In mathematics, more precisely in measure theory, the Lebesgue decomposition theorem provides a way to decompose a measure into two distinct parts based on their relationship with another measure.
Definition
The theorem states that if ( Ω , Σ ) {\displaystyle (\Omega ,\Sigma )} is a measurable space and μ {\displaystyle \mu } and ν {\displaystyle \nu } are σ-finite signed measures on Σ {\displaystyle \Sigma } , then there exist two uniquely determined σ-finite signed measures ν 0 {\displaystyle \nu _{0}} and ν 1 {\displaystyle \nu _{1}} such that:23
- ν = ν 0 + ν 1 {\displaystyle \nu =\nu _{0}+\nu _{1}\,}
- ν 0 ≪ μ {\displaystyle \nu _{0}\ll \mu } (that is, ν 0 {\displaystyle \nu _{0}} is absolutely continuous with respect to μ {\displaystyle \mu } )
- ν 1 ⊥ μ {\displaystyle \nu _{1}\perp \mu } (that is, ν 1 {\displaystyle \nu _{1}} and μ {\displaystyle \mu } are singular).
Refinement
Lebesgue's decomposition theorem can be refined in a number of ways. First, as the Lebesgue-Radon-Nikodym theorem. That is, let ( Ω , Σ ) {\displaystyle (\Omega ,\Sigma )} be a measure space, μ {\displaystyle \mu } a σ-finite positive measure on Σ {\displaystyle \Sigma } and λ {\displaystyle \lambda } a complex measure on Σ {\displaystyle \Sigma } .4
- There is a unique pair of complex measures on Σ {\displaystyle \Sigma } such that λ = λ a + λ s , λ a ≪ μ , λ s ⊥ μ . {\displaystyle \lambda =\lambda _{a}+\lambda _{s},\quad \lambda _{a}\ll \mu ,\quad \lambda _{s}\perp \mu .} If λ {\displaystyle \lambda } is positive and finite, then so are λ a {\displaystyle \lambda _{a}} and λ s {\displaystyle \lambda _{s}} .
- There is a unique h ∈ L 1 ( μ ) {\displaystyle h\in L^{1}(\mu )} such that λ a ( E ) = ∫ E h d μ , ∀ E ∈ Σ . {\displaystyle \lambda _{a}(E)=\int _{E}hd\mu ,\quad \forall E\in \Sigma .}
The first assertion follows from the Lebesgue decomposition, the second is known as the Radon-Nikodym theorem. That is, the function h {\displaystyle h} is a Radon-Nikodym derivative that can be expressed as h = d λ a d μ . {\displaystyle h={\frac {d\lambda _{a}}{d\mu }}.}
An alternative refinement is that of the decomposition of a regular Borel measure567 ν = ν a c + ν s c + ν p p , {\displaystyle \nu =\nu _{ac}+\nu _{sc}+\nu _{pp},} where
- ν a c ≪ μ {\displaystyle \nu _{ac}\ll \mu } is the absolutely continuous part
- ν s c ⊥ μ {\displaystyle \nu _{sc}\perp \mu } is the singular continuous part
- ν p p {\displaystyle \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.
Related concepts
Lévy–Itō decomposition
Main article: 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 ) {\displaystyle X=X^{(1)}+X^{(2)}+X^{(3)}} where:
- X ( 1 ) {\displaystyle X^{(1)}} is a Brownian motion with drift, corresponding to the absolutely continuous part;
- X ( 2 ) {\displaystyle X^{(2)}} is a compound Poisson process, corresponding to the pure point part;
- X ( 3 ) {\displaystyle 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.
See also
- Decomposition of spectrum
- Hahn decomposition theorem and the corresponding Jordan decomposition theorem
- Spectrum (functional analysis) § Classification of points in the spectrum
- Spectral measure
Notes
- 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
This article incorporates material from Lebesgue decomposition theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
References
Hewitt & Stromberg 1965, Chapter V, § 19, (19.42) Lebesgue Decomposition Theorem. - 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 https://archive.org/details/realabstractanal00hewi_0 ↩
Halmos 1974, Section 32, Theorem C. - 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 https://archive.org/details/measuretheory00halm ↩
Swartz 1994, p. 141. - Swartz, Charles (1994), Measure, Integration and Function Spaces, WORLD SCIENTIFIC, doi:10.1142/2223, ISBN 978-981-02-1610-8 https://doi.org/10.1142%2F2223 ↩
Rudin 1974, Section 6.9, The Theorem of Lebesgue-Radon-Nikodym. - 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 https://archive.org/details/realcomplexanaly00rudi_0 ↩
Hewitt & Stromberg 1965, Chapter V, § 19, (19.61) Theorem. - 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 https://archive.org/details/realabstractanal00hewi_0 ↩
Reed & Simon 1981, pp. 22–25. - Reed, Michael; Simon, Barry (1981-01-11), I: Functional Analysis, San Diego, Calif.: Academic Press, ISBN 978-0-12-585050-6 ↩
Simon 2005, p. 43. - 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 https://mathscinet.ams.org/mathscinet-getitem?mr=2105088 ↩