Layer cake representation

From Wikipedia, the free encyclopedia

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

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

where 1E denotes the indicator function of a subset E ⊆ Rn 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.

[edit] References