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Classifying invariant $ \sigma$-ideals with analytic base on good Cantor measure spaces


Authors: Taras Banakh, Robert Rałowski and Szymon Żeberski
Journal: Proc. Amer. Math. Soc. 144 (2016), 837-851
MSC (2010): Primary 03E15, 28A05, 28D05, 54H05
DOI: https://doi.org/10.1090/proc/12709
Published electronically: October 7, 2015
MathSciNet review: 3430858
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Abstract: Let $ X$ be a zero-dimensional compact metrizable space endowed with a strictly positive continuous Borel $ \sigma $-additive measure $ \mu $ which is good in the sense that for any clopen subsets $ U,V\subset X$ with $ \mu (U)<\mu (V)$ there is a clopen set $ W\subset V$ with $ \mu (W)=\mu (U)$. We study $ \sigma $-ideals with Borel base on $ X$ which are invariant under the action of the group $ \mathcal {H}_\mu (X)$ of measure-preserving homeomorphisms of $ (X,\mu )$, and show that any such $ \sigma $-ideal $ \mathcal {I}$ is equal to one of seven $ \sigma $-ideals: $ \{\emptyset \}$, $ [X]^{\le \omega }$, $ \mathcal E$, $ \mathcal {M}\cap \mathcal N$, $ \mathcal {M}$, $ \mathcal N$, or $ [X]^{\le \mathfrak{c}}$. Here $ [X]^{\le \kappa }$ is the ideal consisting of subsets of cardinality $ \le \kappa $ in $ X$, $ \mathcal {M}$ is the ideal of meager subsets of $ X$, $ \mathcal N=\{A\subset X:\mu (A)=0\}$ is the ideal of null subsets of $ (X,\mu )$, and $ \mathcal E$ is the $ \sigma $-ideal generated by closed null subsets of $ (X,\mu )$.


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Additional Information

Taras Banakh
Affiliation: Department of Mathematics, Ivan Franko National University of Lviv, Ukraine — and — Institute of Mathematics, Jan Kochanowski University, Kielce, Poland
Email: t.o.banakh@gmail.com

Robert Rałowski
Affiliation: Faculty of Fundamental Problems of Technology, Institute of Mathematics and Computer Science, Wrocław University of Technology, Wrocław, Poland
Email: robert.ralowski@pwr.wroc.pl

Szymon Żeberski
Affiliation: Faculty of Fundamental Problems of Technology, Wrocław University of Technology, Wrocław, Poland
Email: szymon.zeberski@pwr.wroc.pl

DOI: https://doi.org/10.1090/proc/12709
Keywords: Good Cantor measure space, measure-preserving homeomorphism, invariant $\sigma$-ideal
Received by editor(s): September 13, 2014
Received by editor(s) in revised form: January 3, 2015
Published electronically: October 7, 2015
Additional Notes: This work was partially financed by NCN means granted by decision DEC-2011/01/B/ST1/ 01439.
Communicated by: Mirna Dzamonja
Article copyright: © Copyright 2015 American Mathematical Society