Characterization of Extended Real Sigma-Algebra

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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:

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


Proof

Let $\overline B \in \overline{\mathcal B}$.

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

$\overline B \cap \R \in \mathcal B \left({\R}\right)$


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

$\overline \R \setminus \R = \left\{{+\infty, -\infty}\right\}$

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

Here, $\setminus$ signifies set difference.


By Set Difference Union Intersection:

$\overline B = \left({\overline B \setminus \R}\right) \cup \left({\overline B \cap \R}\right)$

Therefore, any $\overline B \in \overline{\mathcal B}$ is of the purported form $B \cup S$ with $B \in \mathcal B \left({\R}\right)$ and $S \in \mathcal S$.


It remains to show that any such set is in fact an element of $\overline{\mathcal B}$.

Since any $B \in \mathcal B \left({\R}\right)$ is naturally also in $\overline{\mathcal B}$, it suffices to show that:

$\mathcal S \subseteq \overline{\mathcal B}$

by applying Sigma-Algebra Closed under Union.


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

$\varnothing, \left\{{+\infty}\right\}, \left\{{-\infty}\right\}, \left\{{+\infty, -\infty}\right\}$

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.

$\blacksquare$


Sources