---
tag: ZYNY2VFECQ
kind: concept
name: Invariant mean
msc: [43A07, 28C10]
sources:
  - url: https://en.wikipedia.org/wiki/Amenable_group
---

## Summary

An invariant mean on a [[group|G5S1M9]] is a way to average all bounded functions on the group.
The average does not change when the function is translated.
A discrete group has a left-invariant mean exactly when it is [[amenable|CEAT9BESKK]].
Invariant means are the functional form of finitely additive invariant probability measures.

## Explanations

### L1 human

On a finite group one averages a function by adding its values and dividing by the number of elements.
This average does not change when the function is shifted by a group element.
For an infinite group such as the integers, one would like an average of every bounded sequence that ignores shifts.
Such averages exist for the integers, but they cannot be written down explicitly.
For the free group on two generators, no such average exists at all.