1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook