Layer cake representation

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 super-level set

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

The layer cake representation follows easily from observing that

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

and then using the formula

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

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

An important consequence of the layer cake representation is the identity

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

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

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

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

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

References

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