Layer cake representation explained

f defined on a measure space (\Omega, l{A},\mu) is the formulaf(x) = \int ^{inf ty} _{0} 1 _{L(f,} (x) 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 super-level setL(f, t) = \{ y \in \Omega \mid f(y) \geq t \}.

The layer cake representation follows easily from observing that

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

and then using the formula

f(x) = \int ^{f(x)} _{0} 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.^[1]

An important consequence of the layer cake representation is the identity

\int _{\Omega} f(x) d \mu(x) = \int ^{inf ty} _{0} \mu(\{ x \in \Omega \mid f(x) > t \}) d t,

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

An important application is that

L ^{p} for 1 \leq p< +inf ty can be written as follows\int _{\Omega} |f(x)| ^{p} d \mu(x) = p\int ^{inf ty} _{0} s ^{p-1} \mu(\{ x \in \Omega \mid |f(x)| > s \}) 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. . The Brunn–Minkowski inequality . Bull. Amer. Math. Soc. (N.S.) . 39 . 3 . 2002 . 355–405 (electronic) . 10.1090/S0273-0979-02-00941-2 . free .
• Book: Lieb. Elliott. Elliott H. Lieb. Loss. Michael. Michael Loss. Analysis. 2001. 2nd. American Mathematical Society. Graduate Studies in Mathematics. 14. 978-0821827833.

Notes and References

1. Book: Willem . Michel . Functional analysis : fundamentals and applications . 2013 . New York . 978-1-4614-7003-8.