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Counterexample where X is not in the Lebesgue linear space.

  1. Sep 30, 2012 #1
    I'm trying to find a counterexample where [itex] \lim_{n \to +\infty} P(|X|>n) = 0 [/itex] but [itex]X \notin L[/itex] where [itex]L[/itex] is the lebesgue linear space.

    [itex]∫|X|I(|X|>n)dp + ∫|X|I(|X|≤n)dp = ∫|X|dp [/itex] therefore

    [itex]∫nI(|X|>n)dp + ∫|X|I(|X|)dp ≤ ∫|X|dp[/itex]

    Suppose [itex]∫I(|X|>n)dp = 1/(n ln n) [/itex]
    Clearly the hypothesis is satisfied because [itex] \lim_{n \to +\infty} P(|X|>n) = \lim_{n \to +\infty} ∫I(|X|>n)dp = \lim_{n \to +\infty} 1/( ln n) = 0[/itex]
    But I'm not sure how to conclude [itex]∫|X| = ∞[/itex]
    Last edited: Sep 30, 2012
  2. jcsd
  3. Sep 30, 2012 #2

    Stephen Tashi

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    Science Advisor

    Make another try at stating your question. What is [itex] X [/itex] ? Does the use of [itex] P(|X| > n) = 0 [/itex] imply that [itex] X [/itex] is a random variable? What "lebesgue linear space" are you talking about? [itex] L_p [/itex] space? [itex] p = 2 [/itex]?
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