Lebesgue's decomposition theorem


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.

Formal Statement

The theorem states that if is a measurable space and and
are σ-finite signed measures on, then there exist two uniquely determined σ-finite signed measures and such that:
  • .

    Refinement

Lebesgue's decomposition theorem can be refined in a number of ways. First, as the Lebesgue–Radon–Nikodym theorem. That is, let
be a measure space, a σ-finite positive measure on and a complex measure on.
  • There is a unique pair of complex measures on such that If is positive and finite, then so are and.
  • There is a unique such that
The first assertion follows from the Lebesgue decomposition, the second is known as the Radon–Nikodym theorem. That is, the function is a Radon–Nikodym 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 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: