Radon-Nikodym Derivative

The Radon-Nikodym derivative of a positive measure or a complex measure with respect to a positive measure is a measurable function satisfying

for every measurable set . For a positive measure , can be chosen nonnegative. For a complex measure , satisfies . The Radon-Nikodym theorem guarantees such an , unique -almost everywhere, when is sigma-finite, is absolutely continuous with respect to , and is either a sigma-finite positive measure or a complex measure (Hunter 2011, pp. 74-78).

Here, sigma-finiteness means that the measure space is a countable union of measurable sets of finite measure. By analogy with the first fundamental theorem of calculus, is denoted or .