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Generated Sigma Algebra - Help

  1. Jul 5, 2012 #1
    Please help me in understanding Generated Sigma Algebra:
    Consider S = {(H,H), (H,T), (T,H), (T,T)},
    the repeated coin toss. The Sigma-Algebra generated by
    C = {{(HH), (TT)}} is
    σ(C) = { ∅ , S, {(HT), (TH)}, {(HH), (TT)} }

    How does "(H,T), (T,H)" jumped in the σ(C) ? What is the difference between σ-algebra & generated σ-algebra? Why "(HT), (TH)" in σ(C) have been seperated from "(HH), (TT)" with curly brackets?
     
    Last edited by a moderator: Jul 5, 2012
  2. jcsd
  3. Jul 5, 2012 #2

    mathman

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    Science Advisor
    Gold Member

    {(H,T),(T,H)} is the complement of {(H,H),(T,T)}.

    σ-algebra is the general term. Generated σ-algebra is a specific term, refering to the smallest σ-algebra containing the generator.
     
  4. Jul 5, 2012 #3
    This is poorly worded. A sigma algebra is a sigma algebra, regardless of where it comes. I believe your confusion is with the generating set. ##C##. ##C## is not a sigma algebra, but there is a unique minimal sigma algebra containing ##C##, you've denoted it ##\sigma (C)##. In this sense, the sigma algebra (that you have denoted by) ##\sigma (C)## is generated by ##C##.
     
  5. Jul 6, 2012 #4
    thanks
     
  6. Jul 6, 2012 #5
    what does it mean that it is "poorly worded"? I have just posted it from the text, & if you think my questions are stupid then I am trying to learn it & if I have some questions in my mind then shouldn't I clarify them? :(
     
  7. Jul 6, 2012 #6

    berkeman

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    Staff: Mentor

    He didn't say that your questions were stupid. To the contrary, he is trying to help you in your understanding.
     
  8. Jul 6, 2012 #7
    When you asked
    One entity that you named was a sigma algebra, i.e. ##\sigma (C)##. It's a sigma algebra because the set that ##\sigma (C)## represents satisfies the sigma algebra axioms. The second entity you named, ##C## is not a sigma algebra. It's a set with no special properties. However, you can generate a sigma algebra from it by finding the smallest possible sigma algebra that contains ##C##. You chose to name this sigma algebra ##\sigma (C)## because it makes it clear that ##C## is the generating set.

    You could also ask what sigma algebra is generated by ##\sigma (C) ##? You might as well call it ##\sigma ( \sigma (C))##. The answer is of course the same set, ##\sigma (C)## since it was a sigma algebra to begin with, and so, in equations, ##\sigma ( \sigma (C)) = \sigma (C) ##
     
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