- #1
jimholt
- 12
- 0
If we have a partition [tex]\mathcal{P}=\{A_1,A_2\}[/tex] of some set [tex]A[/tex], then we can talk about the sigma-algebra generated by this partition as [tex]\Sigma=\{\emptyset, A_1,A_2,A\}[/tex].
How can I define this concept more generally?
Here is what I have:
A partition [tex]\mathcal{P}[/tex] of some set [tex]A[/tex] generates the sigma-algebra [tex]\Sigma$[/tex] if
i) [tex]\mathcal{P} \subset \Sigma$ [/tex], and
ii) for every set [tex]S \in \Sigma[/tex] and every [tex]\omega \in S[/tex], [tex]\mathcal{P}(\omega) \subseteq S[/tex], where [tex]\mathcal{P}(\omega)[/tex] is the cell of [tex]\mathcal{P}[/tex] containing [tex]\omega[/tex].
Is this complete? I am wondering if it breaks down when [tex]\omega = \emptyset[/tex] (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.
How can I define this concept more generally?
Here is what I have:
A partition [tex]\mathcal{P}[/tex] of some set [tex]A[/tex] generates the sigma-algebra [tex]\Sigma$[/tex] if
i) [tex]\mathcal{P} \subset \Sigma$ [/tex], and
ii) for every set [tex]S \in \Sigma[/tex] and every [tex]\omega \in S[/tex], [tex]\mathcal{P}(\omega) \subseteq S[/tex], where [tex]\mathcal{P}(\omega)[/tex] is the cell of [tex]\mathcal{P}[/tex] containing [tex]\omega[/tex].
Is this complete? I am wondering if it breaks down when [tex]\omega = \emptyset[/tex] (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.
Last edited: