Radon–Nikodym theorem

The Radon–Nikodym theorem answers a deceptively simple question: when can one measure be expressed as the other ‘weighted by a density’? If a measure nu never assigns mass where a reference measure mu sees nothing, the theorem promises a single function — a density — telling you, point by point, how much nu-stuff sits per unit of mu-stuff. It is the rigorous foundation for probability densities, change of variables, and conditional expectation.

Precisely: let mu be sigma-finite and let nu be a sigma-finite (or signed) measure that is absolutely continuous with respect to mu, nu << mu. Then there exists a measurable function f, non-negative when nu is a measure, such that nu(E) = the integral over E of f dmu for every measurable set E. This f is unique up to equality almost everywhere with respect to mu and is called the Radon–Nikodym derivative, written dnu/dmu.

The hypotheses matter. Absolute continuity is necessary — a point mass cannot be written as a density against Lebesgue measure — and sigma-finiteness cannot simply be dropped: against the counting measure on an uncountable set, an absolutely continuous Lebesgue-type measure need not have a density. When all hypotheses hold, the derivative behaves like an ordinary derivative under composition: a chain rule d(lambda)/d(mu) = (d lambda/d nu)(d nu/d mu) holds almost everywhere.