In mathematics, more precisely in measure theory, the Lebesgue decomposition theorem provides a way to decompose a measure into two distinct parts based on their relationship with another measure.
The theorem states that if is a measurable space and and are ÃÂ-finite signed measures on , then there exist two uniquely determined ÃÂ-finite signed measures and such that:
Lebesgue's decomposition theorem can be refined in a number of ways. First, as the LebesgueâÂÂRadonâÂÂNikodym theorem. That is, let be a measure space, a ÃÂ-finite positive measure on and a complex measure on .
The first assertion follows from the Lebesgue decomposition, the second is known as the RadonâÂÂNikodym theorem. That is, the function is a RadonâÂÂNikodym derivative that can be expressed as
An alternative refinement is that of the decomposition of a regular Borel measure
where
The absolutely continuous measures are classified by the RadonâÂÂNikodym 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 LévyâÂÂItà  decomposition: given a Lévy process X, it can be decomposed as a sum of three independent Lévy processes where: