Lebesgue's decomposition theorem

In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem 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:

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