A measure λ is absolutely continuous with respect to another measure μ if λ(E) = 0 for every set with μ(E) = 0. This makes sense as long as μ is a positive measure, such as Lebesgue measure, but λ can be any measure, possibly a complex measure. By the Radon-Nikodym theorem, this is equivalent to saying that λ(E) = integral_E f d μ, 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 d λ/d μ.

complex measure | concentrated | Haar measure | Lebesgue decomposition | Lebesgue measure | mutually singular | singular measure