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 (${\displaystyle \color {red}{\text{strict}}}$) super-level set:

${\displaystyle L(f,t)=\{y\in \Omega \mid f(y)\geq t\}\;\;\;{\color {red}{\text{or}}\;L(f,t)=\{y\in \Omega \mid f(y)>t\}}.}$

The layer cake representation follows easily from observing that

${\displaystyle 1_{L(f,t)}(x)=1_{[0,f(x)]}(t)\;\;\;{\color {red}{\text{or}}\;1_{L(f,t)}(x)=1_{[0,f(x))}(t)}}$

where either integrand gives the same integral:

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

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.^{: cor. 2.2.34}