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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \, \nu = \nu_{\mathrm{cont}} + \nu_{\mathrm{sing}} + \nu_{\mathrm{pp}}}

where

• μ_cont is the absolutely continuous part
• μ_sing is the singular continuous part
• μ_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. The Cantor measure (the probability measure on the real line whose cumulative distribution function is the Cantor function) is an example of a singular continuous measure.

See also

• Decomposition of spectrum

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Lebesgue decomposition theorem at PlanetMath.