Editing Radon–Nikodym theorem (section)

==Formal description==
===Radon–Nikodym theorem===
The '''Radon–Nikodym theorem''' involves a [[measurable space]] <math>(X, \Sigma)</math> on which two [[σ-finite measure]]s are defined, <math>\mu</math> and <math>\nu.</math> 
It states that, if <math>\nu \ll \mu</math> (that is, if <math>\nu</math> is [[Absolute continuity#Absolute continuity of measures|absolutely continuous]] with respect to <math>\mu</math>), then there exists a <math>\Sigma</math>-[[measurable function]] <math>f : X \to [0, \infty),</math> such that for any measurable set <math>A \in \Sigma,</math>
<math display=block>\nu(A) = \int_A f \, d\mu.</math>

===Radon–Nikodym derivative===
The function <math>f</math> satisfying the above equality is {{em|uniquely defined [[up to]] a <math>\mu</math>-[[null set]]}}, that is, if <math>g</math> is another function which satisfies the same property, then <math>f = g</math> {{nowrap|<math>\mu</math>-[[almost everywhere]]}}. The function <math>f</math> is commonly written <math display>\frac{d\nu}{d\mu}</math> and is called the '''{{visible anchor|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).

===Extension to signed or complex measures===
A similar theorem can be proven for [[Signed measure|signed]] and [[complex measure]]s: namely, that if <math>\mu</math> is a nonnegative σ-finite measure, and <math>\nu</math> is a finite-valued signed or complex measure such that <math>\nu \ll \mu,</math> that is, <math>\nu</math> is [[absolutely continuous]] with respect to <math>\mu,</math> then there is a <math>\mu</math>-integrable real- or complex-valued function <math>g</math> on <math>X</math> such that for every measurable set <math>A,</math>
<math display=block>\nu(A) = \int_A g \, d\mu.</math>