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  "body": "\n## Statement\n\nLet $n \\ge 3$ and let $\\mu$ and $\\nu$ be positive [[Radon measures|HZNEQQBCW4]] on $\\R^n$ with compact support.\nFor a positive measure $\\lambda$, let $U^\\lambda(x) = \\int \\abs{x - y}^{2-n} \\, d\\lambda(y)$ be its [[Newtonian potential|DCMFQ01QGS]].\nAssume that $\\mu$ has finite [[energy|SFSRQCM3EG]] $\\int U^\\mu \\, d\\mu \u003c \\infty$, and that $U^\\mu(x) \\le U^\\nu(x)$ for $\\mu$-almost every $x$.\nThen $U^\\mu(x) \\le U^\\nu(x)$ for every $x \\in \\R^n$.\n\n## Remarks\n\nProof idea: standard proofs combine balayage with the positive definiteness of the Newtonian energy.\nBalayage sweeps a measure onto a set without changing its potential there.\nThe finite energy of $\\mu$ is what makes these energy arguments possible.\n\nExample: let $\\nu = \\delta_0$, a unit point mass at the origin of $\\R^3$.\nLet $\\mu$ be the uniform measure of mass $1$ on the sphere of radius $r$ centred at $0$.\nOn the sphere, $U^\\mu = 1/r = U^\\nu$, so the hypothesis holds.\nThe theorem gives $U^\\mu \\le U^\\nu$ everywhere, and indeed $U^\\mu(x) = \\min(1/r, 1/\\abs{x}) \\le 1/\\abs{x} = U^\\nu(x)$.\n\nThe finite energy of $\\mu$ is needed.\nFor $\\mu = \\delta_0$ and $\\nu = \\frac12 \\delta_0$, both potentials are $+\\infty$ at $0$, so $U^\\mu \\le U^\\nu$ holds $\\mu$-almost everywhere, but $U^\\mu(x) = 1/\\abs{x} \u003e U^\\nu(x)$ for $x \\ne 0$.\nTake for $\\nu$ a uniform measure on a large sphere, whose potential equals a constant $M$ inside the sphere and is at most $M$ outside.\nThis gives back the maximum principle for potentials of finite energy with support inside the sphere.\n\n## Explanations\n\n### L1 human\n\nThe potential of a charge $\\mu$ of finite energy is controlled everywhere as soon as it is controlled on the charge itself.\nIf the potential of $\\nu$ is at least as large as that of $\\mu$ wherever $\\mu$ sits, then it is at least as large everywhere.\nSo comparing two potentials only needs a check on the support of one of them.\n\n### L2 human\n\nThe domination principle is one of the basic properties of the Newtonian kernel, and it is taken as an axiom in abstract potential theory.\nIt is used in the theory of balayage and in comparison arguments for equilibrium potentials.\nIt holds for Riesz kernels $\\abs{x - y}^{\\alpha - n}$ with $0 \u003c \\alpha \\le 2$ and $\\alpha \u003c n$.\nIn the plane with the logarithmic kernel, an extra condition on the total masses is needed.\n",
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