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

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

Let $\map {\mathcal B} \R$ be the Borel $\sigma$-algebra on $\R$.

Then:


 * $\overline {\mathcal B}_\R = \map {\mathcal B} \R$

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

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

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