Layer cake representation

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In mathematics, the layer cake representation of a non-negative, real-valued measurable function f defined on n-dimensional Euclidean space Rn is the formula

where 1E denotes the indicator function of a subset E ⊆ Rn and L(ft) 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 f(x) as the sum of contributions from the "layers" L(ft): "layers"/values t below f(x) contribute to the integral, while values t above f(x) do not.

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