Layer cake representation: Difference between revisions
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Cavalieri principle |
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The [[layer cake]] representation takes its name from the representation of the value <math>f(x)</math> as the sum of contributions from the "layers" <math>L(f,t)</math>: "layers"/values ''t'' below <math>f(x)</math> contribute to the integral, while values ''t'' above <math>f(x)</math> do not. |
The [[layer cake]] representation takes its name from the representation of the value <math>f(x)</math> as the sum of contributions from the "layers" <math>L(f,t)</math>: "layers"/values ''t'' below <math>f(x)</math> contribute to the integral, while values ''t'' above <math>f(x)</math> do not. |
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It is a generalization of [[Cavalieri's principle]] and is also known under this name <ref>{{cite book |last1=Willem |first1=Michel |title=Functional analysis : fundamentals and applications |date=2013 |location=New York |isbn=978-1-4614-7003-8}}</ref>{{rp|at=cor. 2.2.34}}. |
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An important consequence of the layer cake representation is the identity |
An important consequence of the layer cake representation is the identity |
Revision as of 11:52, 10 January 2023
In mathematics, the layer cake representation of a non-negative, real-valued measurable function f defined on a measure space is the formula
for all , where denotes the indicator function of a subset and denotes the super-level set
The layer cake representation follows easily from observing that
and then using the formula
The layer cake representation takes its name from the representation of the value as the sum of contributions from the "layers" : "layers"/values t below contribute to the integral, while values t above do not. It is a generalization of Cavalieri's principle and is also known under this name [1]: cor. 2.2.34 .
An important consequence of the layer cake representation is the identity
which follows from it by applying the Fubini-Tonelli theorem.
An important application is that for can be written as follows
which follows immediately from the change of variables in the layer cake representation of .
See also
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.
- ^ Willem, Michel (2013). Functional analysis : fundamentals and applications. New York. ISBN 978-1-4614-7003-8.
{{cite book}}
: CS1 maint: location missing publisher (link)