The measure density on Dedekind domains
Let \(R\) be a Dedekind domain such that, for each non-trivial ideal \(I\subset R\), the factor ring \(R/I\) is finite (e.g., the ring of integers in an algebraic number field). Denote by \({\mathcal I}(R)\) the set of non-zero ideals of \(R\), and let \(N(I)=|R/I|\) . For \(x\in R\) and \(I\in{\mathcal I}(R)\), the density of the set \(x+I\subset R\) is given by \(\mu(x+I)=1/N(I)\). For an arbitrary subset \(S\subset R\), the upper measure is defined by \[ \mu^*(S)=\inf\left\{ \sum_{j=1}^k{1\over N(I_j)}: S\subset\bigcup_{j=1}^k(x_j+I_j), x_j\in R, I_j\in{\mathcal I}(R), k\in{\mathbb N}\right\}. \] This extends a concept of \textit{R. C. Buck} [Am. J. Math. 68, 560-580 (1949)]. The main results of the paper under review are as follows: (1) Let \((I_n)\) be a complete sequence of ideals, i.e., for each \(I\in{\mathcal I}(R)\) there is an index \(n_0\) such that \(I_n\subset I\) for any \(n\geqslant n_0\). Then \[ \mu^*(S)=\lim_{n\to\infty}{|S/I_n|\over |R/I_n|} \qquad\text{for each }S\subset R. \] (2) A set \(S\subset R\) is called measurable if \(\mu^*(S)+\mu^*(R\setminus S)=1\). Denote by \(D_\mu\) the family of measurable sets. Then the restriction \(\mu=\mu^*|_{D_\mu}\) is a finitely additive measure satisfying the so-called Darboux property: If \(A_1\subset A_2\subset R\) are measurable, and if \(\mu(A_1)\leqslant\alpha\leqslant\mu(A_2)\), then there is a set \(B\in D_\mu\) such that \(A_1\subset B\subset A_2\) and \(\mu(B)=\alpha\). This implies that the family \(D_\mu\) is uncountable. (3) The Dedekind domain \(R\) is called abundant if \(\sum_{I\in{\mathcal I}(R)}1/N(I)=\infty\). If \(R\) is abundant and \((P_n)\) is a sequence of prime ideals with \(\sum 1/N(P_n)=\infty\), then \(S\in D_\mu\) if and only if \(S\cap(P_n\setminus P_n^2)\in D_\mu\) for all \(n\). Examples are given to show that some corollaries to theorem 3 are false if \(R\) is not abundant.
scientific article; zbMATH DE number 1293097