We examine measure-theoretic properties of spaces constructed using the technique of Todorcevic (2000). We show that the existence of strictly positive measures on such spaces depends on combinatorial properties of certain families of slaloms. As a corollary, if add(N) = non(M), then there is a non-separable space which supports a measure and which cannot be mapped continuously onto [0,1]ω1. 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 such that its remainder L \ ω is non-separable and the natural copy of c0 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 (1989) of Boolean algebras supporting measures in terms of their chain conditions in generic extensions by a measure algebra.
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© Instytut Matematyczny PAN, 2017.