Lebesgue's decomposition theorem

The theorem states that if $(\backslash Omega,\backslash Sigma)$ is a measurable space and $\backslash mu$ and $\backslash nu$

are σ-finite signed measures on $\backslash Sigma$, then there exist two uniquely determined σ-finite signed measures $\backslash nu\_0$ and $\backslash nu\_1$ such that:{{sfn|Halmos|1974|loc=Section 32, Theorem C}}{{sfn|Swartz|1994|p=141}}

• $\backslash nu=\backslash nu\_0+\backslash nu\_1\backslash ,$
• $\backslash nu\_0\backslash ll\backslash mu$ (that is, $\backslash nu\_0$ is absolutely continuous with respect to $\backslash mu$)
• $\backslash nu\_1\backslash perp\backslash mu$ (that is, $\backslash nu\_1$ and $\backslash mu$ are singular).

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

First, as the Lebesgue-Radon-Nikodym theorem. That is, let

$(\backslash Omega,\backslash Sigma)$ be a measure space, $\backslash mu$ a σ-finite positive measure on $\backslash Sigma$ and $\backslash lambda$ a complex measure on $\backslash Sigma$.{{sfn|Rudin|1974|loc=Section 6.9, The Theorem of Lebesgue-Radon-Nikodym}}

• There is a unique pair of complex measures on $\backslash Sigma$ such that $$\backslash lambda\; =\; \backslash lambda\_a\; +\; \backslash lambda\_s,\; \backslash quad\; \backslash lambda\_a\; \backslash ll\; \backslash mu,\; \backslash quad\; \backslash lambda\_s\; \backslash perp\; \backslash mu.$$ If $\backslash lambda$ is positive and finite, then so are $\backslash lambda\_a$ and $\backslash lambda\_s$.
• There is a unique $h\; \backslash in\; L^1(\backslash mu)$ such that $$\backslash lambda\_a\; (E)\; =\; \backslash int\_E\; h\; d\backslash mu,\; \backslash quad\; \backslash forall\; E\; \backslash in\; \backslash Sigma.$$

The first assertion follows from the Lebesgue decomposition, the second is known as the Radon-Nikodym theorem. That is, the function $h$ is a Radon-Nikodym derivative that can be expressed as

$$h\; =\; \backslash frac\{d\backslash lambda\_a\}\{d\backslash mu\}.$$

An alternative refinement is that of the decomposition of a regular Borel measure{{sfn|Hewitt|Stromberg|1965|loc=Chapter V, § 19, (19.61) Theorem}}{{sfn|Reed|Simon|1981|pp=22-25}}{{sfn|Simon|2005|p=43}}

$$\backslash nu\; =\; \backslash nu\_\{ac\}\; +\; \backslash nu\_\{sc\}\; +\; \backslash nu\_\{pp\},$$

where

• $\backslash nu\_\{ac\}\; \backslash ll\; \backslash mu$ is the absolutely continuous part
• $\backslash nu\_\{sc\}\; \backslash perp\; \backslash mu$ is the singular continuous part
• $\backslash nu\_\{pp\}$ is the pure point part (a discrete measure).

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.

{{main|Lévy–Itō decomposition}}

The analogous{{citation needed|date=January 2017}} 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.

• Decomposition of spectrum
• Hahn decomposition theorem and the corresponding Jordan decomposition theorem
• Spectrum (functional analysis) § Classification of points in the spectrum
• Spectral measure

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{{PlanetMath attribution|id=4003|title=Lebesgue decomposition theorem}}

Category:Integral calculus

Category:Theorems in measure theory