On nonmeasurable images

Robert Rałowski; Szymon Żeberski

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 $𝔞 (ℐ)$ ( $\bar{𝔞} (ℐ)$ ) 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 $𝔞 (𝒵_{\vec{μ}}) = ω$ for any tall density ideal $𝒵_{\vec{μ}}$ including the density zero ideal $𝒵$ . On the other hand, you have $𝔟 \leq \bar{𝔞} (ℐ)$ for any analytic $P$ -ideal $ℐ$ , and $\bar{𝔞} (𝒵_{\vec{μ}}) \leq 𝔞$ for each density ideal $𝒵_{\vec{μ}}$ . For each ideal $ℐ$ on $ω$ denote $𝔟_{ℐ}$ and $𝔡_{ℐ}$ the unbounding and dominating numbers of $〈 ω^{ω}, \leq_{ℐ} 〉$ where $f \leq_{ℐ} g$ iff ${n \in ω : f (n) > g (n)} \in ℐ$ . 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 $E_{x} = y \in Y : (x, y) \in E$ , 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 $U \in ℕ^{*}$ such that for every one-to-one function $f : ℕ \to ℕ$ there exists $U \in U$ 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.

Displaying similar documents to “On nonmeasurable images”

Completely nonmeasurable unions

More on cardinal invariants of analytic P -ideals

Borel sets with σ-compact sections for nonseparable spaces

Ideal limits of sequences of continuous functions

More than a 0-point

More on cardinal invariants of analytic $P$ -ideals