There is a P-measure in the random model

TL;DR

This paper proves the existence of P-measures in models with added random reals, extending the concept of P-points and showing their independence from ZFC, with implications for the structure of ch compactifications.

Contribution

It demonstrates that P-measures exist in models obtained by adding any number of random reals to a CH model, generalizing P-points and their properties.

Findings

Existence of P-measures in models with random reals

P-measures generalize P-points and are ZFC-independent

In the random model, ch space contains a nowhere dense ccc closed P-set

Abstract

We say that a finitely additive probability measure $\mu$ on $\omega$ is \emph{a P-measure} if it vanishes on points and for each 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 in a natural way P-points and it is known that, similarly as in the case of 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 obtain that in the classical random model $\omega^*$ contains a nowhere dense ccc closed P-set.