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Math Help - Liminf to limsup

  1. #1
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    Liminf to limsup

    Suppose that  \mid f_n \mid \leq g \in L^1 and  f_n \rightarrow f in measure.

    Then  \int f = \lim \int f_n .

    Proof so far.

    Now,  g- f_n \geq 0 and  f_n -g \geq 0 .

    So for the first case, then

     \int g - f \leq lim \ inf \int g - f_n = \int g - \ lim \ sup \int f _n

    But how does the lim inf turn into lim sup?

    Thank you!
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  2. #2
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    What does "in measure" mean? If i'ts pointwise convergence a.e. with respect to the measure couldn't you apply Fatou two times (to f_k and -f_k) to get that \limsup \int f_k \leq f \leq \liminf \int f_k

    As for your question \liminf (-f_k)=-\limsup (f_k)
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  3. #3
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    Convergence in measure :

    \forall \epsilon >0, \lim_{n\to\infty} \mu(\{|f_n-f|>\epsilon\})=0
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