Density of sets of natural numbers and the Levy group

Let $\mathbb{N}$ denote the set of positive integers. The asymptotic density of the set $A\subseteq \mathbb{N}$ is $d(A)={\lim }_{n\to \mathrm{\infty }}\mathrm{\mid }A\cap [1,n]\mathrm{\mid }\mathrm{/}n$, if this limit exists. Let $\mathcal{A}\mathcal{D}$ denote the set of all sets of positive integers that have asymptotic density, and let $S_{\mathbb{N}}$ denote the set of all permutations of the positive integers \N. The group ${\mathcal{L}}^{\mathrm{♯}}$ consists of all permutations $f\in S_{\mathbb{N}}$ such that $A\in \mathcal{A}\mathcal{D}$ if and only if $f(A)\in \mathcal{A}\mathcal{D}$, and the group ${\mathcal{L}}^{\ast }$ consists of all permutations $f\in {\mathcal{L}}^{\mathrm{♯}}$ such that $d(f(A)) = d(A)$forall$A \in $\mathcal{A}\mathcal{D}$$\mathrm{.}Let$f:\N \to \N $ be a one-to-one function such that $d(f(\mathbb{N}))=1$ and, if $A\in \mathcal{A}\mathcal{D}$, then $f(A) \in $\mathcal{A}\mathcal{D}$$\mathrm{.}Itisprovedthat$f$mustalsopreservedensity,thatis,$d(f(A)) = d(A)$forall$A \in $\mathcal{A}\mathcal{D}$$\mathrm{.}Thus,thegroups$$\mathcal{L}$^{\sharp}$and$$\mathcal{L}$^{\ast}$ coincide.