Layer Cake Representation

Definition

Let $f$ be a non-negative, real-valued measurable function $f$ defined on a measure space $(\Omega ,\mathcal{A},\mu )$. Define

$$L(f,t)=\{y\in \Omega \mid f(y)⩾t\},$$

then we have

$$f(x)={\int }_0^{\infty }{\chi }_{L(f,t)}(x)dt,$$

which is called the layer cake representation of $f$.

By applying the Fubini-Tonelli theorem (see here), there is

$${\int }_{\Omega }f(x)\mathrm{d}\mu (x)={\int }_0^{\infty }\mu (\{x\in \Omega \mid f(x)>t\})\mathrm{d}t.\text{\hspace{1em}(*)}$$

For any $f\in L^p$ for $1⩽p<\infty$, by $(\ast )$ with the change of variables $t=s^p$ and $f:=|f{|}^p$, we have

\int_{\Omega}|f(x)|^p \mathrm{~d} \mu(x)=p \int_0^{\infty} s^{p-1} \mu(\{x \in \Omega: | f(x) |>s\}) \mathrm{d} s. $$ ^pa9wsk