Definition (Lebesgue $\sigma$-algebra)

Let $m$ be the Lebesgue Measure. We define the Lebesgue σ-algebra as $$\mathscr{M}=\{ A\subseteq \mathbb{R}: m(E)=m(E\cap A)+m(E\cap A^{c}),\forall E\subseteq \mathbb{R}\}$$i.e. the set of