Just my curiosity... Are there proofs the following fact, which does not involve Hall's matching theorem:


A group $\Gamma$ is amenable if and only if it does not admit a paradoxical decomposition.


**Def:** A group $\Gamma$ has a **paradoxical decomposition** if there are pairwise disjoint subsets $F\_1,\ldots, F\_n$, $E\_1,\ldots, E\_m$ of $\Gamma$, and elements $g\_1,\ldots g\_n$, $h\_1,\ldots,h\_m\in \Gamma$ such that $\Gamma$ can be expressed as 
$$\Gamma= \bigsqcup\_{i} g\_i F\_i= \bigsqcup\_{j} h\_j E\_j$$



a paradoxical decomposition of a group
According to a note in Grigorchuk's and Sunic's Self-Similarity and Branching in Group Theory, there is a proof not using the matching theorem in the book The Banch-Tarski Paradox by Stan Wagon.
By the way, you have to mention that $\Gamma$ is also the union of $E\_1,...,E\_m,F\_1,...,F\_n$.


