Layer cake representation

In mathematics, the layer cake representation of a non-negative, real-valued measurable function 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
where either integrand gives the same integral:
The layer cake representation takes its name from the representation of the value as the sum of contributions from the "layers" : "layers"/values below contribute to the integral, while values above do not.
It is a generalization of Cavalieri's principle and is also known under this name.

Applications

The layer cake representation can be used to rewrite the Lebesgue integral as an improper Riemann integral. For the measure space,, let, be a measureable subset and simplifying in terms of the Lebesgue integral of an indicator function, we get the Riemann integral:
This can be used in turn, to rewrite the integral for the L^p-space p-norm, for :
which follows immediately from the change of variables in the layer cake representation of. This representation can be used to prove Markov's inequality and Chebyshev's inequality.