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The General Lebesgue Integral

  1. Mar 1, 2013 #1
    1. The problem statement, all variables and given/known data

    Show that under the hypothesis of Theorem 17 we have [itex]\int |f_n - f| \rightarrow 0[/itex].

    Theorem 7:

    Let [itex]<g_n>[/itex] be a sequence of integrable functions which converges a.e. to an integrable function g. Let [itex]<f_n>[/itex] be a dequence of measurable functions such that [itex]|f_n| \leq g_n[/itex] and [itex]<f_n>[/itex] converges to f a.e.. If [tex]\int g = lim \int g_n[/tex] then [tex]\int f lim \int f_n[/tex].


    2. Relevant equations



    3. The attempt at a solution

    By linearity, [itex]\int (f_n-f) = \int f_n - \int f[/itex]. So if we take the limit of [itex]\int f_n - \int f[/itex], we get [itex] lim_{n \rightarrow \infty} \int f_n - lim_{n \rightarrow \infty} \int f [/itex]. But we know from Theorem 17 that [tex]\int f = lim \int f_n[/tex]. So if we replace [itex] lim_{n \rightarrow \infty} \int f_n[/itex] by [itex]\int f[/itex]. So [itex]\int f - \int f = 0[/itex]. I feel like there is something wrong with my proof, because it's so simple...so can anybody tell me if I'm wrong?

    Thanks in advance
     
  2. jcsd
  3. Mar 1, 2013 #2

    jbunniii

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    You showed that ##\int (f_n - f) \rightarrow 0##, but that was already known from theorem 17. You were asked to show that ##\int |f_n - f| \rightarrow 0##, which is a stronger statement because
    $$\left|\int f_n - f\right| \leq \int |f_n - f|$$
    What you are trying to prove looks like a slight generalization of the dominated convergence theorem, so examining how that proof works should be useful, i.e., you know you will probably have to use Fatou's lemma.
     
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