On ideals of subsets of the plane and on Cohen reals

Cichoń, Jacek; Pawlikowski, Janusz

doi:10.2307/2274013

Cited by 26 publications

(20 citation statements)

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“…These objects have some interesting properties and applications; see e.g. [6], [7], [8], [9], [2], [3], [4]. We show that detailed studies of the structure of the Mendez σ-ideals yield new properties of plane Borel sets and Borel functions of two variables.…”

Section: Some σ-Ideals On the Planementioning

confidence: 93%

“…The mixed product σ-ideals M ⊗ N and N ⊗ M will be called the Mendez σ-ideals. We also study the σ-ideals N * := {∅} ⊗ N and M * := {∅} ⊗ M; they were considered for instance in [6].…”

Section: Some σ-Ideals On the Planementioning

confidence: 99%

“…We follow some ideas from [6,Lemma 2.3]. Let U ∈ Σ Σ Σ 0 α (X 3 ) be a universal set for Σ Σ Σ 0 α (X 2 ), cf.…”

Section: Uniform Versions Of the Nikodym Theoremmentioning

confidence: 99%

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Measure-category properties of Borel plane sets and Borel functions of two variables

Balcerzak

Głąb

2009

Acta Math Hung

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Let I ⊗ J stand for the Fubini-type product of σ-ideals I, J ⊂ P(R). We consider mixed measure-category σ-ideals M ⊗ N and N ⊗ M (called the Mendez σ-ideals), and study some features of their structure. We show that M ⊗ N and N ⊗ M are not invariant with respect to nonzero rotations. Using Fremlin's results, we describe nice Borel bases of M ⊗ N, N ⊗ M, {∅} ⊗ N and {∅} ⊗ M. The rest of the paper is devoted to uniform versions of the Nikodym theorem and the Lusin theorem for Borel functions of two variables. Some σ-ideals on the planeThe monograph of Oxtoby [15] presents several similarities and differences between two families of small subsets of the real line that form σ-ideals: the family N of Lebesgue null sets (i.e. sets of Lebesgue measure zero) and the family M of meager sets (i.e. sets of the first Baire category). Oxtoby continues these studies for the σ-algebras associated with N and M (they consist of Lebesgue measurable sets and of sets with the Baire property, respectively) and real-valued functions measurable with respect to these σ-algebras. The analogous investigations can be conducted for subsets of R 2 or R k with k > 2. Interesting properties arise when one examines small sections of small plane sets -this leads to the theorems of Fubini and of Kuratowski and Ulam.A new idea appears when one considers plane sets whose almost all sections in one sense are small. Families (in fact σ-ideals) of such plane sets fulfilling the "Fubini-type mixed condition" were investigated by Mendez [13], [14]. These objects have some interesting properties and applications; see e.g. [6], [7], [8], [9], [2], [3], [4]. We show that detailed studies of the structure of the Mendez σ-ideals yield new properties of plane Borel sets and Key words and phrases: Lebesgue measure, Baire category, Mendez σ-ideal, nice base of a σ-ideal, Nikodym's theorem, Lusin's theorem.

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Section: Some σ-Ideals On the Planementioning

confidence: 93%

Section: Some σ-Ideals On the Planementioning

confidence: 99%

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Measure-category properties of Borel plane sets and Borel functions of two variables

Balcerzak

Głąb

2009

Acta Math Hung

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“…[5], [4], [7]). For a family F ⊂ P(X) and an ideal I ⊂ P(X) define I|F = {A ⊂ X : A ⊂ B for some B ∈ I ∩ F} (cf.…”

Section: Marek B a L C E R Z A K ( Lódź)mentioning

confidence: 99%

On c-sets and products of ideals

Balcerzak¹

1991

Colloq. Math.

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“…It turned out to be not true. Cichoń and Pawlikowski proved in 6 that the additivity of this ideal, called there Mokobodzki ideal, equals ω 1 in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {ZFC}$\end{document}.…”

Section: Introductionmentioning

confidence: 99%