On Boolean algebras related to σ-ideals generated by compact sets

Pol, Roman; Zakrzewski, Piotr

doi:10.1016/j.aim.2016.04.010

Cited by 3 publications

(8 citation statements)

References 20 publications

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“…The statement in (i) strengthens a result in [13] concerning the "1-1 or constant" property of Sabok and Zapletal [17], [16] (some deep refinements of this result, in another direction, are given in the book by Kanovei, Sabok and Zapletal [6, Section 6.1.1]), cf. Section 7.2.…”

Section: Introductionsupporting

confidence: 65%

“…This was proved in [13] under the additional assumption that the calibrated σ-ideal J 0 pµ h q has the 1-1 or constant property (cf. Section 7.2) but this is now granted by Theorem 1.1 (i).…”

Section: 5mentioning

confidence: 96%

“…The results of this paper provide more information on the topic investigated in our articles [12], [11], [13], which were strongly influenced by the work of Zapletal [19], [21], Farah and Zapletal [3] and Sabok and Zapletal [17].…”

Section: Introductionmentioning

confidence: 95%

“…Also, the σ-ideal J σ pH 1 q of Borel subsets of the Euclidean square r0, 1s 2 that can be covered by countably many compacta of σ-finite H 1 -measure is calibrated, cf. [13]. 7.2.…”

Section: Commentsmentioning

confidence: 99%

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

Pol

Zakrzewski

2018

Trans. Amer. Math. Soc.

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Let µ be a Borel measure on a compactum X. The main objects in this paper are σ-ideals Ipdimq, J 0 pµq, J f pµq 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 µmeasure, respectively. Answering a question of J. Zapletal, we shall show that for the Hilbert cube, the σ-ideal Ipdimq is not homogeneous in a strong way. We shall also show that in some natural instances of measures µ with non-homogeneous σ-ideals J 0 pµq or J f pµq, the completions of the quotient Boolean algebras BorelpXq{J 0 pµq or BorelpXq{J f pµq may be homogeneous.We discuss the topic in a more general setting, involving calibrated σ-ideals.2010 Mathematics Subject Classification. 03E15, 54H05, 28A78, 54F45.

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Section: Introductionsupporting

confidence: 65%

Section: 5mentioning

confidence: 96%

Section: Introductionmentioning

confidence: 95%

“…Also, the σ-ideal J σ pH 1 q of Borel subsets of the Euclidean square r0, 1s 2 that can be covered by countably many compacta of σ-finite H 1 -measure is calibrated, cf. [13]. 7.2.…”

Section: Commentsmentioning

confidence: 99%

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

Pol

Zakrzewski

2018

Trans. Amer. Math. Soc.

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“…must contain a Cohen real over V . This forcing is further studied by [34,36]. Recall that the notion of weak 1-genericity is an effective version of Cohen genericity.…”

Section: 12mentioning

confidence: 99%