  Given an infinite group G, we consider the finitely additive measure defined
on finite unions of cosets of finite index subgroups. We show that this shares
many properties with the size of subsets of a finite group, for instance we can
obtain equivalent results on Ruzsa distance and product free sets. In
particular if G has infinitely many finite index subgroups then it has subsets
S of measure arbitrarily close to 1/2 with the square of S having measure less
than 1.
