Editing Absolute continuity (section)

===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.