Algebraic structures in the family of non-Lebesgue measurable sets

TL;DR

This paper constructs algebraically structured families of non-Lebesgue measurable sets in the real numbers, using Vitali selectors and Bernstein subsets, which are invariant under translations and form semigroups.

Contribution

It introduces new semigroup families of non-measurable sets in alculus, combining Bernstein and Vitali sets with measure-zero sets, invariant under all translations.

Findings

Constructed semigroup of non-measurable sets invariant under translations.

Families include unions of Bernstein and Vitali sets with measure-zero sets.

Proved invariance and non-measurability properties of the constructed families.

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}))*\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.