Inmathematics, more precisely inmeasure theory, theLebesgue decomposition theorem^[1] provides a way to decompose a measure into two distinct parts based on their relationship with another measure.

The theorem states that if${\displaystyle (\mathrm{\Omega },\mathrm{\Sigma })}$ is ameasurable space and${\displaystyle \mu }$ and${\displaystyle \nu }$ areσ-finitesigned measures on${\displaystyle \mathrm{\Sigma }}$, then there exist two uniquely determined σ-finite signed measures${\displaystyle {\nu }_0}$ and${\displaystyle {\nu }_1}$ such that:^[2][3]

• ${\displaystyle \nu ={\nu }_0+{\nu }_1}$
• ${\displaystyle {\nu }_0\ll \mu }$ (that is,${\displaystyle {\nu }_0}$ isabsolutely continuous with respect to${\displaystyle \mu }$)
• ${\displaystyle {\nu }_1\perp \mu }$ (that is,${\displaystyle {\nu }_1}$ and${\displaystyle \mu }$ aresingular).

Lebesgue's decomposition theorem can be refined in a number of ways. First, as theLebesgue–Radon–Nikodym theorem. That is, let${\displaystyle (\mathrm{\Omega },\mathrm{\Sigma })}$ be a measure space,${\displaystyle \mu }$ aσ-finitepositive measure on${\displaystyle \mathrm{\Sigma }}$ and${\displaystyle \lambda }$ acomplex measure on${\displaystyle \mathrm{\Sigma }}$.^[4]

• There is a unique pair of complex measures on${\displaystyle \mathrm{\Sigma }}$ such that$${\displaystyle \lambda ={\lambda }_a+{\lambda }_s,{\lambda }_a\ll \mu ,{\lambda }_s\perp \mu .}$$ If${\displaystyle \lambda }$ is positive and finite, then so are${\displaystyle {\lambda }_a}$ and${\displaystyle {\lambda }_s}$.
• There is a unique${\displaystyle h\in L^1(\mu )}$ such that$${\displaystyle {\lambda }_a(E)={\int }_{E}hd\mu ,E\in \mathrm{\Sigma }.}$$

The first assertion follows from the Lebesgue decomposition, the second is known as theRadon–Nikodym theorem. That is, the function${\displaystyle h}$ is a Radon–Nikodym derivative that can be expressed as$${\displaystyle h=\frac{d{\lambda }_a}{d\mu }.}$$

An alternative refinement is that of the decomposition of a regularBorel measure^[5][6][7]$${\displaystyle \nu ={\nu }_{ac}+{\nu }_{sc}+{\nu }_{pp},}$$where

• ${\displaystyle {\nu }_{ac}\ll \mu }$ is theabsolutely continuous part
• ${\displaystyle {\nu }_{sc}\perp \mu }$ is thesingular continuous part
• ${\displaystyle {\nu }_{pp}}$ is thepure point part (adiscrete measure).

The absolutely continuous measures are classified by theRadon–Nikodym theorem, and discrete measures are easily understood. Hence (singular continuous measures aside), Lebesgue decomposition gives a very explicit description of measures. TheCantor measure (theprobability measure on thereal line whosecumulative distribution function is theCantor function) is an example of a singular continuous measure.

The analogous^{[citation needed]} decomposition for astochastic processes is theLévy–Itō decomposition: given aLévy processX, it can be decomposed as a sum of three independentLévy processes${\displaystyle X=X^{(1)}+X^{(2)}+X^{(3)}}$ where:

• ${\displaystyle X^{(1)}}$ is aBrownian motion with drift, corresponding to the absolutely continuous part;
• ${\displaystyle X^{(2)}}$ is acompound Poisson process, corresponding to the pure point part;
• ${\displaystyle X^{(3)}}$ is asquare integrable pure jumpmartingale 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

• 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
• Reed, Michael; Simon, Barry (1981-01-11),I: Functional Analysis, San Diego, Calif.: Academic Press,ISBN 978-0-12-585050-6
• 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
• Simon, Barry (2005),Orthogonal polynomials on the unit circle. Part 1. Classical theory, American Mathematical Society Colloquium Publications, vol. 54, Providence, R.I.:American Mathematical Society,ISBN 978-0-8218-3446-6,MR 2105088
• Swartz, Charles (1994),Measure, Integration and Function Spaces, WORLD SCIENTIFIC,doi:10.1142/2223,ISBN 978-981-02-1610-8

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