I consider the following set:$$ \mathcal{B} (\mathbb{\bar{R}}):= \{ A \cup E: A \subset \mathbb{R}, A \in \mathcal{B }(\mathbb{R}), E \subset \{-\infty, \infty\} \} $$
$\mathcal{B }(\mathbb{R})$ is the borel-sigma-algebra. Now I want to verify, that $\mathcal{B} (\mathbb{\bar{R}})$ is a sigma-algebra, too.
Obviously $\mathbb{\bar{R}}, \emptyset \in \mathcal{B} (\mathbb{\bar{R}})$
Then: $ A \in \mathcal{B} (\mathbb{\bar{R}}) \Rightarrow A^c \in \mathcal{B} (\mathbb{\bar{R}}) $ Is this the right way? Did I miss something?