I have a question regarding the relationship between measurable Borel sets and the generator of the Borel $\sigma$-algebra. I know that $\sigma$-algebras are stable under countable unions by definition and the set of intervals $\mathcal{E}=\{\, ]r,s]\subset \mathbb{R}: r\leq s\}$ generates $\mathcal{B}(\mathbb{R})$.
So every countable union of elements of $\mathcal{E}$ is measurable. But what about the converse? Is every set $M\in\mathcal{B}(\mathbb{R})$ representable by a countable union of elements of $\mathcal{E}$?