Displaying similar documents to “On nonmeasurable images”

Completely nonmeasurable unions

Robert Rałowski, Szymon Żeberski (2010)

Open Mathematics

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Assume that no cardinal κ < 2ω is quasi-measurable (κ is quasi-measurable if there exists a κ-additive ideal of subsets of κ such that the Boolean algebra P(κ)/ satisfies c.c.c.). We show that for a metrizable separable space X and a proper c.c.c. σ-ideal II of subsets of X that has a Borel base, each point-finite cover ⊆ 𝕀 of X contains uncountably many pairwise disjoint subfamilies , with 𝕀 -Bernstein unions ∪ (a subset A ⊆ X is 𝕀 -Bernstein if A and X A meet each Borel 𝕀 -positive...

More on cardinal invariants of analytic P -ideals

Barnabás Farkas, Lajos Soukup (2009)

Commentationes Mathematicae Universitatis Carolinae

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Given an ideal ℐ on ω let 𝔞 ( ℐ ) ( 𝔞 ¯ ( ℐ ) ) be minimum of the cardinalities of infinite (uncountable) maximal ℐ -almost disjoint subsets of [ ω ] ω . We show that 𝔞 ( ℐ h ) > ω if ℐ h is a summable ideal; but 𝔞 ( 𝒵 μ → ) = ω for any tall density ideal 𝒵 μ → including the density zero ideal 𝒵 . On the other hand, you have 𝔟 ≤ 𝔞 ¯ ( ℐ ) for any analytic P -ideal ℐ , and 𝔞 ¯ ( 𝒵 μ → ) ≤ 𝔞 for each density ideal 𝒵 μ → . For each ideal ℐ on ω denote 𝔟 ℐ and 𝔡 ℐ the unbounding and dominating numbers of 〈 ω ω , ≤ ℐ 〉 where f ≤ ℐ g iff { n ∈ ω : f ( n ) > g ( n ) } ∈ ℐ . We show that 𝔟 ℐ = 𝔟 and 𝔡 ℐ = 𝔡 for each analytic P -ideal ℐ . Given a Borel...

Borel sets with σ-compact sections for nonseparable spaces

Petr Holický (2008)

Fundamenta Mathematicae

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We prove that every (extended) Borel subset E of X × Y, where X is complete metric and Y is Polish, can be covered by countably many extended Borel sets with compact sections if the sections E x = y ∈ Y : ( x , y ) ∈ E , x ∈ X, are σ-compact. This is a nonseparable version of a theorem of Saint Raymond. As a by-product, we get a proof of Saint Raymond’s result which does not use transfinite induction.

Ideal limits of sequences of continuous functions

Miklós Laczkovich, Ireneusz Recław (2009)

Fundamenta Mathematicae

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We prove that for every Borel ideal, the ideal limits of sequences of continuous functions on a Polish space are of Baire class one if and only if the ideal does not contain a copy of Fin × Fin. In particular, this is true for F σ δ ideals. In the proof we use Borel determinacy for a game introduced by C. Laflamme.

More than a 0-point

Jana Flašková (2006)

Commentationes Mathematicae Universitatis Carolinae

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We construct in ZFC an ultrafilter U ∈ ℕ * such that for every one-to-one function f : ℕ → ℕ there exists U ∈ U with f [ U ] in the summable ideal, i.e. the sum of reciprocals of its elements converges. This strengthens Gryzlov’s result concerning the existence of 0 -points.

A characterization of the meager ideal

Piotr Zakrzewski (2015)

Commentationes Mathematicae Universitatis Carolinae

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We give a classical proof of the theorem stating that the σ -ideal of meager sets is the unique σ -ideal on a Polish group, generated by closed sets which is invariant under translations and ergodic.

Pcf theory and cardinal invariants of the reals

Lajos Soukup (2011)

Commentationes Mathematicae Universitatis Carolinae

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The additivity spectrum ADD ( ℐ ) of an ideal ℐ ⊂ 𝒫 ( I ) is the set of all regular cardinals κ such that there is an increasing chain { A α : α < κ } ⊂ ℐ with ⋃ α < κ A α ∉ ℐ . We investigate which set A of regular cardinals can be the additivity spectrum of certain ideals. Assume that ℐ = ℬ or ℐ = 𝒩 , where ℬ denotes the σ -ideal generated by the compact subsets of the Baire space ω ω , and 𝒩 is the ideal of the null sets. We show that if A is a non-empty progressive set of uncountable regular cardinals and pcf ( A ) = A , then ADD ( ℐ ) = A in some c.c.c generic extension...