There is a P-measure in the random model

We say that a finitely additive probability measure $μ$ on $ω$ is \emph{a P-measure} if it vanishes on points and for each decreasing sequence $(E_n)$ of infinite subsets of $ω$ there is $E\subseteqω$ such that $E\subseteq^* E_n$ for each $n\inω$ and $μ(E) = \lim_{n\to\infty}μ(E_n)$. 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 $ω^*$ contains a nowhere dense ccc closed P-set.

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