Layer cake representation - Alchetron, the free social 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 ) = ∫ 0 + ∞ 1 L ( f , t ) ( x ) d t for all x ∈ R n ,

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

L ( f , t ) = { y ∈ R n | f ( y ) ≥ t } .

The layer cake representation follows easily from observing that

1 L ( f , t ) ( x ) = 1 [ 0 , f ( x ) ] ( t )

and then using the formula

f ( x ) = ∫ 0 f ( x ) 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.

References

Layer cake representation Wikipedia

(Text) CC BY-SA