Generated Sigma-Algebra by Generated Monotone Class/Corollary

Corollary to Generated Sigma-Algebra by Generated Monotone Class
Let $X$ be a set, and let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a nonempty collection of subsets of $X$.

Define $\complement_X \left({\mathcal G}\right)$ by:


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

Then:


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

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

Proof
From Subset of Union, $\mathcal G \subseteq \mathcal G \cup \complement_X \left({\mathcal G}\right)$.

Further, as $\sigma \left({\mathcal G}\right)$ is a $\sigma$-algebra, we know:


 * $S \in \sigma \left({\mathcal G}\right) \implies \complement_X \left({X}\right) = X \setminus S \in \sigma \left({\mathcal G}\right)$

from Set Difference Relative Complement.

Since $\mathcal G \subseteq \sigma \left({\mathcal G}\right)$, this means:


 * $\mathcal G \cup \complement_X \left({\mathcal G}\right) \subseteq \sigma \left({\mathcal G}\right)$

Hence, Condition on Equality of Generated Sigma-Algebras applies, and we find:


 * $\sigma \left({\mathcal G}\right) = \sigma \left({\mathcal G \cup \complement_X \left({\mathcal G}\right)}\right)$

Applying Generated Sigma-Algebra by Generated Monotone Class, it follows that:


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