Lebesgue's decomposition theorem

In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem is a theorem which states that given $\mu$ and $\nu$ two σ-finite signed measures in a measurable space $(\Omega ,\Sigma ),$ there exist two σ-finite signed measures $\nu _{0}$ and $\nu _{1}$ such that:

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

These two measures are uniquely determined.

3 Decompositions

Hahn decomposes the Ω into one positive set P and one negative set N, the decoposition is unique. Hence, any nonnull set can be decomposed into one positive set and one negative set (A null set is both positive and negative.). Jordan decopses the a signed measure into the difference of two measures. The decomposition is unique for any set. Hahn decopisiotn and Jordan decomposition is related.Hahn decomposition theorem

While Lebesgue decomposition decomposes a signed measure $\nu$, with respect to $\mu$, into the sum of two signed measures: one absolutely continuous to $\mu$, another one is sigular to $\mu$.

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Lebesgue decomposition theorem at PlanetMath.