In mathematics, the layer cake representation of a non-negative, real-valued measurable function f {\displaystyle f} defined on a measure space ( Ω , A , μ ) {\displaystyle (\Omega ,{\mathcal {A}},\mu )} is the formula
f ( x ) = ∫ 0 ∞ 1 L ( f , t ) ( x ) d t , {\displaystyle f(x)=\int _{0}^{\infty }1_{L(f,t)}(x)\,\mathrm {d} t,}for all x ∈ Ω {\displaystyle x\in \Omega } , where 1 E {\displaystyle 1_{E}} denotes the indicator function of a subset E ⊆ Ω {\displaystyle E\subseteq \Omega } and L ( f , t ) {\displaystyle L(f,t)} denotes the ( strict {\displaystyle \color {red}{\text{strict}}} ) super-level set:
L ( f , t ) = { y ∈ Ω ∣ f ( y ) ≥ t } or L ( f , t ) = { y ∈ Ω ∣ f ( y ) > t } . {\displaystyle 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 ) or 1 L ( f , t ) ( x ) = 1 [ 0 , f ( x ) ) ( t ) {\displaystyle 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 ) = ∫ 0 f ( x ) d t . {\displaystyle f(x)=\int _{0}^{f(x)}\,\mathrm {d} t.}The layer cake representation takes its name from the representation of the value f ( x ) {\displaystyle f(x)} as the sum of contributions from the "layers" L ( f , t ) {\displaystyle L(f,t)} : "layers"/values t {\displaystyle t} below f ( x ) {\displaystyle f(x)} contribute to the integral, while values t {\displaystyle t} above f ( x ) {\displaystyle f(x)} do not. It is a generalization of Cavalieri's principle and is also known under this name.: 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, ( Ω , A , μ ) {\displaystyle (\Omega ,{\mathcal {A}},\mu )} , let S ⊆ Ω {\displaystyle S\subseteq \Omega } , be a measureable subset ( S ∈ Σ ) {\displaystyle S\in \Sigma )} and f {\displaystyle f} a non-negative measureable function. By starting with the Lebesgue integral, then expanding f ( x ) {\displaystyle 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:
∫ S f ( x ) d μ ( x ) = ∫ S ∫ 0 ∞ 1 { x ∈ Ω ∣ f ( x ) > t } ( x ) d t d μ ( x ) = ∫ 0 ∞ ∫ S 1 { x ∈ Ω ∣ f ( x ) > t } ( x ) d μ ( x ) d t = ∫ 0 ∞ ∫ Ω 1 { x ∈ S ∣ f ( x ) > t } ( x ) d μ ( x ) d t = ∫ 0 ∞ μ ( { x ∈ S ∣ f ( x ) > t } ) d t . {\displaystyle {\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 be used in turn, to rewrite the integral for the Lp-space p-norm, for 1 ≤ p < + ∞ {\displaystyle 1\leq p<+\infty } :
∫ Ω | f ( x ) | p d μ ( x ) = p ∫ 0 ∞ s p − 1 μ ( { x ∈ Ω : | f ( x ) | > s } ) d s , {\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 {\displaystyle t=s^{p}} in the layer cake representation of | f ( x ) | p {\displaystyle |f(x)|^{p}} . This representation can be used to prove Markov's inequality and Chebyshev's inequality.
See also
- 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.
References
Willem, Michel (2013). Functional analysis : fundamentals and applications. New York. ISBN 978-1-4614-7003-8.{{cite book}}: CS1 maint: location missing publisher (link) 978-1-4614-7003-8 ↩