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

Theorem

Let $\overline \BB$ be the extended real $\sigma$-algebra.

Let $\map \BB \R$ be the Borel $\sigma$-algebra on $\R$.

Then:

$\overline \BB_\R = \map \BB \R$

where $\overline \BB_\R$ 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.

$\blacksquare$

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $8.2$
• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 8$: Problem $2$