layer cake representation

File:Volume-Alhazen.svg

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

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

The layer cake representation follows easily from observing that

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

where either integrand gives the same integral:

: $f(x) = \int_0^{f(x)} \,\mathrm{d}t.$

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.{{cite book |last1=Willem |first1=Michel |title=Functional analysis : fundamentals and applications |date=2013 |location=New York |isbn=978-1-4614-7003-8}}{{rp|at=cor. 2.2.34}}

Applications

The layer cake representation can be used to rewrite the Lebesgue integral as an improper Riemann integral. For the measure space, $(\Omega,\mathcal{A},\mu)$ , let $S\subseteq\Omega$ , be a measureable subset ( $S\in\mathcal{A})$ 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:

: $\begin{align}
\int_S f(x)\,\text{d}\mu(x)
&= \int_S \int_0^\infty 1_{\{x\in\Omega\mid f(x)>t\}}(x)\,\text{d}t\,\text{d}\mu(x) \\
&= \int_0^\infty\!\! \int_S 1_{\{x\in\Omega\mid f(x)>t\}}(x)\,\text{d}\mu(x)\,\text{d}t\\
&= \int_0^\infty\!\! \int_\Omega 1_{\{x\in S\mid f(x)>t\}}(x)\,\text{d}\mu(x)\,\text{d}t\\
&= \int_0^{\infty} \mu(\{x\in S \mid f(x)>t\})\,\text{d}t.
\end{align}$

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

: $\int_\Omega |f(x)|^p \, \mathrm{d}\mu(x) = p\int_0^{\infty} s^{p-1}\mu(\{ x \in \Omega:|f(x)| > s \}) \mathrm{d}s,$

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.

References

{{cite journal | last=Gardner | first=Richard J. | title=The Brunn–Minkowski inequality | journal=Bull. Amer. Math. Soc. (N.S.) | volume=39 | issue=3 | year=2002 | pages=355–405 (electronic) | doi=10.1090/S0273-0979-02-00941-2 | doi-access=free }}
{{cite book

|last1=Lieb

|first1=Elliott

|authorlink1=Elliott H. Lieb

|last2=Loss

|first2=Michael|author2-link=Michael Loss

|title=Analysis

|year=2001|edition=2nd

|publisher=American Mathematical Society

|series=Graduate Studies in Mathematics|volume=14

|isbn=978-0821827833}}

Category:Real analysis

layer cake representation

Applications

See also

References