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