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

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

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

The layer cake representation follows easily from the formula

:f(x) = int_{0}^{f(x)} mathrm{d} 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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