Lebesgue's decomposition theorem

In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem[1][2][3] states that for every two σ-finite signed measures ${\displaystyle \mu }$ and ${\displaystyle \nu }$ on a measurable space ${\displaystyle (\Omega ,\Sigma ),}$ there exist two σ-finite signed measures ${\displaystyle \nu _{0}}$ and ${\displaystyle \nu _{1}}$ such that:

• ${\displaystyle \nu =\nu _{0}+\nu _{1}\,}$
• ${\displaystyle \nu _{0}\ll \mu }$ (that is, ${\displaystyle \nu _{0}}$ is absolutely continuous with respect to ${\displaystyle \mu }$)
• ${\displaystyle \nu _{1}\perp \mu }$ (that is, ${\displaystyle \nu _{1}}$ and ${\displaystyle \mu }$ are singular).

These two measures are uniquely determined by ${\displaystyle \mu }$ and ${\displaystyle \nu .}$