Layer Cake Representation

In mathematics, the layer cake representation of a non- negative, real-valued measurable function $f$ defined on a measure space $(\Omega,\mathcal,\mu)$ is the formula

:$f(x) = \int_0^\infty 1_ (x) \, \mathrmt,$

for all $x \in \Omega$, where $1_E$ denotes the indicator function of a subset $E\subseteq \Omega$ and $L(f,t)$ denotes the super-level set

:$L(f, t) = \.$

The layer cake representation follows easily from observing that

:$1_(x) = 1_(t)$

and then using the formula

:$f(x) = \int_0^ \,\mathrmt.$

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

$\int_\Omega f(x) \, \mathrm\mu(x) = \int_0^ \mu(\)\,\mathrmt,$

which follows from it by applying the Fubini-Tonelli theorem.

An important application is that $L^p$ for $1\leq p<+\infty$ can be written as follows

$\int_\Omega , f(x), ^p \, \mathrm\mu(x) = p\int_0^ s^\mu(\) \mathrms,$

which follows immediately from the change of variables $t=s^$ in the layer cake representation of $, f(x), ^p$.

See also

* Symmetric decreasing rearrangement

References

*
* {{cite book
, last1=Lieb
, first1=Elliott
, authorlink1=Elliott H. Lieb
, last2=Loss
, first2=Michael, author2-link=Michael Loss
, title=Analysis
, year=2001, edition=2nd
, publisher=American Mathematical Society
, series= Graduate Studies in Mathematics, volume=14
, isbn=978-0821827833

Real analysis