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. 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.

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.