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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