Covering densities. (English) Zbl 0770.11009

Let \(D_ 0\) be the algebra of sets generated by all infinite arithmetic progressions \(\{a+bd\mid d=1,2,\dots;\;a,b\in\mathbb{N}_ 0\}\). R. C. Buck’s measure density is defined by considering natural density as a finitely additive measure on \(D_ 0\) and extending it to a finitely additive outer probability measure \(\mu^*\), defined for all sets \(S\subseteq\mathbb{N}\), which determines a field \(D_ \mu\) of \(\mu^*\)- measurable sets.
The author generalizes this theory by restricting the moduli \(d\) to a fixed (infinite) set \(A\subseteq\mathbb{N}\) of values, assumed to contain all divisors and all l.c.m’s of its elements (and pairs), studying the field \(D_ A\) and the outer measure \(\mu^*_ A\) so obtained. Typical results:
(1) \(\forall S\in D_ A\) \(\forall\alpha\in[0,\mu^*_ A(S)]\) \(\exists S_ 1\subseteq S:S_ 1\in D_ A\), \(\mu^*_ A(S_ 1)=\alpha\) (Darboux property).
(2) For any \(S\subseteq\mathbb{N}\), \(\mu^*(S)=1\) iff a suitable re- arrangement of \(S\) is uniformly distributed to all moduli \(d\in A\): (\(A\)- u.d.).
(3) A set \(S\subseteq\mathbb{N}\) is \(\mu^*_ A\)-measurable if it intersects with each \(A\)-u.d. sequence in a set of natural density \(\mu^*_ A(S)\).


11B05 Density, gaps, topology
11B25 Arithmetic progressions
11B50 Sequences (mod \(m\))
Full Text: EuDML


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