On sequential properties of Banach spaces, spaces of measures and densities

We show that a conjunction of Mazur and Gelfand–Phillips properties of a Banach space E can be naturally expressed in terms of weak ∗ continuity of seminorms on th unit ball of E ∗. We attempt to carry out a construction of a Banach space of the form C(K) which has the Mazur property but does not have the Gelfand–Phillips property. For that purpose we analyze compact spaces on which all regular measures lie in the weak ∗ sequential closure of atomic measures, and set–theoretic properties of generalized densities on the natural numbers. 1