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On Borel maps, calibrated \(\sigma\)-ideals, and homogeneity. (English) Zbl 1522.03199

Summary: 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 [Topology Appl. 167, 31–35 (2014; Zbl 1349.03057)], 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 nonhomogeneous \( \sigma\)-ideals \( J_0(\mu )\) or \( J_f(\mu )\), the completions of the quotient Boolean algebras \(\mathrm{Borel}(X)/J_0(\mu )\) or \(\mathrm{Borel}(X)/J_f(\mu )\) may be homogeneous.
We discuss the topic in a more general setting, involving calibrated \( \sigma\)-ideals.

MSC:

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

Citations:

Zbl 1349.03057
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References:

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