Every null - additive set is meager - additive †

§1. The basic definitions and the main theorem. 1. Definition. (1) We define addition on 2 as addition modulo 2 on each component, i.e., if x, y, z ∈ 2 and x+ y = z then for every n we have z(n) = x(n) + y(n) (mod 2). (2) For A,B ⊆ 2 and x ∈ 2 we set x + A = {x + y : y ∈ A}, and we define A + B similarly. (3) We denote the Lebesgue measure on 2 with μ. We say that X ⊆ 2 is null-additive if for every A ⊆ 2 which is null, i.e. μ(A) = 0, X +A is null too. (4) We say that X ⊆ 2 is meager-additive if for every A ⊆ 2 which is meager also X+A is meager.