# Lebesgue's decomposition theorem Lebesgue's decomposition theorem In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem 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: {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.