There is a P-measure in the random model
Volume 262 / 2023
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 $\mu $ 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.
