---
tag: G9M1H5NNF6
kind: concept
name: Balayage space
msc: [31D05, 60J45]
sources:
  - url: https://en.wikipedia.org/wiki/Balayage
---

## Summary

A balayage space is a locally compact space with a countable base together with a convex cone of positive lower semicontinuous functions.
This cone plays the role of the positive superharmonic functions.
The axioms are chosen so that [[balayage|NTFY23YDEJ]], the sweeping of functions and measures onto sets, can be carried out.
The notion was developed by Jürgen Bliedtner and Wolfhard Hansen as a common frame for [[harmonic spaces|DWN0XEQHHK]] and for the potential theory of Markov processes.
Its theory includes the fine topology, thin sets, and the Dirichlet problem.

## Explanations

### L1 human

In classical potential theory, sweeping a charge onto a set means replacing it by a charge on the set that produces the same potential outside.
Most of this theory uses only the cone of positive superharmonic functions, not the harmonic functions themselves.
A balayage space starts from such a cone of functions and asks only for the properties needed to sweep.
The positive superharmonic functions on a Euclidean domain give an example, and so do the excessive functions of many Markov processes, such as Brownian motion.