIt is used in the proof of Frostman's theorem to show that the equilibrium potential never exceeds its value on the conductor.\nFor Riesz kernels $\\abs{x - y}^{\\alpha - n}$ with $0 \u003c \\alpha \u003c n$ and $\\alpha \u003e 2$, only a weaker version holds, with $M$ replaced by a constant multiple of $M$.\n",
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