Generated Sigma-Algebra by Generated Monotone Class/Corollary

Corollary to Generated Sigma-Algebra by Generated Monotone Class
Let $X$ be a set.

Let $\GG \subseteq \powerset X$ be a non-empty set of subsets of $X$.

Define $\relcomp X \GG$ by:


 * $\relcomp X \GG := \set {\relcomp X A: A \in \GG}$

Then:


 * $\map \sigma \GG = \map {\mathfrak m} {\GG \cup \relcomp X \GG}$

where:
 * $\sigma$ denotes generated $\sigma$-algebra
 * $\mathfrak m$ denotes generated monotone class.

Proof
From Set is Subset of Union:
 * $\GG \subseteq \GG \cup \relcomp X \GG$

Further, as $\map \sigma \GG$ is a $\sigma$-algebra:


 * $S \in \map \sigma \GG \implies \relcomp X s = X \setminus S \in \map \sigma \GG$

from Set Difference as Intersection with Relative Complement.

Since $\GG \subseteq \map \sigma \GG$:


 * $\GG \cup \relcomp X \GG \subseteq \map \sigma \GG$

By Condition on Equality of Generated Sigma-Algebras:


 * $\map \sigma \GG = \map \sigma {\GG \cup \relcomp X \GG}$

Applying Generated Sigma-Algebra by Generated Monotone Class:


 * $\map \sigma \GG = \map {\mathfrak m} {\GG \cup \relcomp X \GG}$