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A discrete group $G$ is amenable when it admits a left-invariant finitely additive probability measure on all subsets of $G$. Equivalently, it admits an invariant mean on the bounded real-valued functions on $G$.

The free group $F_2=⟨a,b⟩$ is not an amenable group. Partitioning reduced words by their first letters gives two translated decompositions of $F_2$ whose invariance equations add to $2=1$, contradicting the existence of an invariant mean.