P-measures in random extensions
Let µ be a finitely additive probability measure on ω which vanishes on points, that is, µ({n})=0 for every n∈ω. It follows immediately that µ is not σ-additive, however it may be almost σ-additive in the following weak sense. We say that µ is a P-measure if for every decreasing sequence (An) of subsets of ω there is a subset A such that A∖ An is finite for every n and µ(A)=limn µ(An). P-measures can be thought of as generalizations of P-points 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 ω* contains a closed nowhere dense ccc P-set, which may be treated as a (weak) partial answer to the open question asking whether there are P-points in the random model.
This is a joint work with Piotr Borodulin-Nadzieja.
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