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Math Help - Fatou's Lemma valid for convergence in measure

  1. #1
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    Fatou's Lemma valid for convergence in measure

    Dears,

    I need the proof shows that the Fatou's Lemma remains valid if convergence almost everywhere is replaced by convergence in measure.

    Best Regards.
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  2. #2
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    Re: Fatou's Lemma valid for convergence in measure

    "There exists a subsequence such that
    Fatou's Lemma valid for convergence in measure-7b4932f9cdf47ab799131bb02a9ea3de.png


    Since this subsequence also converges in measure to f, there exists a further subsequence, which converges pointwise to f almost everywhere, hence the previous variation of Fatou's lemma is applicable to this subsubsequence."
    Click image for larger version. 

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    But I am not understand how we get the subsequence such that
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