# probability space

#### WeeG

##### TS Contributor
hi, I need some help with this one....

Let $$\Omega$$ be a finite set. Show that the set of all subsets of $$\Omega$$ (2^omega), is also finite and that it is a $$\sigma$$ algebra.

thanks !

#### Martingale

##### TS Contributor
hi, I need some help with this one....

Let $$\Omega$$ be a finite set. Show that the set of all subsets of $$\Omega$$ (2^omega), is also finite and that it is a $$\sigma$$ algebra.

thanks !
if $$|\Omega|=n<\infty$$ then $$|\mathcal{P}(\Omega)|=2^n$$ which is finite.

As for the $$\sigma$$-algebra part.

$$\mathcal{P}(\Omega)$$ is nonempty since it contains the empty set.

If we let $$X\in\mathcal{P}(\Omega)$$ then $$X^c=\Omega\backslash X\subset\Omega$$. and since $$\mathcal{P}(\Omega)$$ contains all subsets of $$\Omega$$ we have $$X^c\in\mathcal{P}(\Omega)$$

similarly for countable unions