I have made several attempts at various times to understand the many equivalent definitions of an amenable group. Is the following statement correct?
A group $G$ is amenable if and only if, for any finite subset $X$ of $G$ and any $\epsilon > 0$, there is a finite subset $A$ of the subgroup $\langle X \rangle$ of $G$ generated by $X$, such that $|xA \, \Delta\, A|/|A| < \epsilon$ for all $x \in X$.
Thanks!
Yes, this is correct.
As Yves Cornulier notes, your condition implies the Følner condition which is the same as yours, but with the condition $A \subseteq \langle X \rangle$ removed.
Conversely, suppose $G$ is amenable. Then the subgroup $H = \langle X \rangle$ is amenable and if $(F_n)_{n \in \mathbb{N}}$ is a Følner sequence for $H$ then choosing $n$ large enough allows us to take $A = F_n$.
An advantage of your condition is that it is immediately clear that a subgroup of a group satisfying your condition also satisfies your condition. To show that the usual Følner condition passes to subgroups requires an additional argument.