Density of sets of natural numbers and the Lévy group Original Research Article

Let N denote the set of positive integers. The asymptotic density of the set Asubset of or equal toN is d(A)=limn→∞A∩[1,n]/n, if this limit exists. Let image denote the set of all sets of positive integers that have asymptotic density, and let SN denote the set of all permutations of the positive integers N. The group image consists of all permutations fset membership, variantSN such that image if and only if image, and the group image consists of all permutationsimage such that d(f(A))=d(A) for all image. Let image be a one-to-one function such that d(f(N))=1 and, if image, then image. It is proved that f must also preserve density, that is, d(f(A))=d(A) for all image. Thus, the groups image and image coincide.