Measures and slaloms

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{a}\mathrm{d}\mathrm{d}$($\mathcal{N}$) = $\mathrm{n}\mathrm{o}\mathrm{n}$($\mathcal{M}$)$ 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.