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# Absolutely Continuous

A measure is absolutely continuous with respect to another measure if for every set with . This makes sense as long as is a positive measure, such as Lebesgue measure, but can be any measure, possibly a complex measure.

By the Radon-Nikodym theorem, this is equivalent to saying that

where the integral is the Lebesgue integral, for some integrable function . The function is like a derivative, and is called the Radon-Nikodym derivative .

The measure supported at 0 ( iff ) is not absolutely continuous with respect to Lebesgue measure, and is a singular measure.

Complex Measure, Concentrated, Haar Measure, Lebesgue Decomposition, Lebesgue Measure, Mutually Singular, Polar Representation, Singular Measure

This entry contributed by Todd Rowland

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## Cite this as:

Rowland, Todd. "Absolutely Continuous." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/AbsolutelyContinuous.html