Characterization of Extended Real Sigma-Algebra

Theorem
Let $\mathcal B \left({\R}\right)$ be the Borel $\sigma$-algebra on $\R$.

Let $\overline{\mathcal B}$ be the extended real $\sigma$-algebra.

Define $\mathcal S := \mathcal P \left({\left\{{+\infty, -\infty}\right\}}\right)$, where $\mathcal P$ denotes power set.

Then:


 * $\overline{\mathcal B} = \left\{{B \cup S: B \in \mathcal B \left({\R}\right), S \in \mathcal S}\right\}$