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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 wayP-points and it is known that, similarly as in the case of P-points, theirexistence is independent of $\mathsf{ZFC}$. In this paper we show that there isa P-measure in the model obtained by adding any number of random reals to amodel of $\mathsf{CH}$. As a corollary, we obtain that in the classical randommodel $\omega^*$ contains a nowhere dense ccc closed P-set.

There is a P-measure in the random model