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  "body": "\n## Summary\n\nLet $u$ be a positive [[superharmonic function|X4V2S56YQQ]] and $E$ a set.\nThe reduced function, or réduite, of $u$ on $E$ is the lower envelope of all positive superharmonic functions that dominate $u$ on $E$.\nIts lower semicontinuous regularisation is again superharmonic and equals $u$ on $E$ up to a [[polar set|C95QJ7YH5T]].\nWhen $u$ is the potential of a measure, this regularisation is the potential of the [[balayage|NTFY23YDEJ]] of that measure onto $E$.\nReduced functions give capacities, harmonic measures and thinness in one framework.\n\n## Explanations\n\n### L1 human\n\nStart with a positive superharmonic function $u$ and a set $E$.\nAmong all positive superharmonic functions that are at least $u$ on $E$, take the smallest possible value at each point.\nThe result keeps $u$ on $E$ and is as small as superharmonicity allows elsewhere.\nFor example, in $\\R^3$ with $u = 1$ and $E$ the closed unit ball, the reduced function is $\\min(1, 1/\\abs{x})$, the potential of a charged metal ball.\n",
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