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  "body": "\n## Summary\n\nThe Riesz measure of a [[subharmonic function|Z1MHMGFNFW]] is its [[Laplacian|0E0B6RA8K0]] taken in the sense of distributions, which is always a positive [[Radon measure|HZNEQQBCW4]].\nIt records where and how strongly the function fails to be [[harmonic|0DEB0JMN0W]]: it vanishes exactly on the open sets where the function is harmonic.\nThrough the [[Riesz decomposition theorem|CBZ20SS8RQ]], a subharmonic function is locally the potential of its Riesz measure plus a harmonic function.\nFor $\\log\\abs{f}$ with $f$ holomorphic, the Riesz measure counts the zeros of $f$.\n\n## Explanations\n\n### L1 human\n\nThink of a subharmonic function as being shaped by a distribution of mass, possibly smeared out.\nThe Riesz measure is that mass distribution: it is the Laplacian of the function, which is never negative.\nFor $\\log\\abs{z}$ in the plane the mass is a single point mass $2\\pi$ at the origin.\nFor $\\abs{x}^2$ in $\\R^n$ the mass is spread evenly everywhere, with density $2n$.\n",
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  "name": "Riesz measure of a subharmonic function",
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  "url": "https://tamnd.github.io/math2/wiki/riesz-measure-of-a-subharmonic-function/",
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