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HANDMADE DENSITY SETS

  • GEMMA CAROTENUTO (a1)

Abstract

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.

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[1] Andretta, A. and Camerlo, R., The descriptive set theory of the Lebesgue Density Theorem . Advances in Mathematics, vol. 234 (2013), pp. 142.
[2] Andretta, A., Camerlo, R., and Costantini, C., Lebesgue density and exceptional points, http://arxiv.org/abs/1510.04193, 2015.
[3] Carotenuto, G., On the topological complexity of the density sets of the real line, Ph.D. thesis, Università di Salerno, http://elea.unisa.it:8080/jspui/bitstream/10556/1972/1/tesi_G.Carotenuto.pdf, 2015.
[4] Kechris, A. S., Classical descriptive set theory . Advances in Mathematics, vol. 234 (1995), pp. 142.
[5] Oxtoby, J. C., Measure and Category: A Survey of the Analogies Between Topological and Measure Spaces, Graduate Texts in Mathematics, Springer-Verlag, New York, 1980.
[6] Tacchi, J. P., Points of density of Cantor sets . European Journal of Combinatorics, vol. 16 (1995), no. 6, pp. 645653 (In French).
[7] Wilczyński, W., Density topologies , Handbook of Measure Theory (Pap, E., editor), North-Holland, Amsterdam, 2002, pp. 675702.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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