Can someone give me a hint how to show the equality $\{ B \cup F \mid B\in \sigma(\{(-\infty,a) \mid a\in \mathbb{R}\}),\ F\subseteq\{-\infty,+\infty\} \}=\sigma(\{ [-\infty,a) \mid a \in \mathbb{R} \}) $ ?
The set $\sigma(\{(-\infty,a) \mid a\in \mathbb{R}\})$ represents the usual Borel sets on the real line (and $\{(-\infty,a)\mid a\in \mathbb{R}\}$ is its generator).
What puzzles me is the fact that for example $(-\infty,1)\cup \{+\infty \}$ is in the set on the LHS of the equality, but I don't know how to produce this set on the RHS of the equality (making me wonder, if the above equality is true.)