The Radon-Nikodym theorem asserts that any absolutely continuous complex measure with respect to some positive measure (which could be Lebesgue measure or Haar measure) is given by the integral of some -function ,
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The function is like a density function for the measure.
A closely related theorem says that any complex measure decomposes into an absolutely continuous measure and a singular measure . This is the Lebesgue decomposition,
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One consequence of the Radon-Nikodym theorem is that any complex measure has a polar representation
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with .