Følner's characterization of Amenability says that a group $G$ is amenable if there exists a directed set $(I,\leq)$ and a net {$F\_i:i\in I$} of finite subsets of $G$ such that for every $γ ∈ G$,
$$\lim\_{i\in I}\frac{|γF\_i\Delta F\_i|}{|F\_i|}\ \ \ \rightarrow 0\ ,$$
where $\Delta$ is the symmetric difference of two sets. It is also known that, if $G$ is countable, the word "net" can be substituted by "sequence" (that is $I=\mathbb N$ with the usual order). 


Is it true that for countable (or at least finitely generated) groups we can always find a Følner sequence as above, which satisfies the following conditions:


(1) $F\_{n}\subseteq F\_{n+1}$, for all $n\in \mathbb N$;


(2) $\bigcup\_{n\in\mathbb N}F\_n=G$.


The motivation for my question comes from the paper "The Abramov-Rokhlin Entropy Addition Formula for Amenable Group Actions" by Ward and Zhang (Mh. Math, 1992). In fact, their Theorem 2.6 (that they attribute to Ornstein and Weiss) is proved for a Følner sequence as the one above but it is applied to actions of arbitrary countable Amenable groups. So... it seems that such sequences always exist.



Another question about amenability and Følner sequences
Yes - see Lemma 5.1 of Gabor Pete's book on probability and groups: <http://www.math.bme.hu/~gabor/PGG.html> (by the way, these are excellent lecture notes)


