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Homework Help: Define the sigma-algebra generated by a partition

  1. Aug 27, 2010 #1
    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.
     
    Last edited: Aug 27, 2010
  2. jcsd
  3. Aug 29, 2010 #2
    Really? No thoughts, suggestions, opinions?
     
  4. Aug 29, 2010 #3
    Yes, that should work.
     
  5. Aug 29, 2010 #4
    Cool, thanks a bunch. Just wanted to have another set of eyes look at it.
     
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