On measures induced by forcing names for ultrafilters

TL;DR

This paper explores how measure-theoretic properties of Boolean algebras relate to forcing names for ultrafilters, providing new characterizations and applications, including a reproof of a theorem on towers in the random model.

Contribution

It introduces natural characterizations of measure properties of Boolean algebras within the forcing framework and applies these to classical set-theoretic results.

Findings

Characterization of measure properties via forcing names

Reproof of Kunen's theorem on towers in the random model

Connections between measure support and forcing properties

Abstract

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 $\omega_2$.