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  "body": "\n## Summary\n\nBalayage, French for \"sweeping out\", replaces a measure by a measure on a given set.\nThe new measure has the same potential as the old one outside that set, and on it up to small exceptional sets.\nPoincaré introduced it to solve the Dirichlet problem: the charge of a measure in a domain is swept to the boundary through [[harmonic measure|CFD7BPM51W]].\nIn modern potential theory the balayage of a measure onto any set is defined through the [[reduced function|909B1WYQ4Q]] of its potential.\nIt is one of the basic operations of classical and axiomatic potential theory.\n\n## Explanations\n\n### L1 human\n\nThink of a point charge inside a hollow metal ball.\nSeen from outside the ball, the charge cannot be told apart from a suitable charge spread over the sphere.\nBalayage is the rule that produces this spread-out charge.\nFor a unit charge at the centre of a ball in space, the swept charge is the uniform distribution of total mass $1$ on the sphere.\nIts potential agrees with that of the point charge outside the ball and is smaller inside.\n",
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