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

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- by Taras Banakh, Robert Rałowski and Szymon Żeberski PDF
- Proc. Amer. Math. Soc.
**144**(2016), 837-851 Request permission

## 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 $(X,\mu )$.## References

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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
- MR Author ID: 249694
- 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
- MR Author ID: 362782
- 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
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 837-851 - MSC (2010): Primary 03E15, 28A05, 28D05, 54H05
- DOI: https://doi.org/10.1090/proc/12709
- MathSciNet review: 3430858