Lebesgue's decomposition theorem

In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem^[1]^[2]^[3] states that for every two σ-finite signed measures $\mu$ and $\nu$ on a measurable space $(\Omega ,\Sigma ),$ there exist two σ-finite signed measures $\nu _{0}$ and $\nu _{1}$ such that:

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

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

Refinement[edit]

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:^[4]

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

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.

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

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

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

Halmos, Paul R. (1974) [1950], Measure Theory, Graduate Texts in Mathematics, vol. 18, New York, Heidelberg, Berlin: Springer-Verlag, ISBN 978-0-387-90088-9, MR 0033869, Zbl 0283.28001
Hewitt, Edwin; Stromberg, Karl (1965), Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable, Graduate Texts in Mathematics, vol. 25, Berlin, Heidelberg, New York: Springer-Verlag, ISBN 978-0-387-90138-1, MR 0188387, Zbl 0137.03202
Rudin, Walter (1974), Real and Complex Analysis, McGraw-Hill Series in Higher Mathematics (2nd ed.), New York, Düsseldorf, Johannesburg: McGraw-Hill Book Comp., ISBN 0-07-054233-3, MR 0344043, Zbl 0278.26001

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