to define a measure on , which philosophically speaking says how much of any subset takes up. The natural definition that uses our Følner sequence would be Of course, this limit doesn't necessarily exist. To overcome this technicality, we take an ultrafilter on the natural numbers that contains intervals . Then we use an ultralimit instead of the regular limit: It turns out ultralimits have all the properties we need. Namely, is a probability measure. That is, , since the ultralimit coincides with the regular limit when it exists. is finitely additive. This is since ultralimits commute with addition just as regular limits do. is left invariant. This is since by the Følner sequence definition. References Geometric group theory