Lebesgue's decomposition theorem

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 {displaystyle mu } and {displaystyle nu } on a measurable space {displaystyle (Omega ,Sigma ),} there exist two σ-finite signed measures {displaystyle nu _{0}} and {displaystyle nu _{1}} such that: {displaystyle nu =nu _{0}+nu _{1},} {displaystyle nu _{0}ll mu } (that is, {displaystyle nu _{0}} is absolutely continuous with respect to {displaystyle mu } ) {displaystyle nu _{1}perp mu } (that is, {displaystyle nu _{1}} and {displaystyle mu } are singular).

These two measures are uniquely determined by {displaystyle mu } and {displaystyle nu .} Contents 1 Refinement 2 Related concepts 2.1 Lévy–Itō decomposition 3 See also 4 Citations 5 References 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] {displaystyle ,nu =nu _{mathrm {cont} }+nu _{mathrm {sing} }+nu _{mathrm {pp} }} where νcont is the absolutely continuous part νsing is the singular continuous part νpp 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 Main article: Lévy–Itō decomposition The analogous[citation needed] 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 {displaystyle X=X^{(1)}+X^{(2)}+X^{(3)}} where: {displaystyle X^{(1)}} is a Brownian motion with drift, corresponding to the absolutely continuous part; {displaystyle X^{(2)}} is a compound Poisson process, corresponding to the pure point part; {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 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, 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 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 This article incorporates material from Lebesgue decomposition theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Categories: Integral calculusTheorems in measure theory