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 (strict) super-level set:

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

The layer cake representation follows easily from observing that

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

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.

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:

${\begin{aligned}\int _{S}f(x)\,{\text{d}}\mu (x)&=\int _{S}\int _{0}^{\infty }1_{\{x\in \Omega \mid f(x)>t\}}(x)\,{\text{d}}t\,{\text{d}}\mu (x)\\&=\int _{0}^{\infty }\!\!\int _{S}1_{\{x\in \Omega \mid f(x)>t\}}(x)\,{\text{d}}\mu (x)\,{\text{d}}t\\&=\int _{0}^{\infty }\!\!\int _{\Omega }1_{\{x\in S\mid f(x)>t\}}(x)\,{\text{d}}\mu (x)\,{\text{d}}t\\&=\int _{0}^{\infty }\mu (\{x\in S\mid f(x)>t\})\,{\text{d}}t.\end{aligned}}$

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

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

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.