Let $\mathscr{J}$ be any proper ideal of subsets of the real line $R$ which contains all finite subsets of $R$. We define an ideal $\mathscr{J}^\ast\mid\mathscr{B}$ as follows: $X \in \mathscr{J}^\ast\mid\mathscr{B}$ if there exists a Borel set $B \subset R \times R$ such that $X \subset B$ and for any $x \in R$ 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 $\omega_1$ 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.
@article{1183742155,
author = {Cichon, Jacek and Pawlikowski, Janusz},
title = {On Ideals of Subsets of the Plane and on Cohen Reals},
journal = {J. Symbolic Logic},
volume = {51},
number = {1},
year = {1986},
pages = { 560-569},
language = {en},
url = {http://dml.mathdoc.fr/item/1183742155}
}
Cichon, Jacek; Pawlikowski, Janusz. On Ideals of Subsets of the Plane and on Cohen Reals. J. Symbolic Logic, Tome 51 (1986) no. 1, pp. 560-569. http://gdmltest.u-ga.fr/item/1183742155/