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I'm looking for a reference (article or book) of the following equivalence of amenability. Let $G$ be a countable group.

  1. There exists a left invariant mean $m : \ell^{\infty}(G) \to \mathbb{R}$
  2. For every finite set $S \subseteq G$ and every $\epsilon >0$ there exists a non empty set finite $A \subseteq G $ such that $ \frac{\left| s A \Delta A \right|}{\left| A \right|} < \epsilon$ for all $s \in S$
  3. There exists a sequence $\Phi_n$ of non empty finite sets such that $$ \lim_n \frac{ \left| \Phi_n \cap \Phi_n g \right|}{ \left| \Phi_n \right|} =1 $$ for all $g \in G$.

Thank you in advance.

3m0o
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    These are all standard. Which books did you check? – Moishe Kohan Dec 22 '23 at 02:21
  • I know these are all standard, but unfortunately, I don't know any beautiful book about amenable groups. I have only the notes of a past course I took. So I was looking for a reference for all these standard facts. – 3m0o Dec 22 '23 at 03:18

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