Layer cake representation
In mathematics, the layer cake representation of a non-negative, real-valued measurable function f defined on a measure space is the formula
for all, where denotes the indicator function of a subset and denotes the super-level set
The layer cake representation follows easily from observing that
and then using the formula
The layer cake representation takes its name from the representation of the value as the sum of contributions from the "layers" : "layers"/values t below contribute to the integral, while values t above do not.
An important consequence of the layer cake representation is the identity
which follows from it by applying the Fubini-Tonelli theorem.
An important application is that for can be written as follows
which follows immediately from the change of variables in the layer cake representation of.
