Lebesgue's decomposition theorem

In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem states that for every two σ-finite signed measures and on a measurable space there exist two σ-finite signed measures and such that: (that is, is absolutely continuous with respect to ) (that is, and are singular). These two measures are uniquely determined by and Lebesgue's decomposition theorem can be refined in a number of ways. First, the decomposition of the singular part of a regular Borel measure on the real line can be refined: where νcont is the absolutely continuous part νsing is the singular continuous part νpp is the pure point part (a discrete measure). 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. Lévy–Itō decomposition 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 where: is a Brownian motion with drift, corresponding to the absolutely continuous part; is a compound Poisson process, corresponding to the pure point part; is a square integrable pure jump martingale that almost surely has a countable number of jumps on a finite interval, corresponding to the singular continuous part.