The Radon-Nikodym theorem asserts that any absolutely continuous complex measure λ with respect to some positive measure μ (which could be Lebesgue measure or Haar measure) is given by the integral of some L^1(μ)-function f, λ(E) = integral_E f d μ. The function f is like a density function for the measure. A closely related theorem says that any complex measure λ decomposes into an absolutely continuous measure λ_a and a singular measure λ_c. This is the Lebesgue decomposition, λ = λ_a + λ_c.

absolutely continuous | complex measure | Haar measure | Lebesgue decomposition | Lebesgue measure | polar representation | singular measure

Johann Radon