I want to prove that the complement of a non-measurable set is also not measurable.
Let $N \subset \mathbb{R}$ be non-measurable. Then $N \notin \sigma$-Algebra.
Sps. $\mathbb{R}$ \ $N$ is measurable, then $\mathbb{R}$ \ $N \in \sigma$-Algebra. But by definition of a $\sigma$-Algebra $A \in \sigma$-Algebra $\implies \mathbb{R}$ \ $A \in \sigma$-Algebra.
But this means that $\mathbb{R} \setminus(\mathbb{R} \setminus N)$ = $N \in \sigma-$Algebra. But this contradicts to our assumption that $N \notin \sigma - $Algebra.
Is this argumentation correct?