Condition on Equality of Generated Sigma-Algebras

Theorem
Let $X$ be a set, and let $\mathcal G$, $\mathcal H$ be collections of subsets of $X$.

Suppose that:


 * $\mathcal G \subseteq \mathcal H \subseteq \sigma \left({\mathcal G}\right)$

where $\sigma$ denotes generated $\sigma$-algebra.

Then:


 * $\sigma \left({\mathcal G}\right) = \sigma \left({\mathcal H}\right)$

Proof
From Generated Sigma-Algebra Preserves Subset, it follows that:


 * $\sigma \left({\mathcal G}\right) \subseteq \sigma \left({\mathcal H}\right)$

Since $\sigma \left({\mathcal G}\right)$ is a $\sigma$-algebra containing $\mathcal H$:


 * $\sigma \left({\mathcal H}\right) \subseteq \sigma \left({\mathcal G}\right)$

from the definition of generated $\sigma$-algebra.

Hence the result, from the definition of set equality.