In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem states that for every two σ-finite signed measures
\mu
\nu
(\Omega,\Sigma),
\nu0
\nu1
\nu=\nu0+\nu1
\nu0\ll\mu
\nu0
\mu
\nu1\perp\mu
\nu1
\mu
These two measures are uniquely determined by
\mu
\nu.
Lebesgue's decomposition theorem can be refined in a number of ways.
First, the decomposition of a regular Borel measure on the real line can be refined:
\nu=\nucont+\nusing+\nupp
Second, 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 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
X=X(1)+X(2)+X(3)
X(1)
X(2)
X(3)