Editing Absolute continuity (section)

==Absolute continuity of measures==

===Definition===
A [[Measure (mathematics)|measure]] <math>\mu</math> on [[Borel set|Borel subsets]] of the real line is absolutely continuous with respect to the [[Lebesgue measure]] <math>\lambda</math> if for every <math>\lambda</math>-measurable set <math>A,</math> <math>\lambda(A) = 0</math> implies <math>\mu(A) = 0</math>. Equivalently,  <math>\mu(A) > 0</math> implies <math>\lambda(A) > 0</math>. This condition is written as <math>\mu \ll \lambda.</math> We say <math>\mu</math> is ''dominated'' by <math>\lambda.</math>

In most applications, if a measure on the real line is simply said to be absolutely continuous — without specifying with respect to which other measure it is absolutely continuous — then absolute continuity with respect to the Lebesgue measure is meant.

The same principle holds for measures on Borel subsets of <math>\mathbb{R}^n, n \geq 2.</math>

===Equivalent definitions===
The following conditions on a finite measure <math>\mu</math> on Borel subsets of the real line are equivalent:<ref>Equivalence between (1) and (2) is a special case of {{harvnb|Nielsen|1997|loc=Proposition 15.5 on page 251}} (fails for σ-finite measures); equivalence between (1) and (3) is a special case of the [[Radon–Nikodym theorem]], see {{harvnb|Nielsen|1997|loc=Theorem 15.4 on page 251}} or {{harvnb|Athreya|Lahiri|2006|loc=Item (ii) of Theorem 4.1.1 on page 115}} (still holds for σ-finite measures).</ref>

# <math>\mu</math> is absolutely continuous;
# For every positive number <math>\varepsilon</math> there is a positive number <math>\delta > 0</math> such that <math>\mu(A) < \varepsilon</math> for all Borel sets <math>A</math> of Lebesgue measure less than <math>\delta;</math>
# There exists a Lebesgue integrable function <math>g</math> on the real line such that: <math display="block">\mu(A) = \int_A g \,d\lambda</math> for all Borel subsets <math>A</math> of the real line.

For an equivalent definition in terms of functions see the section [[#Relation between the two notions of absolute continuity|Relation between the two notions of absolute continuity]].

Any other function satisfying (3) is equal to <math>g</math> almost everywhere. Such a function is called [[Radon–Nikodym derivative]], or density, of the absolutely continuous measure <math>\mu.</math>

Equivalence between (1), (2) and (3) holds also in <math>\R^n</math> for all <math>n = 1, 2, 3, \ldots.</math>

Thus, the absolutely continuous measures on <math>\R^n</math> are precisely those that have densities; as a special case, the absolutely continuous probability measures are precisely the ones that have [[probability density function]]s.

===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>

===Singular measures===
Via [[Lebesgue's decomposition theorem]],<ref>{{harvnb|Royden|1988|loc=Proposition 11.24 on page 278}}; {{harvnb|Nielsen|1997|loc=Theorem 15.14 on page 262}}; {{harvnb|Athreya|Lahiri|2006|loc=Item (i) of Theorem 4.1.1 on page 115}}.</ref> every σ-finite measure can be decomposed into the sum of an absolutely continuous measure and a singular measure with respect to another σ-finite measure. See [[singular measure]] for examples of measures that are not absolutely continuous.