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| In [[mathematics]], more precisely in [[measure theory]], '''Lebesgue's decomposition theorem'''<ref>{{harv|Halmos|1974|loc=Section 32, Theorem C}}</ref><ref>{{harv|Hewitt|Stromberg|1965|loc=Chapter V, § 19, (19.42) Lebesque Decomposition Theorem}}</ref><ref>{{harv|Rudin|1974|loc=Section 6.9, The Theorem of Lebesgue-Radon-Nikodym}}</ref> states that for every two [[sigma-finite measure|σ-finite]] [[signed measure]]s <math>\mu</math> and <math>\nu</math> on a [[measurable space]] <math>(\Omega,\Sigma),</math> there exist two σ-finite signed measures <math>\nu_0</math> and <math>\nu_1</math> such that:
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| * <math>\nu=\nu_0+\nu_1\, </math>
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| * <math>\nu_0\ll\mu</math> (that is, <math>\nu_0</math> is [[absolutely continuous]] with respect to <math>\mu</math>)
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| * <math>\nu_1\perp\mu</math> (that is, <math>\nu_1</math> and <math>\mu</math> are [[singular measure|singular]]).
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| These two measures are uniquely determined by <math>\mu</math> and <math>\nu</math>.
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| ==Refinement==
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| Lebesgue's decomposition theorem can be refined in a number of ways.
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| First, the decomposition of the [[singular measure|singular]] part of a regular [[Borel measure]] on the [[real line]] can be refined:<ref>{{harv|Hewitt|Stromberg|1965|loc=Chapter V, § 19, (19.61) Theorem}}</ref>
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| :<math>\, \nu = \nu_{\mathrm{cont}} + \nu_{\mathrm{sing}} + \nu_{\mathrm{pp}}</math>
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| where | |
| * ''ν''<sub>cont</sub> is the '''absolutely continuous''' part
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| * ''ν''<sub>sing</sub> is the '''singular continuous''' part
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| * ''ν''<sub>pp</sub> is the '''pure point''' part (a [[discrete measure]]).
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| 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.
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| == Related concepts ==
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| === Lévy–Itō decomposition ===
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| {{main|Lévy–Itō decomposition}}
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| 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 process|Lévy processes]] <math>X=X^{(1)}+X^{(2)}+X^{(3)}</math> where: | |
| * <math>X^{(1)}</math> is a [[Brownian motion]] with drift, corresponding to the absolutely continuous part;
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| * <math>X^{(2)}</math> is a [[compound Poisson process]], corresponding to the pure point part;
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| * <math>X^{(3)}</math> is a [[square integrable]] pure jump [[Martingale (probability theory)|martingale]] that almost surely has a countable number of jumps on a finite interval, corresponding to the singular continuous part.
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| ==See also==
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| * [[Decomposition_of_spectrum_(functional_analysis)|Decomposition of spectrum]]
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| * [[Hahn decomposition theorem]] and the corresponding Jordan decomposition theorem
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| ==Citations==
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| {{Reflist|2}}
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| ==References==
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| * {{Citation
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| | last = Halmos
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| | first = Paul R.
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| | author-link = Paul Halmos
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| | title = Measure Theory
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| | place = New York, Heidelberg, Berlin | |
| | publisher = Springer-Verlag
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| | series = Graduate Texts in Mathematics
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| | volume = 18
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| | origyear = 1950
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| | year = 1974
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| | isbn = 978-0-387-90088-9
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| | mr = 0033869
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| | zbl = 0283.28001}}
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| * {{Citation
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| | last = Hewitt
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| | first = Edwin
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| | author-link = Edwin Hewitt
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| | last2 = Stromberg
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| | first2 = Karl
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| | title = Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable
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| | place = Berlin, Heidelberg, New York
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| | publisher = Springer-Verlag
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| | series = Graduate Texts in Mathematics
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| | volume = 25
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| | year = 1965
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| | isbn = 978-0-387-90138-1
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| | mr = 0188387
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| | zbl = 0137.03202}}
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| * {{Citation
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| | last = Rudin
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| | first = Walter
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| | author-link = Walter Rudin
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| | title = Real and Complex Analysis
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| | place = New York, Düsseldorf, Johannesburg
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| | publisher = McGraw-Hill Book Comp.
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| | series = McGraw-Hill Series in Higher Mathematics
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| | edition = 2nd
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| | year = 1974
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| | isbn = 0-07-054233-3
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| | mr = 0344043
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| | zbl = 0278.26001}}
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| ==External links==
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| * [http://www.encyclopediaofmath.org/index.php/Lebesgue_decomposition Lebesgue decomposition] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| * [http://www.encyclopediaofmath.org/index.php/Absolutely_continuous_measures Absolutely continuous measures] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| {{PlanetMath attribution|id=4003|title=Lebesgue decomposition theorem}}
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| [[Category:Integral calculus]]
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| [[Category:Theorems in measure theory]]
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