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Lebesgue's decomposition theorem

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In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem 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 ν 0 and ν 1 such that:

Contents

  • ν = ν 0 + ν 1
  • ν 0 ≪ μ (that is, ν 0 is absolutely continuous with respect to μ )
  • ν 1 ⊥ μ (that is, ν 1 and μ are singular).

    • 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 the singular part of a regular Borel measure on the real line can be refined:

    ν = ν c o n t + ν s i n g + ν p p

    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.

    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 ) where:

  • X ( 1 ) is a Brownian motion with drift, corresponding to the absolutely continuous part;
  • X ( 2 ) is a compound Poisson process, corresponding to the pure point part;
  • 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.

    • References

    Lebesgue's decomposition theorem Wikipedia
    (Text) CC BY-SA


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