Define the sigma-algebra generated by a partition

  • Thread starter jimholt
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  • #1
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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.
 
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Answers and Replies

  • #2
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Really? No thoughts, suggestions, opinions?
 
  • #3
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Yes, that should work.
 
  • #4
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Cool, thanks a bunch. Just wanted to have another set of eyes look at it.
 

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