I am curious on different ways to prove that a real-valued function $f$ is Borel measurable. If we denote the Borel $\sigma$-algebra by $\mathcal{B}$, I'm aware that TFAE:
- $f^{-1}(-\infty, c)\in \mathcal{B}$ $ \forall c\in \mathbb{R}$
- $f^{-1}(c, \infty) \in \mathcal{B}$ $ \forall c \in \mathbb{R}$
- $f^{-1}(a,b) \in \mathcal{B}$ $ \forall a,b \in \mathbb{R}$
- $f$ is Borel measurable.
Are there any other ways, or does one just play games with the first three to get the fourth?