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.
Formal Statement 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:
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 } .
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 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 measure
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
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