Measures and slaloms

Abstract

We examine measure-theoretic properties of spaces constructed using certain technique of Todorčević. 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 $[0,1]^{ω_1}$. Also, without any additional axioms we prove that there is a non-separable growth of $ω$ supporting a measure and that there is a compactification $L$ of $ω$ 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.