Amenable locally compact group

A locally compact Hausdorff group $G$ is amenable if there exists a left-invariant mean $m$ on $L^{\mathrm{\infty }}(G)$, using the fixed left Haar measure and the translation action from Left-invariant means on $L^{\mathrm{\infty }}$ of a locally compact group. Thus $m$ is a positive complex-linear functional with $m(1_G)=1$ and $m(L_{g}f)=m(f)$ for every $g\in G$ and $f\in L^{\mathrm{\infty }}(G)$.

This definition imposes no countability, discreteness, compactness, or unimodularity assumption. It does not depend on the normalization of Haar measure: replacing $\mu$ by $c\mu$ for $c>0$ preserves exactly the same null sets, so it gives the same almost-everywhere classes and essential-supremum norm on $L^{\mathrm{\infty }}(G)$. The mean and its left-invariance condition are therefore unchanged.