---
tag: 909B1WYQ4Q
kind: concept
name: Reduced function
msc: [31B05, 31C15, 31D05]
sources:
  - url: https://en.wikipedia.org/wiki/Balayage
---

## Summary

Let $u$ be a positive [[superharmonic function|X4V2S56YQQ]] and $E$ a set.
The reduced function, or réduite, of $u$ on $E$ is the lower envelope of all positive superharmonic functions that dominate $u$ on $E$.
Its lower semicontinuous regularisation is again superharmonic and equals $u$ on $E$ up to a [[polar set|C95QJ7YH5T]].
When $u$ is the potential of a measure, this regularisation is the potential of the [[balayage|NTFY23YDEJ]] of that measure onto $E$.
Reduced functions give capacities, harmonic measures and thinness in one framework.

## Explanations

### L1 human

Start with a positive superharmonic function $u$ and a set $E$.
Among all positive superharmonic functions that are at least $u$ on $E$, take the smallest possible value at each point.
The result keeps $u$ on $E$ and is as small as superharmonicity allows elsewhere.
For 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.