algebraic structures in the family of non-lebesgue measurable sets

In the additive topological group (r, +) of real numbers,we construct families of sets for which elements are not measurable in the lebesgue sense. the constructed families have algebraicstructures of being semigroups (i.e., closed under finite unionsof sets), and invariant under the action of the group φ(r) of alltranslations of r onto itself. those semigroups are constructed by using vitali selectors and bernstein subsets on r. in particular, we prove that the family (s(b) ∨ s(v)) ∗n0 := {((u1 ∪ u2) n) ∪m :u1 ∈ s(b),u2 ∈ s(v),n,m ∈ n0} is a semigroup of sets, invariantunder the action of φ(r) and consists of sets which are not measurable in the lebesgue sense. here, s(b) is the collection of all finite unions of some type of bernstein subsets of r, s(v) is thecollection of all finite unions of vitali selectors of r, and n0 is theσ-ideal of all subsets of r having the lebesgue measure zero.