Extended Real Sigma-Algebra Induces Borel Sigma-Algebra on Reals

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

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

Then:


 * $\overline{\mathcal B}_\R = \mathcal B \left({\R}\right)$

where $\overline{\mathcal B}_\R$ denotes a trace $\sigma$-algebra.

Proof
We have Euclidean Space Subspace of Extended Real Number Space.

The result follows from Borel Sigma-Algebra of Subset is Trace Sigma-Algebra.