Handmade density sets. (English) Zbl 1422.03096
Summary: Given a metric space \((X,d)\), equipped with a locally finite Borel measure, a measurable set \(A \subseteq X\) is a density set if the points where \(A\) has density 1 are exactly the points of \(A\). We study the topological complexity of the density sets of the real line with Lebesgue measure, with the tools – and from the point of view – of descriptive set theory. In this context a density set is always in \(\Pi_3^0\). We single out a family of true \(\Pi_3^0\) density sets, an example of true \(\Sigma_2^0\) density set and finally one of true \(\Pi_2^0\) density set.
MSC:
| 03E15 | Descriptive set theory |
| 28A05 | Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets |
References:
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