Lebesgue decomposition theorem
Let $\mu $ and $\nu $ be two $\sigma $finite signed measures in the measurable space^{} $(\mathrm{\Omega},\mathcal{S})$. There exist two $\sigma $finite (http://planetmath.org/SigmaFinite) signed measures ${\nu}_{0}$ and ${\nu}_{1}$ such that:

1.
$\nu ={\nu}_{0}+{\nu}_{1}$;

2.
${\nu}_{0}\ll \mu $ (i.e. ${\nu}_{0}$ is absolutely continuous^{} with respect to $\mu $;)

3.
${\nu}_{1}\u27c2\mu $ (i.e. ${\nu}_{1}$ and $\mu $ are singular.)
These two measures^{} are uniquely determined.
Title  Lebesgue decomposition theorem 

Canonical name  LebesgueDecompositionTheorem 
Date of creation  20130322 13:26:28 
Last modified on  20130322 13:26:28 
Owner  Koro (127) 
Last modified by  Koro (127) 
Numerical id  8 
Author  Koro (127) 
Entry type  Theorem 
Classification  msc 28A12 