Asymptotic densities of sets of positive integers. (English) Zbl 0516.10052

A density measure is any finitely additive measure which extends the asymptotic density \(d\) to all subsets of \(\mathbb N\). Let \(A(n) = \sum_{a\in A,\ a\le n} 1\). A set function \(\nu\) is said to have property P if for every pair \(A,B \subset \mathbb N\) with \(\lim A(n)/B(n) = t\) then \(\nu(A)/\nu(B) =t\). The authors show that any density measure on \(2^{\mathbb N}\) has property P. By identifying elements of \(2^{\mathbb N}\) with points in \([0,1]\) they show that if \(\nu\) is any set function on \(2^{\mathbb N}\) with property P then if \(A_x\subset A = A_1\), \(\nu(A_x) = \nu(A)/2\) for almost all \(x\in [0,1]\).
Other measure theoretic and topological properties associated with this identification are also discussed.


11B05 Density, gaps, topology
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
11B83 Special sequences and polynomials
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