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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 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 below contribute to the integral, while values 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 .

This representation can be used to prove Markov's inequality and Chebyshev's inequality.

See also

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References

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  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)