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 1_E denotes the indicator function of a subset and L(f,t) 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 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. It is a generalization of Cavalieri's principle and is also known under this name. 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 L^p for can be written as follows which follows immediately from the change of variables t=s^{p} in the layer cake representation of |f(x)|^p. This representation can be used to prove Markov's inequality and Chebyshev's inequality.