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Layer cake representation

Concept in mathematics From Wikipedia, the free encyclopedia

Layer cake representation
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In mathematics, the layer cake representation of a non-negative, real-valued measurable function defined on a measure space is the formula

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Layer cake representation.

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

where either integrand gives the same integral:

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

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Applications

The layer cake representation can be used to rewrite the Lebesgue integral as an improper Riemann integral. For the measure space, , let , be a measureable subset ( and a non-negative measureable function. By starting with the Lebesgue integral, then expanding , 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:

This can be used in turn, to rewrite the integral for the Lp-space p-norm, for :

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.

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See also

References

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