Generated Sigma-Algebra Contains Generated Sigma-Algebra of Subset
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Theorem
Let $\map \sigma \FF$ be the $\sigma$-algebra generated by $\EE$.
Let $\map \sigma \FF$ contain a set of sets $\EE$.
Let $\map \sigma \EE$ be the $\sigma$-algebra generated by $\EE$.
Then $\map \sigma \EE \subseteq \map \sigma \FF$.
Proof
Apply Sigma-Algebra Contains Generated Sigma-Algebra of Subset to $\map \sigma \FF$.
$\blacksquare$
Sources
- 1984: Gerald B. Folland: Real Analysis: Modern Techniques and their Applications : $\S 1.2$