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Partial densities on the group of integers. (English) Zbl 1052.11054
Let \(G\) be a locally compact abelian group, \(\{H_\alpha \,| \,\alpha \in A\}\) be the set of all subgroups of \(G\) such that the quotient group \(G_\alpha =G/H_\alpha \) is compact. A system \[ D=\{\mu _\alpha \,| \,\mu _\alpha \text{ is a probability measure on } G_\alpha , \alpha \in A\} \] is called a density on \(G\) if it satisfies the following condition: If \(\psi :\,G_\beta \to G_\alpha \) is the natural homomorphism from \(G_\beta \) to a quotient \(G_\alpha \) of \(G_\beta \), then for any Borel set \(B\) in \(G_\alpha \), \(\mu _\alpha (B)=\mu _\beta (\psi ^{-1}(B))\). Generalizing this notion, H. Niederreiter [Astérisque 24–25, 243–261 (1975; Zbl 0306.10035)] defined a partial density on \(G\).
In this paper the question is solved when a partial density on the group of integers with the discrete topology can be extended to a density.
11K06 General theory of distribution modulo \(1\)
28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
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