Proposition 13.1.10. The following families of sets generate the Borel $\sigma$-algebra on $\ol \real$:
$\bracs{[-\infty, a]| a \in \real}$.
$\bracs{(a, \infty]|a \in \real}$.
$\bracs{[a, \infty]| a \in \real}$.
$\bracs{[-\infty, a)| a \in \real}$.
Proof. (1) $\to$ (2): For any $a \in \real$, $(a, \infty] = [-\infty, a]^{c}$.
(2) $\to$ (3): For any $a \in \real$, $[a, \infty] = \bigcap_{n \in \natp}(a - 1/n, \infty]$.
(3) $\to$ (4): For any $a \in \real$, $[-\infty, a) = [a, \infty]^{c}$.
(4) $\to$ (1): For any $a \in \real$, $[-\infty, a] = \bigcap_{n \in \natp}[-\infty, a + 1/n)$.
By definition, all elements of (1), (2), (3), and (4) belong to $\cb_{\ol{\real}}$. Let $\cm$ be the $\sigma$-algebra generated by (1), (2), (3), and (4), then
are elements of $\cm$. For any $a \in \real$, $(a, \infty) = (a, \infty] \setminus \bracs{\infty}$, so $\cm \supset \cb_{\real}$ by Proposition 13.1.9.
In addition, for any $E \in \cb_{\ol{\real}}$, $E = (E \cap \real) \cup (E \setminus \real)$, where $E \cap \real \in \cb_{\real}$. Since $\bracs{\infty}, \bracs{-\infty}\in \cm$ and $\cb_{\real} \subset \cm$, $E \in \cm$ and $\cm = \cb_{\ol \real}$.$\square$