Definition of amenability by Følner sequences I am a little bit confused with the definition of amenability using the existence of Følner sequences. It seems that some people define a Følner sequence for certain classes of groups only. For example, Prof. Terence Tao in his notes writes:

Let $G = (G,\cdot)$ be a discrete, at most countable, group.  A Følner sequence is a sequence $F_1, F_2, F_3, \ldots$ of finite subsets of $G$ with $\bigcup_{N=1}^\infty F_N = G$ with the property that $\lim_{N \to \infty} \frac{|g F_N \triangle F_N|}{|F_N|} = 0$ for all $g \in G$, where $\triangle$ denotes symmetric difference.

On the other hand, Prof. Cornelia Drutu gives a more straightforward definition, for any group $G$:

A sequence of non-empty subsets $F_n$ of $G$ is called a Følner sequence in $G$ if for every $g$ in $G$,
$$\lim\limits_{n \to \infty} \frac{|g F_n \triangle F_n|}{|F_n|}.$$

In both cases, the definition of amenability would be that $G$ (necessarily discrete and at most countable in the first case only) is amenable if it has at least one Følner sequence.
My questions are as follows.

*

*How are these two definitions related? Is there any reason to consider the topology on $G$ and limit oneself to countability? How are these assumptions relevant?

*If $G$ is countable and dicrete, are these two definitions equivalent?

*Is the condition $\bigcup_{N=1}^\infty F_N = G$ in the first definition necessary?

*Is the second definition equivalent to $G$ being non-paradoxical, under $\mathrm{AC}$? If so, is there a direct proof, without referring to other equivalent definitions of amenability?

Any help would be largely appreciated!

The condition that $\bigcup F_n=G$ is unusual, for instance $(\{1,\dots,n\})_{n\ge 0}$ is a reasonable Følner sequence in $\mathbf{Z}$.
If $(F_n)$ is Følner, however, one can modify it to get a covering sequence (i.e. $\bigcup F_n=G$). Just fix an enumeration $(g_n)$ of $G$ and set $F'_n=F_n\cup\{g_n\}$. It's still Følner.
Indeed if $G$ is finite, necessarily $F_n=G$ for large $n$. Otherwise, the added singleton makes a negligible contribution to the ratio.
For instance, $(\{-n\}\cup\{1,\dots,n\})_{n\ge 1}$ is Følner and covers $\mathbf{Z}$. This example illustrates that the covering condition is somewhat artificial.
