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Homework Help: A problem based on Fubini's theorem

  1. Nov 30, 2007 #1

    quasar987

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    [SOLVED] A problem based on Fubini's theorem

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

    Let [tex]1<p<+\infty[/tex] and [tex]f:\mathbb{R}^2\rightarrow [0,
    +\infty[[/tex] a measurable function. Set

    [tex]f_n=\inf \{f,n\}\mathbb{I}_{[-n,n]\times [-n,n]}[/tex]

    and

    [tex]F_n(x)=\int_{-\infty}^{+\infty}f_n(x,y)dy[/tex]

    Show that

    [tex]\left(\int_{-\infty}^{+\infty}F_n(x)^p dx\right)^{1/p}\leq\int_{-\infty}^{+\infty}\left(\int_{-\infty}^{+\infty}f_n(x,y)^pdx \right)^{1/p}dy[/tex]

    3. The attempt at a solution

    In a somewhat different language, we are asked to show that

    [tex]||F_n||_p\leq \int_{-\infty}^{+\infty}||f_n||_pdy[/tex]

    Aside from this sad recasting of the problem, I have no lead! :grumpy:
     
  2. jcsd
  3. Nov 30, 2007 #2

    Hurkyl

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    That last inequality looks like the triangle inequality, doesn't it?
     
    Last edited: Nov 30, 2007
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