On ideals of subsets of the plane and on Cohen reals

Journal of Symbolic Logic 51 (3):560-569 (1986)
Let J be any proper ideal of subsets of the real line R which contains all finite subsets of R. We define an ideal J * ∣B as follows: X ∈ J * ∣B if there exists a Borel set $B \subset R \times R$ such that $X \subset B$ and for any x ∈ R we have $\{y \in R: \langle x,y\rangle \in B\} \in \mathscr{J}$ . We show that there exists a family $\mathscr{A} \subset \mathscr{J}^\ast\mid\mathscr{B}$ of power ω 1 such that $\bigcup\mathscr{A} \not\in \mathscr{J}^\ast\mid\mathscr{B}$ . In the last section we investigate properties of ideals of Lebesgue measure zero sets and meager sets in Cohen extensions of models of set theory.
Keywords Lebesgue measure   Baire category   cardinal indices   Cohen reals
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DOI 10.2307/2274013
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Jörg Brendle (2006). Van Douwen's Diagram for Dense Sets of Rationals. Annals of Pure and Applied Logic 143 (1):54-69.

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