In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem states that for every two σ-finite signed measures \mu and \nu on a measurable space there exist two σ-finite signed measures \nu_0 and \nu_1 such that: These two measures are uniquely determined by \mu and \nu.

Refinement

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: where 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.

Related concepts

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:

Citations