Let T be the standard Cantor-Lebesgue function that maps the Cantor space onto the unit interval ⟨0,1⟩. We prove within ZFC that for every , X is meager additive in iff T(X) is meager additive in ⟨0,1⟩. As a consequence, we deduce that the cartesian product of meager additive sets in ℝ remains meager additive in ℝ × ℝ. In this note, we also study the relationship between null additive sets in and ℝ.
We prove in ZFC that there is a set and a surjective function H: A → ⟨0,1⟩ such that for every null additive set X ⊆ ⟨0,1), is null additive in . This settles in the affirmative a question of T. Bartoszyński.
We prove among other theorems that it is consistent with that there exists a set which is not meager additive, yet it satisfies the following property: for each measure zero set , belongs to the intersection ideal .
We prove in ZFC that every additive set is additive, thus we solve Problem 20 from paper [Weiss T., A note on the intersection ideal , Comment. Math. Univ. Carolin. 54 (2013), no. 3, 437-445] in the negative.
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