Layer cake representation

In mathematics, the layer cake representation of a non-negative, real-valued measurable function f defined on n-dimensional Euclidean space $\mathbb {R} ^{n}$ is the formula

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

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

L(f,t)=\{y\in {\mathbb {R}}^{n}|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)}dt.

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.

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. 14 (2nd ed.). American Mathematical Society. ISBN 978-0821827833.

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Layer cake representation

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References