Sigma-Algebra Contains Generated Sigma-Algebra of Subset
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Theorem
Let $\sigma_\FF$ be a be a $\sigma$-algebra on a set $\FF$.
Let $\sigma_\FF$ contain a set of sets $\EE$.
Let $\map \sigma \EE$ be the $\sigma$-algebra generated by $\EE$.
Then $\map \sigma \EE \subseteq \sigma_\FF$
Proof
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$\sigma_\FF$ is a $\sigma$-algebra containing $\EE$.
$\map \sigma \EE$ is a subset of all $\sigma$-algebras containing $\EE$, by definition of a generated $\sigma$-algebra.
Therefore it contains $\map \sigma \EE$.
$\blacksquare$
Sources
- 1984: Gerald B. Folland: Real Analysis: Modern Techniques and their Applications: $\S 1.2$