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Lebesgue Decomposition

The Lebesgue decomposition of a complex measure lambda, with respect to a positive measure mu on the same measurable space, is its decomposition into an absolutely continuous measure lambda_a and a singular measure lambda_c,

lambda=lambda_a+lambda_c,

where both absolute continuity and singularity are with respect to mu.

See also

Absolutely Continuous, Complex Measure, Fundamental Theorems of Calculus, Lebesgue Measure, Polar Representation, Radon-Nikodym Theorem, Singular Measure

This entry contributed by Todd Rowland

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Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland. "Lebesgue Decomposition." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LebesgueDecomposition.html

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