On Meager Additive and Null Additive Sets in the Cantor Space $2^{\omega}$ and in $\mathbb{R}$

Tomasz Weiss¹

¹ Instytut Matematyki Akademia Podlaska 08-110 Siedlce, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 57 (2009) no. 2, pp. 91-99

Résumé

Let $T$ be the standard Cantor–Lebesgue function that maps the Cantor space $2^\omega$ onto the unit interval $\langle0,1\rangle$. We prove within ZFC that for every $X\subseteq 2^{\omega}$, $X$ is meager additive in $2^\omega$ if{f} $T(X)$ is meager additive in $\langle0,1\rangle$. As a consequence, we deduce that the cartesian product of meager additive sets in $\mathbb R$ remains meager additive in $\mathbb R\times \mathbb R$. In this note, we also study the relationship between null additive sets in $2^{\omega}$ and $\mathbb R$.

Détail

DOI : 10.4064/ba57-2-1

Keywords: standard cantor lebesgue function maps cantor space omega unit interval langle rangle prove within zfc every subseteq omega meager additive omega meager additive langle rangle consequence deduce cartesian product meager additive sets mathbb remains meager additive mathbb times mathbb note study relationship between null additive sets omega nbsp mathbb