Layer cake representation

In mathematics, the layer cake representation of a non-negative, real-valued measurable function ${\displaystyle f}$ defined on a measure space ${\displaystyle (\Omega ,{\mathcal {A}},\mu )}$ is the formula

${\displaystyle f(x)=\int _{0}^{\infty }1_{L(f,t)}(x)\,\mathrm {d} t,}$

for all ${\displaystyle x\in \Omega }$, where ${\displaystyle 1_{E}}$ denotes the indicator function of a subset ${\displaystyle E\subseteq \Omega }$ and ${\displaystyle L(f,t)}$ denotes the super-level set

${\displaystyle L(f,t)=\{y\in \Omega \mid f(y)\geq t\}.}$

The layer cake representation follows easily from observing that

${\displaystyle 1_{L(f,t)}(x)=1_{[0,f(x)]}(t)}$

and then using the formula

${\displaystyle f(x)=\int _{0}^{f(x)}\,\mathrm {d} t.}$

The layer cake representation takes its name from the representation of the value ${\displaystyle f(x)}$ as the sum of contributions from the "layers" ${\displaystyle L(f,t)}$: "layers"/values ${\displaystyle t}$ below ${\displaystyle f(x)}$ contribute to the integral, while values ${\displaystyle t}$ above ${\displaystyle f(x)}$ do not. It is a generalization of Cavalieri's principle and is also known under this name.[1]^{: cor. 2.2.34}

An important consequence of the layer cake representation is the identity

${\displaystyle \int _{\Omega }f(x)\,\mathrm {d} \mu (x)=\int _{0}^{\infty }\mu (\{x\in \Omega \mid f(x)>t\})\,\mathrm {d} t,}$

which follows from it by applying the Fubini-Tonelli theorem.

An important application is that ${\displaystyle L^{p}}$ for ${\displaystyle 1\leq p<+\infty }$ can be written as follows

${\displaystyle \int _{\Omega }|f(x)|^{p}\,\mathrm {d} \mu (x)=p\int _{0}^{\infty }s^{p-1}\mu (\{x\in \Omega \mid \,|f(x)|>s\})\mathrm {d} s,}$

which follows immediately from the change of variables ${\displaystyle t=s^{p}}$ in the layer cake representation of ${\displaystyle |f(x)|^{p}}$.