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Math Help - showing convergence in lebesgue integral

  1. #1
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    Question showing convergence in lebesgue integral

    hi all
    i need help in doing this problem,

    Let f_{n}, f be integrable functions and f_{n} \to f a.e.
    show that \int |f_{n} - f| \to 0 if and only if \int |f_{n}| \to \int |f|

    i don't know where should i start.
    any hint would be greatly appreciated
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  2. #2
    Senior Member Shanks's Avatar
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    this is a problem from the stanford analysis textbook.
    (1) \bigl|\left|f_n\right|-\left|f\right|\bigr|\leq \left|f_n-f\right|
    gives the only if part.
    (2)as for the if part, Hint: A={x:f(x)>0}, B={x:f(x)<0}, C={x:f(x)=0}.
    Last edited by Opalg; December 21st 2009 at 01:40 AM. Reason: Fixed LaTeX
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  3. #3
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    Opalg's Avatar
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    For the "if" part, I would use Fatou's lemma on the sequence of positive functions |f| + |f_n| - |f-f_n|.
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