L\'evy group and density measures

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