Two Constructions of Sierpiński and Some Cardinal Invariants of Ideals

Abstract

Let $\mathcal{N}$ denote the ideal of null sets in $\mathbb{R} .$ Using CH Sierpiński constructed a set $A\subset \mathbb{R}$ satisfying $A\notin \mathcal{N} , \mathbb{R} \setminus A \notin \mathcal{N} ,$ and $(A+h)\setminus A \in \mathcal{N}$ for every $h\in \mathbb{R} .$ He also constructed a set $H\subset \mathbb{R}$ such that each horizontal section of $H$ and the complement of each vertical section of $H$ belong to $\mathcal{N} .$ In this note we investigate the existence of sets with analogous properties when $\mathcal{N}$ is replaced by an arbitrary invariant ideal. We also discuss the relationship among several related statements, including some inequalities between the cardinal invariants of the ideal in question. In the case of null sets in $\mathbb{R}$ we show that the nonexistence of $A\subset \mathbb{R}$ with $A\notin \mathcal{N}, \ \mathbb{R} \setminus A \notin \mathcal{N} ,$ and $(A+h)\setminus A \in \mathcal{N}\ (h\in \mathbb{R})$ is equivalent to the difference property of the class $\mathcal {L}$ of Lebesgue measurable functions defined on $\mathbb{R} .$ As an application we obtain that the difference property of the class $\mathcal{L}$ is consistent with ZFC.