Absolutely Continuous

A measure lambda is absolutely continuous with respect to another measure mu if lambda(E)=0 for every set with mu(E)=0. This makes sense as long as mu is a positive measure, such as Lebesgue measure, but lambda 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 f. The function f is like a derivative, and is called the Radon-Nikodym derivative dlambda/dmu.

The measure supported at 0 (mu(E)=1 iff 0 in E) is not absolutely continuous with respect to Lebesgue measure, and is a singular measure.

See also

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.

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