Theorem

equivalent definition of $\sigma(X)$ for a random variable $X$

If $X$ is a random variable and $\mathcal{C}$ is a class of subsets of $\mathbb{R}$ such that $\sigma(\mathcal{C}) = \mathcal{B}(\mathbb{R})$ then $\sigma(X) = \sigma([X \in B],\, B \in \mathcal{C})$.