Lebesgue decomposition theorem

Let $\mu$ and $\nu$ be two $\sigma$-finite signed measures in the measurable space $(\mathrm{\Omega },𝒮)$. 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. ${\nu }_1⟂\mu$ (i.e. ${\nu }_1$ and $\mu$ are singular.)

These two measures are uniquely determined.