  We study the interplay between properties of measures on a Boolean algebra A
and forcing names for ultrafilters on A. We show that several well known
measure theoretic properties of Boolean algebras (such as supporting a strictly
positive measure or carrying only separable measures) have quite natural
characterizations in the forcing language. We show some applications of this
approach. In particular, we reprove a theorem of Kunen saying that in the
classical random model there are no towers of height $\omega_2$.
