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Math Help - probability space and non-random variables

  1. #1
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    probability space and non-random variables

    Hello! on my book there is this exercise, but I don't understand what it is asking and clearly I don't know how to solve it.

    It says: Give an example of a probability space and a real-valued function on Ω= {0,1,2} that is not a random variable.


    Can you please help me understand it? Thank you so much!
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  2. #2
    Member HappyJoe's Avatar
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    Just to get a feel on what level this is, what is your definition of a random variable?
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  3. #3
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    well I'm just trying to come up with an algebra A and a function X such that there exists a B Borel for which X^-1 does not belong to A.
    But I can't make up one
    Any ides?
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  4. #4
    Member HappyJoe's Avatar
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    How about equipping your set \Omega=\{0,1,2\} with the trivial sigma-algebra, containing only all of \Omega and the empty set.

    Then whenever you have a map from \Omega, it won't be a random variable if the pre-image of a measurable set is not the entire \Omega or the empty set.
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  5. #5
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    you are basically saying that I should be taking the case where A is made of omega and the empty set; so that basically A is not a measurable sigma-algebra and so I cannot get X r.v.
    did I get it correctly?
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  6. #6
    Member HappyJoe's Avatar
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    I am not sure what you mean by a "measurable sigma-algebra". But you are right in that I meant A to consist of both Omega and the empty set. Then most maps X into the real numbers (or other measure spaces) will not be measurable.
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  7. #7
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    Well what about if I build the A={emptyset, 1, 2, {1U2}, Omega}
    what would be X such that the counter image is not mapping into A ?
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  8. #8
    Member HappyJoe's Avatar
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    You could define X : Omega -> R by X(1) = 1, X(2) = 2, X(3) = 3.

    Then take an open interval I in R that contains 3, but not 1 or 2. Then the preimage of I under X is {3}, which is not in the sigma-algebra.
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