Characterization of Extended Real Sigma-Algebra

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

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

Define $\SS := \powerset {\set {-\infty, +\infty} }$, where $\PP$ denotes power set.

Then:


 * $\overline \BB = \set {B \cup S: B \in \map \BB \R, S \in \SS}$

Proof
Let $\overline B \in \overline \BB$.

Then by Extended Real Sigma-Algebra Induces Borel Sigma-Algebra on Reals, we have:


 * $\overline B \cap \R \in \map \BB \R$

We also have, by definition of the extended real numbers $\overline \R$, that:


 * $\overline \R \setminus \R = \set {-\infty, +\infty}$

and therefore, $\overline B \setminus \R \subseteq \set {-\infty, +\infty}$.

Here, $\setminus$ signifies set difference.

By Set Difference Union Intersection:


 * $\overline B = \paren {\overline B \setminus \R} \cup \paren {\overline B \cap \R}$

Therefore, any $\overline B \in \overline \BB$ is of the purported form $B \cup S$ with $B \in \map \BB \R$ and $S \in \SS$.

It remains to show that any such set is in fact an element of $\overline \BB$.

Since any $B \in \map \BB \R$ is naturally also in $\overline \BB$, it suffices to show that:


 * $\SS \subseteq \overline \BB$

by applying Sigma-Algebra Closed under Union.

From Closed Set Measurable in Borel Sigma-Algebra, it will now suffice to show that:


 * $\O, \set {-\infty}, \set {+\infty}, \set {-\infty, +\infty}$

are all closed sets in $\overline \R$.

That they are follows from Extended Real Number Space is Hausdorff and Finite Subspace of Hausdorff Space is Closed.