# What is a “finite $\sigma$-algebra”?

I have an exersice which is outlined as follows

Suppose $G_{i}$ where $i=0 \ldots n$ is a disjoint union of $\Omega$. Prove that the family of unions of these $G_{i}$ is a sigma algebra on $\Omega$. Also prove that any "finite sigma algebra" $\mathcal{F}$ on $\Omega$ is of this form.

My guess is that a finite sigma algebra is a sigma algebra with finite number of sets but I am not sure.

I cant find the definition anywhere, does anyone know where I can find it?

• $\{\emptyset, \Omega\}$ is an example of a finite sigma-algebra. – Gabriel Romon Sep 24 '17 at 11:16
• @GabrielRomon good example – user123124 Sep 24 '17 at 11:24
• @GabrielRomon Indeed. And $\Omega$ is allowed to be infinite here. – drhab Sep 24 '17 at 11:25
• @drhab right, that kinda bothers me, but ill go with your suggestion for now – user123124 Sep 24 '17 at 11:25

## 2 Answers

A $\sigma$-algebra on space $\Omega$ is a subset of $\wp(\Omega)$ with special properties.

That means that a finite $\sigma$-algebra is a finite subset of $\wp(\Omega)$ with these properties.

As you guessed, a finite $\sigma$-algebra is just a $\sigma$-algebra that is finite.

As mentioned in the comments, a $\sigma$-algebra is a set, so when we describe a $\sigma$-algebra as "finite" we are using the standard definition of a finite set.

• It is wierd that one cannot find it written down anywhere. When one has infinite operations one can always get supprised. – user123124 Sep 24 '17 at 11:21
• @user1 see here and realize that a $\sigma$-algebra is a set. In fact you allready did that. Your guess is fine. – drhab Sep 24 '17 at 11:22
• @drhab Thanks ill go with that for now. – user123124 Sep 24 '17 at 11:23
• @user1 a sigma-algebra is a set, first and foremost ! – Gabriel Romon Sep 24 '17 at 11:31
• @drhab u wanna write that in answer or let littleO add it to his? I think this comment is vital. – user123124 Sep 24 '17 at 11:35