(adsbygoogle = window.adsbygoogle || []).push({}); [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:

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# Homework Help: A problem based on Fubini's theorem

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