Sigma-Algebra Contains Generated Sigma-Algebra of Subset

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

$\sigma_\FF$ is a $\sigma$-algebra containing $\EE$.

$\map \sigma \EE$ is a subset of all $\sigma$-algebras containing $\FF$, 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 : $\S 1.2$