# Lebesgue decomposition theorem

Let $\mu$ and $\nu$ be two $\sigma$-finite signed measures in the measurable space $(\Omega,\mathscr{S})$. There exist two $\sigma$-finite (http://planetmath.org/SigmaFinite) signed measures $\nu_{0}$ and $\nu_{1}$ such that:

1. 1.

$\nu=\nu_{0}+\nu_{1}$;

2. 2.

$\nu_{0}\ll\mu$ (i.e. $\nu_{0}$ is absolutely continuous with respect to $\mu$;)

3. 3.

$\nu_{1}\perp\mu$ (i.e. $\nu_{1}$ and $\mu$ are singular.)

These two measures are uniquely determined.

Title Lebesgue decomposition theorem LebesgueDecompositionTheorem 2013-03-22 13:26:28 2013-03-22 13:26:28 Koro (127) Koro (127) 8 Koro (127) Theorem msc 28A12