---
tag: 64F9VMG27D
kind: definition
name: Invariant mean
msc: [43A07, 28C10]
defines: ZYNY2VFECQ
uses: [G5S1M9, 07YQ1C78G0, CEAT9BESKK, KFWQ24D9M6]
sources:
  - url: https://en.wikipedia.org/wiki/Amenable_group
---

## Definition

Let $G$ be a [[group|G5S1M9]] and let $\ell^\infty(G)$ be the real vector space of bounded functions $f \colon G \to \R$.
For $g \in G$ and $f \in \ell^\infty(G)$, let $g \cdot f$ be the function $x \mapsto f(g^{-1} x)$.
A *left-invariant mean* on $G$ is a [[linear form|07YQ1C78G0]] $m \colon \ell^\infty(G) \to \R$ with these three properties.

1. Positivity: $m(f) \ge 0$ for every $f \ge 0$.
2. Normalization: $m(\mathbf{1}) = 1$, where $\mathbf{1}$ is the constant function $1$.
3. Invariance: $m(g \cdot f) = m(f)$ for every $g \in G$ and $f \in \ell^\infty(G)$.

## Remarks

Right-invariant and two-sided invariant means are defined in the same way.
A group has a left-invariant mean if and only if it is [[amenable|CEAT9BESKK]].
A left-invariant mean $m$ gives a left-invariant [[finitely additive measure|KFWQ24D9M6]] $\nu(A) = m(\mathbf{1}_A)$ on all subsets of $G$, with $\nu(G) = 1$, and every such $\nu$ arises in this way.

Example: for a finite group $G$, the average $m(f) = \frac{1}{\abs{G}} \sum_{x \in G} f(x)$ is a left-invariant mean.
Positivity and normalization are clear, and invariance holds because $x \mapsto g^{-1} x$ is a bijection of $G$.

Non-example: on $G = \Z$, the evaluation $m(f) = f(0)$ is positive and normalized but not invariant.
For $f = \mathbf{1}_{\set{0}}$ and $g = 1$, the function $g \cdot f$ is $\mathbf{1}_{\set{1}}$, so $m(g \cdot f) = 0 \ne 1 = m(f)$.

On $\Z$, invariant means exist by the Hahn–Banach theorem, for example Banach limits, but none can be given by an explicit formula.
For a locally compact group, one uses $L^\infty(G)$ with respect to Haar measure in place of $\ell^\infty(G)$.

## Explanations

### L1 human

An invariant mean is an averaging rule for all bounded functions on a group that ignores translations.
It always exists on finite groups and abelian groups, and never on free groups of rank two.