Algebraic structures in the family of non-Lebesgue measurable sets

Abstract

In the additive topological group $(\mathbb{R},+)$ of real numbers, we construct families of sets for which elements are not measurable in the Lebesgue sense. The constructed families have algebraic structures of being semigroups (i.e., closed under finite unions of sets), and invariant under the action of the group $\Phi (\mathbb{R})$ of all translations of $\mathbb{R}$ onto itself. Those semigroups are constructed by using Vitali selectors and Bernstein subsets on $\mathbb{R}$. In particular, we prove that the family $(\mathcal{S}(\mathcal{B})\vee \mathcal{S}(\mathcal{V}))\ast {\mathcal{N}}_0:=\{((U_1\cup U_2)\setminus N)\cup M:U_1\in \mathcal{S}(\mathcal{B}),U_2\in \mathcal{S}(\mathcal{V}),N,M\in {\mathcal{N}}_0\}$ is a semigroup of sets, invariant under the action of $\Phi (\mathbb{R})$ and consists of sets which are not measurable in the Lebesgue sense. Here, $\mathcal{S}(\mathcal{B})$ is the collection of all finite unions of some type of Bernstein subsets of $\mathbb{R}$, $\mathcal{S}(\mathcal{V})$ is the collection of all finite unions of Vitali selectors of $\mathbb{R}$, and ${\mathcal{N}}_0$ is the $\sigma$-ideal of all subsets of $\mathbb{R}$ having the Lebesgue measure zero.