---
tag: PR8KWEA3MH
kind: concept
name: Riesz measure of a subharmonic function
msc: [31B05, 31A05]
sources:
  - url: https://en.wikipedia.org/wiki/Subharmonic_function
---

## Summary

The 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]].
It records where and how strongly the function fails to be [[harmonic|0DEB0JMN0W]]: it vanishes exactly on the open sets where the function is harmonic.
Through the [[Riesz decomposition theorem|CBZ20SS8RQ]], a subharmonic function is locally the potential of its Riesz measure plus a harmonic function.
For $\log\abs{f}$ with $f$ holomorphic, the Riesz measure counts the zeros of $f$.

## Explanations

### L1 human

Think of a subharmonic function as being shaped by a distribution of mass, possibly smeared out.
The Riesz measure is that mass distribution: it is the Laplacian of the function, which is never negative.
For $\log\abs{z}$ in the plane the mass is a single point mass $2\pi$ at the origin.
For $\abs{x}^2$ in $\R^n$ the mass is spread evenly everywhere, with density $2n$.