Lebesgue's decomposition theorem

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In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem[1][2][3] states that for every two σ-finite signed measures μ and ν on a measurable space (Ω,Σ), there exist two σ-finite signed measures ν0 and ν1 such that:

These two measures are uniquely determined by μ and ν.

Refinement

Lebesgue's decomposition theorem can be refined in a number of ways. First, the decomposition of a regular Borel measure on the real line can be refined:[4]

ν=νcont+νsing+νpp

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

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 X=X(1)+X(2)+X(3) where:

See also

Citations

  1. (Halmos 1974, Section 32, Theorem C)
  2. (Hewitt & Stromberg 1965, Chapter V, § 19, (19.42) Lebesgue Decomposition Theorem)
  3. (Rudin 1974, Section 6.9, The Theorem of Lebesgue-Radon-Nikodym)
  4. (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

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