  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.
