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

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:^[4]

\,\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 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.

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

Citations

^ (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)

References

Halmos, Paul R. (1974) [1950], Measure Theory, Graduate Texts in Mathematics, 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, 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

[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)

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