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

These two measures are uniquely determined.

Type of Math Object:

Theorem

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Reference

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