On Meager Additive and Null Additive Sets in the Cantor Space $2^{\omega}$ and in $\mathbb{R}$
Volume 57 / 2009
Bulletin Polish Acad. Sci. Math. 57 (2009), 91-99
MSC: 03E05, 03E15, 03E35.
DOI: 10.4064/ba57-2-1
Abstract
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 $\mathbb R$.
