Condition on Equality of Generated Sigma-Algebras
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Theorem
Let $X$ be a set.
Let $\GG$, $\HH$ be sets of subsets of $X$.
Suppose that:
- $\GG \subseteq \HH \subseteq \map \sigma \GG$
where $\sigma$ denotes generated $\sigma$-algebra.
Then:
- $\map \sigma \GG = \map \sigma \HH$
Proof
From Generated Sigma-Algebra Preserves Subset, it follows that:
- $\map \sigma \GG \subseteq \map \sigma \HH$
Since $\map \sigma \GG$ is a $\sigma$-algebra containing $\HH$:
- $\map \sigma \HH \subseteq \map \sigma \GG$
from the definition of generated $\sigma$-algebra.
Hence the result, from the definition of set equality.
$\blacksquare$