Semigroups of sets generated by non-Lebesgue measurable subsets of the real line

Abstract

Let (R,+) be the additive group of real numbers. The collection P(R) of all subsets of R can be decomposed into two disjoint subfamilies, namely, the family L(R) of all Lebesgue measurable subsets of R and the family Lc(R) of all non-Lebesgue measurable subsets of R. The algebraic structure, from the set-theoretical point of view, of the family L(R) is well known. On the other hand, the family Lc(R) does not have a well-defined structure from the set-theoretic point of view. In this thesis, we construct subfamilies of the collection Lc(R), having an algebraic structure of being semigroups of sets. These semigroups are constructed by using the two classical examples of sets that are not measurable in the Lebesgue sense: Vitali selectors of R and Bernstein subsets of R. In particular, we show 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, which is invariant under translations, 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 the collection of all finite unions of Vitali selectors of R; and N0 is the σ-ideal of all subsets of R having the Lebesgue measure.