Measures and slaloms

Piotr Borodulin-Nadzieja¹ ; Tanmay Inamdar²

¹ Instytut Matematyczny Uniwersytet Wrocławski 50-384 Wrocław, Poland
² School of Mathematics University of East Anglia Norwich, NR4 7TJ, UK

Fundamenta Mathematicae, Tome 239 (2017) no. 2, pp. 149-176

Résumé

We examine measure-theoretic properties of spaces constructed using the technique of Todorčević (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 $\mathrm {add}(\mathcal {N}) = \mathrm {non}(\mathcal {M})$, 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 $ such that its remainder $L\setminus \omega $ is non-separable and 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 (1989) of Boolean algebras supporting measures in terms of their chain conditions in generic extensions by a measure algebra.

Détail

DOI : 10.4064/fm318-10-2016

Keywords: examine measure theoretic properties spaces constructed using technique todor evi existence strictly positive measures spaces depends combinatorial properties certain families slaloms corollary mathrm mathcal mathrm mathcal there non separable space which supports measure which cannot mapped continuously omega without additional axioms prove there non separable growth omega supporting measure there compactification omega its remainder setminus omega non separable natural copy complemented finally discuss examples spaces supporting measures satisfying quite strong chain conditions main tool characterization due kamburelis boolean algebras supporting measures terms their chain conditions generic extensions measure algebra