Amenable group

Summary

An amenable group is a group with a finitely additive, translation-invariant way to measure the size of all of its subsets, with total size $1$. Finite groups, abelian groups and solvable groups are amenable; groups containing a non-abelian free group are not. Amenability has many equivalent forms, such as the Følner condition. It governs the existence of paradoxical decompositions such as the one in the Banach–Tarski paradox.