# Existence and Uniqueness of Sigma-Algebra Generated by Collection of Subsets

## Theorem

Let $X$ be a set.

Let $\GG \subseteq \powerset X$ be a collection of subsets of $X$.

Then $\map \sigma \GG$, the $\sigma$-algebra generated by $\GG$, exists and is unique.

## Proof

### Existence

By Power Set is Sigma-Algebra, there is at least one $\sigma$-algebra containing $\GG$.

Next, let $\Bbb E$ be the collection of $\sigma$-algebras containing $\GG$:

- $\Bbb E := \set {\Sigma': \GG \subseteq \Sigma', \text{$\Sigma'$ is a $\sigma$-algebra} }$

By Intersection of Sigma-Algebras, $\Sigma := \bigcap \Bbb E$ is a $\sigma$-algebra.

Also, by Set Intersection Preserves Subsets:

- $\GG \subseteq \Sigma$

Now let $\Sigma'$ be a $\sigma$-algebra containing $\GG$.

By construction of $\Sigma$, and Intersection is Subset: General Result:

- $\Sigma \subseteq \Sigma'$

$\Box$

### Uniqueness

Suppose both $\Sigma_1$ and $\Sigma_2$ are $\sigma$-algebras generated by $\GG$.

Then property $(2)$ for these $\sigma$-algebras implies both $\Sigma_1 \subseteq \Sigma_2$ and $\Sigma_2 \subseteq \Sigma_1$.

By definition of set equality:

- $\Sigma_1 = \Sigma_2$

$\blacksquare$

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## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $3.4 \ \text{(ii)}$ - 2013: Donald L. Cohn:
*Measure Theory*(2nd ed.) ... (previous) ... (next): $1.1$: Algebras and Sigma-Algebras