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.

MSC:

 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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References:

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