On sequences of homomorphisms into measure algebras and the Efimov Problem

TL;DR

This paper explores the convergence of Boolean homomorphisms into measure algebras, linking topological properties of ultrafilters and measure sequences to the Efimov problem and the existence of Efimov spaces in set theory.

Contribution

It introduces new topologies on homomorphism spaces into measure algebras and studies their convergence, connecting these to Efimov spaces and measure-theoretic properties.

Findings

Established connections between homomorphism convergence and Efimov spaces.

Analyzed topologies on homomorphism spaces with measure algebra targets.

Linked ultrafilter convergence to measure sequence properties.

Abstract

For given Boolean algebras $\mathbb{A}$ and $\mathbb{B}$ we endow the space $\mathcal{H}(\mathbb{A},\mathbb{B})$ of all Boolean homomorphisms from $\mathbb{A}$ to $\mathbb{B}$ with various topologies and study convergence properties of sequences in $\mathcal{H}(\mathbb{A},\mathbb{B})$. We are in particular interested in the situation when $\mathbb{B}$ is a measure algebra as in this case we obtain a natural tool for studying topological convergence properties of sequences of ultrafilters on $\mathbb{A}$ in random extensions of the set-theoretical universe. This appears to have strong connections with Dow and Fremlin's result stating that there are Efimov spaces in the random model. We also investigate relations between topologies on $\mathcal{H}(\mathbb{A},\mathbb{B})$ for a Boolean algebra $\mathbb{B}$ carrying a strictly positive measure and convergence properties of sequences of measures on $\mathbb{A}$.