Measures and slaloms

TL;DR

This paper investigates measure-theoretic properties of certain topological spaces constructed via Todorčević's technique, linking measure existence to combinatorial properties of slaloms and exploring implications under various set-theoretic assumptions.

Contribution

It establishes new connections between measure support, combinatorial properties of slaloms, and set-theoretic axioms, including results on non-separable spaces and chain conditions.

Findings

Existence of measures depends on combinatorial properties of slaloms.

Under certain axioms, non-separable measure-supporting spaces exist.

Spaces with strong chain conditions may not support measures.

Abstract

We examine measure-theoretic properties of spaces constructed using certain technique of Todor\v{c}evi\'{c}. We show that the existence of strictly positive measures on such spaces depends on combinatorial properties of certain families of slaloms. As a corollary we get that if $\mathrm{add}(\mathcal{N}) = \mathrm{non}(\mathcal{M})$ then there is a non-separable space which supports a measure and which cannot be mapped continuously onto $[0,1]^{\omega_1}$. Also, without any additional axioms we prove that there is a non-separable growth of $\omega$ supporting a measure and that there is a compactification $L$ of $\omega$ with growth of such properties and such that the natural copy of $c_0$ is complemented in $C(L)$. Finally, we discuss examples of spaces not supporting measures but satisfying quite strong chain conditions. Our main tool is a characterization due to Kamburelis of Boolean algebras supporting measures in terms of their chain conditions in generic extensions by a measure algebra.