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Continuity of the integral

  1. Jun 10, 2008 #1
    Hi,

    I was wondering if the Lebesgue-integral is continuos with respect to the [itex]L^\infty[/itex] norm.

    More precisely, assume there is a space of functions

    [tex]
    \mathcal{P}=\{f\in L^1 :||f||_{L^1}=1\}\cap L^\infty
    [/tex]

    endowed with the essential supremum norm [itex]||\cdot||_{L^\infty}[/itex]. If there is then a Cauchy sequence [itex]f_1,f_2,\dots[/itex] in [itex]\mathcal{P}[/itex], can one conclude that the limit (which exists in [itex]L^\infty[/itex]) is also integrable with [itex]L^1[/itex] norm one?

    Thank you very much.
    Pere
     
  2. jcsd
  3. Jun 13, 2008 #2

    morphism

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    How about something like [itex]f_n = \frac{1}{n} \chi_{[0,n]}[/itex] in [itex]L^1(\mathbb{R})[/itex]? Here we have [itex]\|f_n \|_{L^1} = 1[/itex] and [itex]\|f_n - 0\|_{L^\infty} = \frac{1}{n} \to 0[/itex], while [itex]\| 0 \|_{L^1} = 0[/itex].
     
  4. Jun 14, 2008 #3
    Thanks morphism, that seems to answer my question.
     
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