Editing Absolute continuity (section)

===Generalizations===
If <math>\mu</math> and <math>\nu</math> are two [[Measure (mathematics)|measure]]s on the same [[measurable space]] <math>(X, \mathcal{A}),</math> <math>\mu</math> is said to be '''{{visible anchor|Absolutely continuous measure|text=absolutely continuous}} with respect to <math>\nu</math>''' if <math>\mu(A) = 0</math> for every set <math>A</math> for which <math>\nu(A) = 0.</math><ref>{{harvnb|Nielsen|1997|loc=Definition 15.3 on page 250}}; {{harvnb|Royden|1988|loc=Sect. 11.6, page 276}}; {{harvnb|Athreya|Lahiri|2006|loc=Definition 4.1.1 on page 113}}.</ref> This is written as "<math>\mu\ll\nu</math>". That is:
<math display=block>\mu \ll \nu \qquad \text{ if and only if } \qquad \text{ for all } A\in\mathcal{A}, \quad (\nu(A) = 0\ \text{ implies } \ \mu (A) = 0).</math>

When <math>\mu\ll\nu,</math> then <math>\nu</math> is said to be '''{{visible anchor|Domination (measure theory)|text=dominating}}''' <math>\mu.</math>

Absolute continuity of measures is [[Reflexive relation|reflexive]] and [[Transitive relation|transitive]], but is not [[Antisymmetric relation|antisymmetric]], so it is a [[preorder]] rather than a [[partial order]]. Instead, if <math>\mu \ll \nu</math> and <math>\nu \ll \mu,</math> the measures <math>\mu</math> and <math>\nu</math> are said to be [[Equivalence (measure theory)|equivalent]].  Thus absolute continuity induces a partial ordering of such [[equivalence class]]es.

If <math>\mu</math> is a [[Signed measure|signed]] or [[complex measure]], it is said that <math>\mu</math> is absolutely continuous with respect to <math>\nu</math> if its variation <math>|\mu|</math> satisfies <math>|\mu| \ll \nu;</math> equivalently, if every set <math>A</math> for which <math>\nu(A) = 0</math> is <math>\mu</math>-[[Null set|null]].

The [[Radon–Nikodym theorem]]<ref>{{harvnb|Royden|1988|loc=Theorem 11.23 on page 276}}; {{harvnb|Nielsen|1997|loc=Theorem 15.4 on page 251}}; {{harvnb|Athreya|Lahiri|2006|loc=Item (ii) of Theorem 4.1.1 on page 115}}.</ref> states that if <math>\mu</math> is absolutely continuous with respect to <math>\nu,</math> and both measures are [[σ-finite]], then <math>\mu</math> has a density, or "Radon-Nikodym derivative", with respect to <math>\nu,</math> which means that there exists a <math>\nu</math>-measurable function <math>f</math> taking values in <math>[0, +\infty),</math> denoted by <math>f = d\mu / d\nu,</math> such that for any <math>\nu</math>-measurable set <math>A</math> we have:
<math display=block>\mu(A) = \int_A f \,d\nu.</math>