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Radon-Nikodym Theorem


The Radon-Nikodym theorem asserts that any absolutely continuous complex measure lambda with respect to some positive measure mu (which could be Lebesgue measure or Haar measure) is given by the integral of some L^1(mu)-function f,

 lambda(E)=int_Efdmu.
(1)

The function f is like a density function for the measure.

A closely related theorem says that any complex measure lambda decomposes into an absolutely continuous measure lambda_a and a singular measure lambda_c. This is the Lebesgue decomposition,

 lambda=lambda_a+lambda_c.
(2)

One consequence of the Radon-Nikodym theorem is that any complex measure has a polar representation

 dmu=hd|mu|,
(3)

with |h|=1.


See also

Absolutely Continuous, Complex Measure, Haar Measure, Lebesgue Decomposition, Lebesgue Measure, Polar Representation, Singular Measure

This entry contributed by Todd Rowland

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References

Nagy, G. "Radon-Nikodym Theorems." §4.4 in Real Analysis. pp. 300-321. http://www.math.ksu.edu/~nagy/real-an/.Rudin, W. Real and Complex Analysis. New York: McGraw-Hill, pp. 116-134, 1986.

Referenced on Wolfram|Alpha

Radon-Nikodym Theorem

Cite this as:

Rowland, Todd. "Radon-Nikodym Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Radon-NikodymTheorem.html

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