# Sigma-Algebra Generated by Finite Partition is Finite Sub-Sigma-Algebra

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## Theorem

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\xi$ be a finite partition of $\Omega$.

Let $\map \sigma \xi$ the generated $\sigma$-algebra by $\xi$.

Then, $\map \sigma \xi$ is a finite sub-$\sigma$-algebra of $\Sigma$.

Furthermore:

- $\ds \map \sigma \xi = \set {\bigcup S: S \subseteq \xi}$

## Proof

Let:

- $\ds \Gamma := \set {\bigcup S: S \subseteq \xi}$

By Power Set of Finite Set is Finite, $\Gamma$ is finite.

Therefore, it suffices to show:

- $\map \sigma \xi = \Gamma$

As $\xi$ is finite, from (SA3) follows:

- $\map \sigma \xi \supseteq \Gamma$

To conclude the equality, by definition of $\map\sigma\xi$, we need to show that $\Gamma$ is a $\sigma$-algebra.

Although this article appears correct, it's inelegant. There has to be a better way of doing it.In particular: The arguments below are trivial but need more detailsYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Improve}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

(SA1) follows from:

- $\ds X = \bigcup \xi \in \Gamma$

(SA2) follows from:

- $\ds \forall S\subseteq\xi : X\setminus\bigcup S = \bigcup \paren {\xi\setminus S} \in \Gamma$

(SA3) follows from:

- $\ds \forall S_1, S_2,\ldots\subseteq\xi : \bigcup _{i=1}^\infty \paren {\bigcup S_i} = \bigcup \paren {\bigcup _{i=1}^\infty S_i} \in \Gamma$

$\blacksquare$