Invariant mean

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. A left-invariant mean $m$ gives a left-invariant finitely additive measure $\nu(A) = m(\mathbf{1}_A)$ 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 $\ell^\infty(G)$.