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

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.

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(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:

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

$\blacksquare$