Generated Sigma-Algebra by Generated Monotone Class

Theorem
Let $X$ be a set, and let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a nonempty collection of subsets of $X$.

Suppose that $\mathcal G$ satisfies the following condition:


 * $(1):\quad A \in \mathcal G \implies \complement_X \left({A}\right) \in \mathcal G$

that is, $\mathcal G$ is closed under complement in $X$.

Then $\mathfrak m \left({\mathcal G}\right) = \sigma \left({\mathcal G}\right)$.

Here, $\mathfrak m$ denotes generated monotone class, and $\sigma$ denotes generated $\sigma$-algebra.

Corollary
For any $\mathcal G$ (not necessarily satisfying $(1)$), define $\complement_X \left({\mathcal G}\right)$ by:


 * $\complement_X \left({\mathcal G}\right) := \left\{{A \subseteq X: \complement_X \left({A}\right) \in \mathcal G}\right\}$

Then $\sigma \left({\mathcal G}\right) = \mathfrak m \left({\mathcal G \cup \complement_X \left({\mathcal G}\right)}\right)$.