P-measures in models without P-points

We answer in negative the problem if the existence of a P-measure implies the existence of a P-point. Namely, we show that if we add random reals to a certain unique P-point model, then in the resulting model we will have a P-measure but not P-points. Also, we investigate the question if there is a P-measure in the Silver model. We show that rapid filters cannot be extended to a P-measure in the extension by $\omega$ product of Silver forcings and that in the model obtained by the product of $\omega_2$ many Silver forcings there are no P-measures of countable Maharam type