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On nonparadoxical sets. (English) Zbl 0763.04005
Summary: Subsets \(A\), \(B\) of \(\mathbb{R}^ n\) are countably equidecomposable if there is a partition \(\{A_ m: m\in\omega\}\) of \(A\) and isometries \(g_ m\) of \(\mathbb{R}^ n\) such that \(\{g_ m A_ m: m\in\omega\}\) is a partition of \(B\). A set \(A\) is paradoxical if \(A\) contains two disjoint subsets each countably equidecomposable with \(A\). We show the existence of nonparadoxical sets of full Lebesgue measure. We also prove that every set of positive measure contains an uncountable paradoxial subset of full measure. A subset \(A\) of \(\mathbb{R}^ n\) is hereditarily nonparadoxical if \(A\) has no uncountable paradoxical subsets. It is shown that the family of hereditarily nonparadoxical sets is a proper ideal and that, under \(\neg\text{CH}\), the union of countably many hereditarily nonparadoxical sets has inner measure zero. This generalizes a result by Erdős and Kunen. We answer related questions concerning sets without repeated distances.

03E05 Other combinatorial set theory
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
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