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Set thory: sigma-fields

  1. Aug 22, 2012 #1
    I have started to solve exercises given on a previous exam, but typically I do not have the answers.

    1. The problem statement, all variables and given/known data
    The question is: Which (if any) of these collections are potentially σ-fields over some sample space such that probability functions could be defined over them? Explain briefly


    2. Relevant equations
    a) A={∅, {A,B,C}, {A}, {B}, {C}}
    b) B={∅, {A,B,C}, {A}, {B,C}}
    c) C={∅, {1,2,3}, {4,5}}

    3. The attempt at a solution
    I started by setting up the conditions for a these sets to be σ-fields. Let B be a collection of subsets of ℂ, then B is a σ-field is:
    1) The empty set is part of the subset
    2) If x [itex]\in[/itex] B, then [itex]x^{C}[/itex][itex]\in[/itex] B
    3) If the sequence of sets {X_1, X_2, X_3,...} is in B, then [itex]\bigcup[/itex]X_i is a part of ℂ

    The first condition is satisfied for all the collections.

    a) If I have understood the theory of complements right we have that:
    [itex]∅^{C}[/itex]={A,B,C} - OK
    [itex]{(A,B,C)}^{C}[/itex]=∅ - OK
    [itex]{A}^{C}[/itex]={B,C} - not in the collection -> not a σ-field

    b) In the same manner, I find that all the complements exist in this collection. For the third part i have writte [itex]\bigcup{(A,B,C)}\cup{(A)}\cup{(B,C)}=∅[/itex]. Will that be correct?
    Anyway, I conclude that this is a potentially σ-field.

    c) Here I find that the complement of the empty set does not exist, and thus this is not a σ
    -field
     
    Last edited: Aug 22, 2012
  2. jcsd
  3. Aug 22, 2012 #2

    micromass

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    For (b), are you sure that every complement is again in the set??
     
  4. Aug 22, 2012 #3
    Well, I believe so.
    The complement of the empty set is {A,B,C}.
    The complement of {A,B,C} is the empty set.
    The complement of {A} is {B,C}
    The complement of {B,C} is {A},

    Am I missing something?
     
  5. Aug 22, 2012 #4

    micromass

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    I don't see {B,C} in the set...
     
  6. Aug 22, 2012 #5
    Sorry, my mistake, it should be {B,C}, not {A,B}.
     
  7. Aug 22, 2012 #6

    micromass

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    OK, then you're right that (b) is sigma-field. However, your explanation

    does not seem right. It's a bit weird that the union of all these things could be empty.
     
  8. Aug 22, 2012 #7
    Your right, I have been confusing it with the intersections. So, the union of all these things will be {A,B,C}, right? and that should of course be a part of the larger set.
     
  9. Aug 22, 2012 #8

    micromass

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    Yes, but that's only the union of all the sets. That is, you then know that

    [tex]\emptyset\cup \{A,B,C\}\cup \{A\}\cup \{B,C\}[/tex]

    is in the sigma-field.

    But you also need to check the other unions, like

    [tex]\{A\}\cup\{B,C\}[/tex]

    and

    [tex]\{A\}\cup\{A,B,C\}[/tex]

    and so on.

    Of course, these will lie trivially in the sigma-field again, but you need to check it.
     
  10. Aug 22, 2012 #9
    Thankyou very much. This has been very helpful!
     
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