The author considers \({}^ \omega 2\) with Lebesgue measure and its usual topology. Addition of sequences is defined component-wise modulo 2. A subset \(X\subseteq {}^ \omega 2\) is null-additive if for every \(A\subseteq {}^ \omega 2\) which has Lebesgue measure 0, \(X+A\) has measure 0 too. The meager-additivity of \(X\) is defined similarly. The author proves: Every null-additive set is meager-additive. He goes on to characterize null-additivity and meager-additivity. Using the Continuum Hypothesis, he proves that there is an uncountable null- additive set. It is reported that Haim Judah has given a model of ZFC in which every null-additive set is countable, and in which there exist uncountable meager-additive sets.