On Q- and selective measures

We will present some generalizations of well-known definitions of types of ultrafilters to the realm of finitely additive measures on $\omega$. We will show a few results similar to the ones for ultrafilters: measure is selective if and only if it is a P-measure and a Q-measure, and that selective measures (Q-measures, respectively) are minimal in the Rudin-Keisler (Rudin-Blass) ordering. We will also show an example of a selective non-atomic measure. The second part will be focused on the integration: we will briefly describe Lebesgue integral with respect to finitely additive measures on $\omega$ and prove that it is a generalization of an ultralimit. Finally, we will present an idea of further generalizations of these definitions for functionals on $\ell^{\infty}$.