Let \(\mu\) be a finitely additive probability measure on \(\omega\) which vanishes on points, that is, \(\mu(\{n\})=0\) for every \(n\). It follows immediately that \(\mu\) is not \(\sigma\)-additive, however it may be almost \(\sigma\)-additive in the following weak sense. We say that \(\mu\) is a P‑measure if for every decreasing sequence \((A_n)\) of subsets of \(\omega\) there is a subset \(A\) such that \(A\setminus A_n\) is finite for every \(n\) and \(\mu(A)=\lim_n \mu(A_n)\). It follows immediately that, e.g., an ultrafilter \(\mathcal{U}\) on \(\omega\) is a P‑point if and only if the one-point measure \(\delta_\mathcal{U}\) is a P‑measure. And similarly as in the case of P‑points the existence of P‑measures is independent of ZFC.

During my talk I will discuss basic properties of P‑measures and show, at least briefly, that using old ideas of Solovay and Kunen one can obtain a non-atomic P‑measure in the random model. The latter result implies that in this model there exists a nowhere dense ccc P‑set in \(\omega^*\), which may be treated as a (weak) partial answer to the question asking whether there are P‑points in the random model.

This is a joint work with Piotr Borodulin-Nadzieja.