Lebesgue's decomposition theorem

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In mathematics, more precisely in measure theory, the Lebesgue decomposition theorem [1] provides a way to decompose a measure into two distinct parts based on their relationship with another measure.

Contents

Definition

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: [2] [3]

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 . [4]

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 [5] [6] [7] where

The 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[ citation needed ] 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:

See also

Notes

  1. Hewitt & Stromberg 1965, Chapter V, § 19, (19.42) Lebesgue Decomposition Theorem.
  2. Halmos 1974, Section 32, Theorem C.
  3. Swartz 1994, p. 141.
  4. Rudin 1974, Section 6.9, The Theorem of Lebesgue–Radon–Nikodym.
  5. Hewitt & Stromberg 1965, Chapter V, § 19, (19.61) Theorem.
  6. Reed & Simon 1981, pp. 22–25.
  7. Simon 2005, p. 43.

References

This article incorporates material from Lebesgue decomposition theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.