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  "body": "\n## Definition\n\nLet $n \\ge 2$, let $\\Omega \\subseteq \\R^n$ be an [[open set|02BH3MCBQB]], and let $u$ be [[subharmonic|Z1MHMGFNFW]] on $\\Omega$ and not identically $-\\infty$ on any connected component of $\\Omega$.\nThen $u$ is locally integrable on $\\Omega$.\nMoreover, there is a unique positive [[Radon measure|HZNEQQBCW4]] $\\mu_u$ on $\\Omega$ such that\n$$\\int_\\Omega u \\, \\Delta\\psi \\, d\\lambda = \\int_\\Omega \\psi \\, d\\mu_u \\qquad \\text{for all } \\psi \\in C_c^\\infty(\\Omega),$$\nwhere $\\lambda$ is Lebesgue measure and $\\Delta$ is the [[Laplacian|0E0B6RA8K0]].\nThe measure $\\mu_u$ is the *Riesz measure* of $u$.\nIn other words, $\\mu_u = \\Delta u$ in the sense of distributions.\n\n## Remarks\n\nSome authors normalise differently, for example by $\\frac{1}{2\\pi}\\Delta u$ in the plane, so that $\\log\\abs{z}$ has Riesz measure $\\delta_0$.\nIf $u$ is of class $C^2$, then $\\mu_u = (\\Delta u)\\, \\lambda$.\nThe function $u$ is [[harmonic|0DEB0JMN0W]] on an open subset $V$ of $\\Omega$ if and only if $\\mu_u(V) = 0$.\n\nExample: let $u(x) = \\abs{x}^2$ on $\\R^n$.\nIntegration by parts gives $\\int \\abs{x}^2 \\Delta\\psi \\, d\\lambda = \\int \\Delta(\\abs{x}^2)\\, \\psi \\, d\\lambda = 2n \\int \\psi \\, d\\lambda$, so $\\mu_u = 2n\\,\\lambda$.\nExample: for $u(z) = \\log\\abs{z}$ on $\\C = \\R^2$, Green's formula gives $\\mu_u = 2\\pi\\,\\delta_0$.\n\nNon-example: for $v(x) = -\\abs{x}^2$, the distribution $\\Delta v = -2n\\,\\lambda$ is a negative measure.\nThis matches the fact that $v$ is not subharmonic, so the hypothesis of the definition fails and $v$ has no Riesz measure.\n",
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