Define the sigma-algebra generated by a partition

[jimholt]

If we have a partition $$\mathcal{P}=\{A_1,A_2\}$$ of some set $$A$$, then we can talk about the sigma-algebra generated by this partition as $$\Sigma=\{\emptyset, A_1,A_2,A\}$$.

How can I define this concept more generally?

Here is what I have:

A partition $$\mathcal{P}$$ of some set $$A$$ generates the sigma-algebra $$\Sigma$$ if

i) $$\mathcal{P} \subset \Sigma$$, and

ii) for every set $$S \in \Sigma$$ and every $$\omega \in S$$, $$\mathcal{P}(\omega) \subseteq S$$, where $$\mathcal{P}(\omega)$$ is the cell of $$\mathcal{P}$$ containing $$\omega$$.

Is this complete? I am wondering if it breaks down when $$\omega = \emptyset$$ (or whether this possibility is precluded by the definition). Or if anyone knows the "standard" definition, I would be glad to hear it.

Thanks for any help, folks.

 

[jimholt]

Really? No thoughts, suggestions, opinions?

 

[Raskolnikov]

Yes, that should work.

 

[jimholt]

Cool, thanks a bunch. Just wanted to have another set of eyes look at it.