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Need help with part of a proof (measure theory class)

  1. May 29, 2012 #1
    1. The problem statement, all variables and given/known data

    As part of a larger proof, I'm trying to show that ||fn|| --> ||f|| implies <fn> --> f in Lp, where <fn> is a sequence of functions in Lp, 1≤p<∞, which converge a.e. to a function f in Lp.

    2. Relevant equations

    ||f|| = (∫|f|p)1/p.

    3. The attempt at a solution

    There's some string of inequalities I need to obtain ||f-fn||≤|||f||-||fn|||<∂. Any ideas?
     
  2. jcsd
  3. May 29, 2012 #2

    micromass

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    Apply Fatou's lemma on

    [tex]\lim_{n\rightarrow +\infty} {2^p(|f_n|^p+|f|^p)-|f_n-f|^p}[/tex]
     
  4. May 29, 2012 #3
    Genius!
     
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