Layer cake representation

File:Volume-Alhazen.svg

Layer cake representation.

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

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

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

$L(f,t)=\{y\in \mathrm{\Omega }\mid f(y)\ge t\}\text{or}L(f,t)=\{y\in \mathrm{\Omega }\mid f(y)>t\}.$

The layer cake representation follows easily from observing that

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

where either integrand gives the same integral:

$f(x)=\underset{0}{\overset{f(x)}{\int }}\mathrm{d}t.$

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

Applications

The layer cake representation can be used to rewrite the Lebesgue integral as an improper Riemann integral. For the measure space, $(\mathrm{\Omega },{𝒜},\mu )$, let $S\subseteq \mathrm{\Omega }$, be a measureable subset ($S\in {𝒜})$ and $f$ a non-negative measureable function. By starting with the Lebesgue integral, then expanding $f(x)$, then exchanging integration order (see Fubini-Tonelli theorem) and simplifying in terms of the Lebesgue integral of an indicator function, we get the Riemann integral:

$\begin{array}{rl}\underset{S}{\int }f(x)\text{d}\mu (x)& =\underset{S}{\int }\underset{0}{\overset{\mathrm{\infty }}{\int }}{1}_{\{x\in \mathrm{\Omega }\mid f(x)>t\}}(x)\text{d}t\text{d}\mu (x)\\ & =\underset{0}{\overset{\mathrm{\infty }}{\int }}\underset{S}{\int }{1}_{\{x\in \mathrm{\Omega }\mid f(x)>t\}}(x)\text{d}\mu (x)\text{d}t\\ & =\underset{0}{\overset{\mathrm{\infty }}{\int }}\underset{\mathrm{\Omega }}{\int }{1}_{\{x\in S\mid f(x)>t\}}(x)\text{d}\mu (x)\text{d}t\\ & =\underset{0}{\overset{\mathrm{\infty }}{\int }}\mu (\{x\in S\mid f(x)>t\})\text{d}t.\end{array}$

This can be used in turn, to rewrite the integral for the L^p-space p-norm, for $1\le p<+\mathrm{\infty }$:

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

which follows immediately from the change of variables $t=s^p$ in the layer cake representation of $|f(x){|}^p$. This representation can be used to prove Markov's inequality and Chebyshev's inequality.

See also

• Symmetric decreasing rearrangement