Computing the $L_p$ norm

Question

Let $1\leq p < \infty$ and $E\in \mathcal{M}(\mathbb{R}^n)$ with $m(E)>0$.

For any $f\in \mathcal{L}^p(E)$ we define the distribution function associated to $|f|$ to be $$m_{|f|}:\left(0,\infty\right)\ni t \mapsto m_{|f|}(t)=m\left(\{x\in E:|f(x)|>t \}\right).$$

Now we consider the $p$-power of the $L_p$ norm of $f$, that is:$$||f||_p^p=\int_E|f|^pdm.$$

My claim is the equality:

$$||f||_p^p=\int_E|f|^pdm=p\int_0^{+\infty} t^{p-1}m_{|f|}(t)dt$$

From Tchebychev inequality we have:

$$t^pm\left(\{x\in E:|f(x)|>t \}\right)=t^pm_{|f|}(t)\leq \int_{E}|f|^pdm$$

and I think one should be able to conclude from here; but I lack ideas as to how.

Can you give me a proof or some reference in the literature for the claim above?

Answer 1

You cannot prove the equality from your inequality. The reslt is an easy consequence of Fubini/Tonelli Theorem. $p\int_0^{\infty} t^{p-1} m_{|f|} (t) dt=p\int_0^{\infty} \int I_{\{x: |f(x)| >t\}}t^{p-1} dmdt=p \int \int_0^{\infty}I_{\{x: |f(x)| >t\}} t^{p-1} dmdt=\int |f|^{p} dm$ since $\int_0^{\infty} I_{\{x: t <|f(x)|\} }t^{p-1} dt =|f(x)|^{p}$.

Answer 2

The classical way is to use Tonelli Theorem: \begin{align*} \int|f|^{p}dm(x)&=\int\int_{0}^{\infty}p\chi_{t<|f(x)|}t^{p-1}dtdm(x)\\ &=\int_{0}^{\infty}\int\chi_{t<f(x)}dm(x)pt^{p-1}dt\\ &=\cdots \end{align*} The other way is to test $f=\chi_{S}$ for a measurable set $S$ and then use Monotone Convergence Theorem to approximate nonnegative $f$ by simple functions. Note that a simple function can be written as finite sum of characteristic functions of disjoint measurable sets.