Layer cake representation
In mathematics, the layer cake representation of a non-negative, real-valued measurable function [math]\displaystyle{ f }[/math] defined on a measure space [math]\displaystyle{ (\Omega,\mathcal{A},\mu) }[/math] is the formula
- [math]\displaystyle{ f(x) = \int_0^\infty 1_{L(f, t)} (x) \, \mathrm{d}t, }[/math]
for all [math]\displaystyle{ x \in \Omega }[/math], where [math]\displaystyle{ 1_E }[/math] denotes the indicator function of a subset [math]\displaystyle{ E\subseteq \Omega }[/math] and [math]\displaystyle{ L(f,t) }[/math] denotes the super-level set
- [math]\displaystyle{ L(f, t) = \{ y \in \Omega \mid f(y) \geq t \}. }[/math]
The layer cake representation follows easily from observing that
- [math]\displaystyle{ 1_{L(f, t)}(x) = 1_{[0, f(x)]}(t) }[/math]
and then using the formula
- [math]\displaystyle{ f(x) = \int_0^{f(x)} \,\mathrm{d}t. }[/math]
The layer cake representation takes its name from the representation of the value [math]\displaystyle{ f(x) }[/math] as the sum of contributions from the "layers" [math]\displaystyle{ L(f,t) }[/math]: "layers"/values [math]\displaystyle{ t }[/math] below [math]\displaystyle{ f(x) }[/math] contribute to the integral, while values [math]\displaystyle{ t }[/math] above [math]\displaystyle{ f(x) }[/math] 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
[math]\displaystyle{ \int_\Omega f(x) \, \mathrm{d}\mu(x) = \int_0^{\infty} \mu(\{ x \in \Omega \mid f(x) \gt t \})\,\mathrm{d}t, }[/math]
which follows from it by applying the Fubini-Tonelli theorem.
An important application is that [math]\displaystyle{ L^p }[/math] for [math]\displaystyle{ 1\leq p\lt +\infty }[/math] can be written as follows
[math]\displaystyle{ \int_\Omega |f(x)|^p \, \mathrm{d}\mu(x) = p\int_0^{\infty} s^{p-1}\mu(\{ x \in \Omega \mid\, |f(x)| \gt s \}) \mathrm{d}s, }[/math]
which follows immediately from the change of variables [math]\displaystyle{ t=s^{p} }[/math] in the layer cake representation of [math]\displaystyle{ |f(x)|^p }[/math].
This representation can be used to prove Markov's inequality and Chebyshev's inequality.
See also
References
- ↑ Willem, Michel (2013). Functional analysis : fundamentals and applications. New York. ISBN 978-1-4614-7003-8.
- 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 (2001). Analysis. Graduate Studies in Mathematics. 14 (2nd ed.). American Mathematical Society. ISBN 978-0821827833.
 |  |
