On Borel maps, calibrated $\sigma$-ideals and homogeneity

DOI10.1090/TRAN/7462zbMATH Open1522.03199arXiv1706.04773MaRDI QIDQ6287875

Roman Pol, Piotr Zakrzewski

Publication date: 15 June 2017

Abstract: Let $mu$ be a Borel measure on a compactum $X$. The main objects in this paper are $sigma$-ideals $I(dim)$, $J_0(mu)$, $J_f(mu)$ of Borel sets in $X$ that can be covered by countably many compacta which are finite-dimensional, or of $mu$-measure null, or of finite $mu$-measure, respectively. Answering a question of J. Zapletal, we shall show that for the Hilbert cube, the $sigma$-ideal $I(dim)$ is not homogeneous in a strong way. We shall also show that in some natural instances of measures $mu$ with non-homogeneous $sigma$-ideals $J_0(mu)$ or $J_f(mu)$, the completions of the quotient Boolean algebras $Borel(X)/J_0(mu)$ or $Borel(X)/J_f(mu)$ may be homogeneous. We discuss the topic in a more general setting, involving calibrated $sigma$-ideals.

Mathematics Subject Classification ID

Descriptive set theory (03E15) Dimension theory in general topology (54F45) Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) (54H05) Hausdorff and packing measures (28A78)

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