A problem based on Fubini's theorem

[quasar987]

[SOLVED] A problem based on Fubini's theorem

Homework Statement

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

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

and

$$F_n(x)=\int_{-\infty}^{+\infty}f_n(x,y)dy$$

Show that

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

The Attempt at a Solution

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

$$||F_n||_p\leq \int_{-\infty}^{+\infty}||f_n||_pdy$$

Aside from this sad recasting of the problem, I have no lead!

 

[Hurkyl]

That last inequality looks like the triangle inequality, doesn't it?