Invariant mean

An invariant mean on a group $G$ is a positive linear functional $m:{\ell }^{\infty }(G)\to \mathbb{R}$ with $m(1)=1$ and $m(gf)=m(f)$ for every $g\in G$. Applied to indicator functions, it is a left-invariant finitely additive probability measure on all subsets of $G$.