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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?

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

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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.

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

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