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:
These two measures are uniquely determined by and.

Refinement

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
Second, absolutely continuous measures are classified by the Radon–Nikodym theorem, and discrete measures are easily understood. Hence, Lebesgue decomposition gives a very explicit description of measures. The Cantor measure 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: