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There is a P-measure in the random model

Volume 262 / 2023

Piotr Borodulin-Nadzieja, Damian Sobota Fundamenta Mathematicae 262 (2023), 235-257 MSC: Primary 03E05; Secondary 03E75, 03E35, 28E15. DOI: 10.4064/fm277-3-2023 Published online: 15 May 2023

Abstract

We say that a finitely additive probability measure on \omega is a P-measure if it vanishes on points, and for each \subseteq -decreasing sequence (E_n) of infinite subsets of \omega there is E\subseteq \omega such that E\,\subseteq ^* E_n for each n\in \omega and \mu (E) = \lim _{n\to \infty}\mu (E_n). Thus, P-measures generalize P-points and it is known that, similarly to P-points, their existence is independent of \mathsf{ZFC}. In this paper we show that there is a P-measure in the model obtained by adding any number of random reals to a model of \mathsf{CH}. As a corollary, we deduce that in the classical random model, \omega ^* contains a nowhere dense ccc closed P-set.

Authors

  • Piotr Borodulin-NadziejaMathematical Institute
    University of Wrocław
    50-384 Wrocław, Poland
    e-mail
  • Damian SobotaKurt Gödel Research Center for Mathematical Logic
    Department of Mathematics
    University of Vienna
    1090 Wien, Austria
    e-mail

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