Layer cake representation

Layer cake representation

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

:$f\left(x\right) = int_\left\{0\right\}^\left\{+ infty\right\} 1_\left\{L\left(f, t\right)\right\} \left(x\right) , mathrm\left\{d\right\} t mbox\left\{ for all \right\} x in mathbb\left\{R\right\}^\left\{n\right\},$

where 1"E" denotes the indicator function of a subset "E" &sube; R"n" and "L"("f", "t") denotes the super-level set

:$L\left(f, t\right) = \left\{ x in mathbb\left\{R\right\}^\left\{n\right\} | f\left(x\right) geq t \right\}.$

The layer cake representation follows easily from the formula

:$f\left(x\right) = int_\left\{0\right\}^\left\{f\left(x\right)\right\} mathrm\left\{d\right\} t.$

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.

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