Radon–Nikodym deri
Radon–Nikodym derivativ
The function f satisfying the above equality is uniquely defined up to a μ-null set, that is, if g is another function which satisfies the same property, then f = g μ-almost everywhere. f is commonly written 
and is called the Radon–Nikodym derivative. The choice of notation and the name of the function reflects the fact that the function is analogous to a derivative in calculus in the sense that it describes the rate of change of density of one measure with respect to another (the way the Jacobian determinant is used in multivariable integration). A similar theorem can be proven for signed and complex measures: namely, that if μ is a nonnegative σ-finite measure, and ν is a finite-valued signed or complex measure such that ν ≪ μ, i.e. ν is absolutely continuous with respect to μ, then there is a μ-integrable real- or complex-valued function g on X such that for every measurable set A,
