Abstract
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.
| Original language | English |
|---|---|
| Pages (from-to) | 149-176 |
| Number of pages | 28 |
| Journal | Fundamenta Mathematicae |
| Volume | 239 |
| Issue number | 2 |
| DOIs | |
| State | Published - 2017 |
| Externally published | Yes |
Bibliographical note
Publisher Copyright:© Instytut Matematyczny PAN, 2017.
Funding
This research was partially done whilst the authors were visiting fellows at the Isaac Newton Institute for Mathematical Sciences in the programme "Mathematical, Foundational and Computational Aspects of the Higher Infinite" (HIF) funded by EPSRC grant EP/K032208/1. We would like to thank Tomek Bartoszynski, Andreas Blass, Mirna Džamonja, Barnabás Farkas, Osvaldo Guzmán and Grzegorz Plebanek for their helpful remarks concerning the subject of this paper. The first author was partially supported by Polish National Science Center grant 2013/11/B/ST1/03596 (2014-2017).
| Funders | Funder number |
|---|---|
| Engineering and Physical Sciences Research Council | EP/K032208/1 |
| Narodowe Centrum Nauki | 2013/11/B/ST1/03596 |