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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.
##### MSC:
 11K06 General theory of distribution modulo $$1$$ 28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
##### Keywords:
partial density; extension to density
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