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Layer cake representation

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Layer cake representation.


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 super-level set

L ( f , t ) = { y ∈ Ω ∣ f ( y ) ≥ t } . {\displaystyle L(f,t)=\{y\in \Omega \mid f(y)\geq t\}.}

The layer cake representation follows easily from observing that

1 L ( f , t ) ( x ) = 1 [ 0 , f ( x ) ] ( t ) {\displaystyle 1_{L(f,t)}(x)=1_{}(t)}

and then using the formula

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.

An important consequence of the layer cake representation is the identity

∫ Ω f ( x ) d μ ( x ) = ∫ 0 ∞ μ ( { x ∈ Ω ∣ f ( x ) > t } ) d t , {\displaystyle \int _{\Omega }f(x)\,\mathrm {d} \mu (x)=\int _{0}^{\infty }\mu (\{x\in \Omega \mid f(x)>t\})\,\mathrm {d} t,}

which follows from it by applying the Fubini-Tonelli theorem.

An important application is that L p {\displaystyle L^{p}} for 1 ≤ p < + ∞ {\displaystyle 1\leq p<+\infty } can be written as follows

∫ 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 \mid \,|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

References

  1. Willem, Michel (2013). Functional analysis : fundamentals and applications. New York. ISBN 978-1-4614-7003-8.{{cite book}}: CS1 maint: location missing publisher (link)
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