Lebesgue's decomposition theorem
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In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem is a theorem which states that given <math>\mu</math> and <math>\nu</math> two σ-finite signed measures in a measurable space <math>(\Omega,\Sigma),</math> there exist two σ-finite signed measures <math>\nu_0</math> and <math>\nu_1</math> such that:
- <math>\nu=\nu_0+\nu_1\, </math>
- <math>\nu_0\ll\mu</math> (that is, <math>\nu_0</math> is absolutely continuous with respect to <math>\mu</math>)
- <math>\nu_1\perp\mu</math> (that is, <math>\nu_1</math> and <math>\mu</math> are singular).
These two measures are uniquely determined.
This article incorporates material from Lebesgue decomposition theorem on PlanetMath, which is licensed under the GFDL.
