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

$${\int }_{\Omega }|f(x){|}^p\mathrm{d}\mu (x)=p{\int }_0^{\infty }{s}^{p-1}\mu (\{x\in \Omega :|f(x)|>s\})\mathrm{d}s.$$