On Borel maps, calibrated σ-ideals and homogeneity

2017/06/15 by Pol, Roman, Zakrzewski, Piotr
#03E15 #28A78 #54F45 #54H05 #FOS: Mathematics #General Topology (math.GN) #Logic (math.LO)

paper · doi:10.48550/arxiv.1706.04773

Abstract

Let μ be a Borel measure on a compactum X. The main objects in this paper are σ-ideals I(dim), J₀(μ), J_f(μ) of Borel sets in X that can be covered by countably many compacta which are finite-dimensional, or of μ-measure null, or of finite μ-measure, respectively. Answering a question of J. Zapletal, we shall show that for the Hilbert cube, the σ-ideal I(dim) is not homogeneous in a strong way. We shall also show that in some natural instances of measures μ with non-homogeneous σ-ideals J₀(μ) or J_f(μ), the completions of the quotient Boolean algebras Borel(X)/J₀(μ) or Borel(X)/J_f(μ) may be homogeneous. We discuss the topic in a more general setting, involving calibrated σ-ideals.

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