In mathematics, more precisely in measure theory, the Lebesgue decomposition theorem[1] provides a way to decompose a measure into two distinct parts based on their relationship with another measure.
is a measurable space and
are σ-finite signed measures on
, then there exist two uniquely determined σ-finite signed measures
such that:[2][3] Lebesgue's decomposition theorem can be refined in a number of ways.
a σ-finite positive measure on
[4] The first assertion follows from the Lebesgue decomposition, the second is known as the Radon-Nikodym theorem.
is a Radon-Nikodym derivative that can be expressed as
An alternative refinement is that of the decomposition of a regular Borel measure[5][6][7]
{\displaystyle \nu =\nu _{ac}+\nu _{sc}+\nu _{pp},}
where The absolutely continuous measures are classified by the Radon–Nikodym theorem, and discrete measures are easily understood.
Hence (singular continuous measures aside), Lebesgue decomposition gives a very explicit description of measures.
The Cantor measure (the probability measure on the real line whose cumulative distribution function is the Cantor function) is an example of a singular continuous measure.
The analogous[citation needed] decomposition for a stochastic processes is the Lévy–Itō decomposition: given a Lévy process X, it can be decomposed as a sum of three independent Lévy processes
where: This article incorporates material from Lebesgue decomposition theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.