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Homework Help: Probability - Infinite Union of Subsets of a Sample Space

  1. Apr 10, 2010 #1
    1. The problem statement, all variables and given/known data

    This is a question about mathematical probability, using the sigma-algebra, measure and probability space approach.

    Define A(t) = {all outcomes, w, in the sample space such that Y(w) < or = t}
    where Y is a random variable and t is any real number.

    Fix a real number X.

    Consider an increasing sequence {Xn} such that Xn tend to X from below/left as n tend to infinity. Therefore, Xn < or = X for all n.

    So, A(Xn) is a subset of A(X) for all n.

    2. Relevant equations

    N.A.

    3. The attempt at a solution

    This isn't a homework question but I thought it fits in this forum. It is stated in my text that the infinite union of A(Xn) is not A(X).

    I think this is because Xn can tend to X in such a way that X is not equal to X all for n. So their infinite union will always not contain some element in X.

    But how do I show this mathematically? A little nudge in the right direction so I can solve this myself? Thanks in advance!
     
  2. jcsd
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