- #1
jimholt
- 12
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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$, and
ii) for every set S \in \Sigma and every \omega \in S, \mathcal{P}(\omega) \subseteq S, where \mathcal{P}(\omega) is the cell of \mathcal{P} containing \omega.
Is this complete? I am wondering if it breaks down when \omega = \emptyset (or whether this possibility is precluded by the definition). Or if anyone knows the "standard" definition, I would be glad to hear it.
Thanks for any help, folks.
How can I define this concept more generally?
Here is what I have:
A partition \mathcal{P} of some set A generates the sigma-algebra \Sigma$ 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.
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