On Boolean algebras with strictly positive measures

TL;DR

This paper explores the properties of Boolean algebras with strictly positive measures, demonstrating that certain large algebras lack such measures despite all their smaller subalgebras possessing them.

Contribution

It constructs a Boolean algebra in the constructible universe that is not in SPM but has all large subalgebras with strictly positive measures, extending understanding of measure reflection.

Findings

Existence of a Boolean algebra outside SPM with all large subalgebras in SPM

Demonstrates reflection properties of measures in Boolean algebras

Builds on prior work by Farah and Velickovic

Abstract

We investigate reflection-type problems on the class SPM, of Boolean algebras carrying strictly positive finitely additive measures. We show, in particular, that in the constructible universe there is a Boolean algebra $\mathfrak A$ which is not in SPM but every subalgebra of $\mathfrak A$ of cardinality $\mathfrak c$ admits a strictly positive measure. This result is essentially due to Farah and Velickovic.