# A problem based on Fubini's theorem

1. Nov 30, 2007

### quasar987

[SOLVED] A problem based on Fubini's theorem

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

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

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

3. 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! :grumpy:

2. Nov 30, 2007

### Hurkyl

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

Last edited: Nov 30, 2007