TITLE: a paradoxical decomposition of a group
QUESTION [6 upvotes]: 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 
$$\Gamma= \bigsqcup_{i} g_i F_i= \bigsqcup_{j} h_j E_j$$

REPLY [3 votes]: 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 $E_1,...,E_m,F_1,...,F_n$.