Lebesgue's decomposition theorem

In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem is a theorem which states that given $μ$ and $ν$ two σ-finite signed measures on a measurable space $(Ω,Σ),$ there exist two σ-finite signed measures $ν 0$ and $ν 1$ such that:

These two measures are uniquely determined.

[edit] Refinement

Lebesgue's decomposition theorem can be refined in a number of ways.

First, the decomposition of the singular part can refined:

$\, \nu = \nu_{\mathrm{cont}} + \nu_{\mathrm{sing}} + \nu_{\mathrm{pp}}$

where

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.