Generated Sigma-Algebra by Generated Monotone Class/Corollary
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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:
- $\map \sigma \GG = \map {\mathfrak m} {\GG \cup \relcomp X \GG}$
$\blacksquare$