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.

Refinement

Lebesgue's decomposition theorem can be refined in a number of ways.

First, the decomposition of the singular part can refined:

$\,\nu =\nu _{\mathrm {cont} }+\nu _{\mathrm {sing} }+\nu _{\mathrm {pp} }$

where

• $\nu _{\mathrm {cont} }$ is the absolutely continuous part
• $\nu _{\mathrm {sing} }$ is the singular continuous part
• $\nu _{\mathrm {pp} }$ is the pure point part (a discrete measure)

Second, absolutely continuous measures are classified by the Radon–Nikodym theorem, and discrete measures are easily understood. Hence (singular continuous measures aside), Lebesgue decomposition gives a very explicit description of measures.

See also

• Decomposition of spectrum

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Lebesgue decomposition theorem at PlanetMath.