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Lebesgue integral proof, prove integral is zero

  1. Dec 15, 2011 #1
    1. The problem statement, all variables and given/known data

    Suppose S is a set with finite measure, f>=0 on S and f is Lebesgue measurable. For each n in N define E_n = {x \in S | f(x) > n} Prove that \lim_{n \rightarrow \infty} \int_{E_n} f = 0

    2. Relevant equations

    Definition of Lebesgue measurability for unbounded functions

    3. The attempt at a solution

    Honestly, I'm not entirely sure where to even begin with this one. Would I want to somehow incorporate the epsilon definition of limit? Or should I define f_n(x) = f(x) if f(x) <= n and n if f(x) > n, and show that limit will equal 0? Or none of these ideas? Just a little push in the right direction would be really helpful.

    Last edited: Dec 15, 2011
  2. jcsd
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