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Algebras and Sigma-Algebras

  1. Sep 10, 2009 #1
    I'm currently taking a college level probability course and I am stuck on a couple questions involving algebras and sigma-algebras.

    Let S be a fixed set.

    1. What is the difference between an algebra on S and a sigma-algebra on S?

    2. Why do we require an event space to be a sigma algebra instead of an algebra?

    3. Find a set S and an algebra A on S such that A is not a sigma-algebra on S.


    Also, I have a proof that I could use some hints on how to start and the general form in which I should go about it.

    Prove that every sigma-algebra on S is an algebra on S.

    Thanks.
     
  2. jcsd
  3. Sep 10, 2009 #2
    I suspect it has something to do with sigma-algebras being limited to only finite intersections.

    Continuous random variables cause problems of you assume countable additivity of probability for point events.

    --Elucidus
     
  4. Sep 11, 2009 #3
    A sigma algebra is an algebra closed wrt countable unions.
    Because we want probability/measure to be sigma-additive.
    The system of finite and co-finite subsets of any infinite set forms an algebra which is not a sigma algebra.
    What definitions of algebra and sigma-algebra are you using? Under most definitions, this would be trivial (perhaps requiring the use of de Morgan's laws).
     
  5. Sep 11, 2009 #4
    Why do we require an event space to be a sigma algebra instead of an algebra?

    Perhaps not entirely responsive... the next question would be "why do we want that?"

    So ... How about:

    In order to do probability calculations involving limits of sequences. Either sequences of events or more generally sequences of random variables.
     
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