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Addendum to “On Meager Additive and Null Additive Sets in the Cantor space 2ω and in R” (Bull. Polish Acad. Sci. Math. 57 (2009), 91–99)

Weiss T.

Bulletin of the Polish Academy of Sciences. Mathematics

2014

Vol. 62, no 1

1--9

We prove in ZFC that there is a set A⊆2ω and a surjective function H:A→(0,1) such that for every null additive set X⊆(0,1), H-1(X) is null additive in 2ω. This settles in the affirmative a question of T. Bartoszyński.

On meager additive and null additive sets in the Cantor space 2ω and in R

Weiss T.

Bulletin of the Polish Academy of Sciences. Mathematics

2009

Vol. 57, no 2

91-99

Let T be the standard Cantor-Lebesgue function that maps the Cantor space 2ω onto the unit interval (0,1). We prove within ZFC that for every X ⊆ 2ω, X is meager additive in 2ω 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 x R. In this note, we also study the relationship between null additive sets in 2ω and R.