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 $(\Omega ,{\mathcal {A}},\mu )$ is the formula

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

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

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

The layer cake representation follows easily from observing that

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

and then using the formula

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

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.

An important consequence of the layer cake representation is the identity

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

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

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

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

which follows immediately from the change of variables $t=s^{p}$ 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

• Gardner, Richard J. (2002). "The Brunn–Minkowski inequality". Bull. Amer. Math. Soc. (N.S.). 39 (3): 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2.
• Lieb, Elliott; Loss, Michael (2001). Analysis. Graduate Studies in Mathematics. Vol. 14 (2nd ed.). American Mathematical Society. ISBN 978-0821827833.