Lebesgue's decomposition theorem

In

${\displaystyle \mu }$ and ${\displaystyle \nu }$ on a measurable space ${\displaystyle (\Omega ,\Sigma ),}$ there exist two σ-finite signed measures ${\displaystyle \nu _{0}}$ and ${\displaystyle \nu _{1}}$ such that:

• ${\displaystyle \nu =\nu _{0}+\nu _{1}\,}$
• ${\displaystyle \nu _{0}\ll \mu }$ (that is, ${\displaystyle \nu _{0}}$ is
  absolutely continuous
  with respect to ${\displaystyle \mu }$)
• ${\displaystyle \nu _{1}\perp \mu }$ (that is, ${\displaystyle \nu _{1}}$ and ${\displaystyle \mu }$ are singular).

These two measures are uniquely determined by ${\displaystyle \mu }$ and ${\displaystyle \nu .}$

Refinement

Lebesgue's decomposition theorem can be refined in a number of ways.

First, the decomposition of a regular

${\displaystyle \,\nu =\nu _{\mathrm {cont} }+\nu _{\mathrm {sing} }+\nu _{\mathrm {pp} }}$

where

• ν_cont is the absolutely continuous part
• ν_sing is the singular continuous part
• ν_pp is the pure point part (a discrete measure).

Second, absolutely continuous measures are classified by the

) is an example of a singular continuous measure.

Related concepts

Lévy–Itō decomposition

Main article:

Lévy–Itō decomposition

The analogous^[

${\displaystyle X=X^{(1)}+X^{(2)}+X^{(3)}}$ where:

• ${\displaystyle X^{(1)}}$ is a Brownian motion with drift, corresponding to the absolutely continuous part;
• ${\displaystyle X^{(2)}}$ is a compound Poisson process, corresponding to the pure point part;
• ${\displaystyle X^{(3)}}$ 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.

See also

• Decomposition of spectrum
• Hahn decomposition theorem and the corresponding Jordan decomposition theorem

Citations

1. ^ (Halmos 1974, Section 32, Theorem C)
2. ^ (Hewitt & Stromberg 1965, Chapter V, § 19, (19.42) Lebesgue Decomposition Theorem)
3. ^ (Rudin 1974, Section 6.9, The Theorem of Lebesgue-Radon-Nikodym)
4. ^ (Hewitt & Stromberg 1965, Chapter V, § 19, (19.61) Theorem)

References

• Zbl 0283.28001
• Zbl 0137.03202
• Zbl 0278.26001

This article incorporates material from Lebesgue decomposition theorem on

Creative Commons Attribution/Share-Alike License

.