On the Families of Sets Without the Baire Property Generated by the Vitali Sets

Let A be the family of all meager sets of the real line R, V be the family of all Vitali sets of R, V-1 be the family of all finite unions of elements of V and V-2 = {(C \ A(1)) boolean OR A(2) : C is an element of V-1; A(1), A(2) is an element of A}. We show that the families V, V-1, V-2 are invariant under translations of R, and V-1, V-2 are abelian semigroups with the respect to the operation of union of sets. Moreover, V subset of V-1 subset of V-2 and V-2 consists of zero-dimensional sets without the Baire property. Then we extend the results above to the Euclidean spaces R-n, n > 2, and their products with the finite powers of the Sorgenfrey line.