Lévy group and density measures

We will deal with finitely additive measures on integers extending the asymptotic density. We will study their relation to the Lévy group $\mathcal{G}$ of permutations of $\mathbb N$. Using a new characterization of the Lévy group $\mathcal G$ we will prove that a finitely additive measure extends density if and only if it is $\mathcal{G}$-invariant.

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