View source for Absolute continuity

{{Short description|Form of continuity for functions}}
In [[calculus]] and [[real analysis]], '''absolute continuity''' is a [[smoothness (mathematics)|smoothness]] property of [[function (mathematics)|function]]s that is stronger than [[continuous function|continuity]] and [[uniform continuity]]. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of [[calculus]]—[[derivative|differentiation]] and [[integral|integration]]. This relationship is commonly characterized (by the [[fundamental theorem of calculus]]) in the framework of [[Riemann integration]], but with absolute continuity it may be formulated in terms of [[Lebesgue integration]]. For real-valued functions on the [[real line]], two interrelated notions appear: '''absolute continuity of functions''' and '''absolute continuity of measures'''. These two notions are generalized in different directions. The usual derivative of a function is related to the ''[[Radon–Nikodym derivative]]'', or ''density'', of a measure. We have the following chains of inclusions for functions '''over a [[compact space|compact]] subset''' of the real line:

: ''absolutely continuous'' ⊆ ''[[uniformly continuous]]'' <math>=</math> ''[[Continuous function|continuous]]''<!--All continuous functions on a compact domain are uniformly continuous-->

and, for a compact interval,

: ''[[continuously differentiable]]'' ⊆ ''[[Lipschitz continuous]]'' ⊆ ''absolutely continuous'' ⊆ ''[[bounded variation]]''  ⊆ ''[[Differentiable function|differentiable]] [[almost everywhere]]''.

==Absolute continuity of functions==

A continuous function fails to be absolutely continuous if it fails to be [[uniformly continuous]], which can happen if the domain of the function is not compact – examples are tan(''x'') over {{closed-open|0, ''π''/2}}, ''x''<sup>2</sup> over the entire real line, and sin(1/''x'') over (0, 1]. But a continuous function ''f'' can fail to be absolutely continuous even on a compact interval. It may not be "differentiable almost everywhere" (like the [[Weierstrass function]], which is not differentiable anywhere). Or it may be [[Differentiable function|differentiable]] almost everywhere and its derivative ''f''&nbsp;′ may be [[Lebesgue integration|Lebesgue integrable]], but the integral of ''f''&nbsp;′ differs from the increment of ''f'' (how much ''f'' changes over an interval). This happens for example with the [[Cantor function]].

===Definition===
Let <math>I</math> be an [[Interval (mathematics)|interval]] in the [[real line]] <math>\R</math>. A function <math>f\colon I \to \R</math> is '''absolutely continuous''' on <math>I</math> if for every positive number <math>\varepsilon</math>, there is a positive number <math>\delta</math> such that whenever a finite sequence of [[pairwise disjoint]] sub-intervals <math>(x_k, y_k)</math> of <math>I</math> with <math>x_k < y_k</math> satisfies<ref>{{harvnb|Royden|1988|loc=Sect. 5.4, page 108}}; {{harvnb|Nielsen|1997|loc=Definition 15.6 on page 251}}; {{harvnb|Athreya|Lahiri|2006|loc=Definitions 4.4.1, 4.4.2 on pages 128,129}}. The interval <math>I</math> is assumed to be bounded and closed in the former two books but not the latter book.</ref>
:<math>\sum_{k=1}^{N} (y_k - x_k) < \delta </math>
then
:<math> \sum_{k=1}^{N} | f(y_k) - f(x_k) | < \varepsilon.</math>

The collection of all absolutely continuous functions on <math>I</math> is denoted <math>\operatorname{AC}(I)</math>.

===Equivalent definitions===

The following conditions on a real-valued function ''f'' on a compact interval [''a'',''b''] are equivalent:<ref>{{harvnb|Nielsen|1997|loc=Theorem 20.8 on page 354}}; also {{harvnb|Royden|1988|loc=Sect. 5.4, page 110}} and {{harvnb|Athreya|Lahiri|2006|loc=Theorems 4.4.1, 4.4.2 on pages 129,130}}.</ref>

# ''f'' is absolutely continuous;
# ''f'' has a derivative ''f''&nbsp;′ [[almost everywhere]], the derivative is Lebesgue integrable, and <math display="block"> f(x) = f(a) + \int_a^x f'(t) \, dt </math> for all ''x'' on [''a'',''b''];
# there exists a Lebesgue integrable function ''g'' on [''a'',''b''] such that <math display="block"> f(x) = f(a) + \int_a^x g(t) \, dt </math> for all ''x'' in [''a'',''b''].

If these equivalent conditions are satisfied, then necessarily any function ''g'' as in condition 3. satisfies ''g'' = ''f''&nbsp;′ almost everywhere.

Equivalence between (1) and (3) is known as the '''fundamental theorem of Lebesgue integral calculus''', due to [[Lebesgue]].<ref>{{harvnb|Athreya|Lahiri|2006|loc=before Theorem 4.4.1 on page 129}}.</ref>

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

===Properties===
* The sum and difference of two absolutely continuous functions are also absolutely continuous. If the two functions are defined on a bounded closed interval, then their product is also absolutely continuous.<ref>{{harvnb |Royden|1988|loc=Problem 5.14(a,b) on page 111}}.</ref>
* If an absolutely continuous function is defined on a bounded closed interval and is nowhere zero then its reciprocal is absolutely continuous.<ref>{{harvnb |Royden|1988|loc=Problem 5.14(c) on page 111}}.</ref>
* Every absolutely continuous function (over a compact interval) is [[uniform continuity|uniformly continuous]] and, therefore, [[Continuous function|continuous]]. Every (globally) [[Lipschitz continuity|Lipschitz-continuous]] [[function (mathematics)|function]] is absolutely continuous.<ref>{{harvnb |Royden|1988|loc=Problem 5.20(a) on page 112}}.</ref>
* If ''f'': [''a'',''b''] → '''R''' is absolutely continuous, then it is of [[bounded variation]] on [''a'',''b''].<ref>{{harvnb|Royden|1988|loc=Lemma 5.11 on page 108}}.</ref>
*  If ''f'': [''a'',''b''] → '''R''' is absolutely continuous, then it can be written as the difference of two monotonic nondecreasing absolutely continuous functions on [''a'',''b''].
* If ''f'': [''a'',''b''] → '''R''' is absolutely continuous, then it has the [[Luzin N property|Luzin ''N'' property]] (that is, for any <math>N \subseteq [a,b]</math> such that <math>\lambda(N) = 0</math>, it holds that <math>\lambda(f(N)) = 0</math>, where <math>\lambda</math> stands for the [[Lebesgue measure]] on '''R''').
* ''f'': ''I'' → '''R''' is absolutely continuous if and only if it is continuous, is of bounded variation and has the Luzin ''N'' property. This statement is also known as the Banach-Zareckiǐ theorem.<ref>{{harvnb |Bruckner|Bruckner|Thomson|1997|loc=Theorem 7.11}}.</ref>
* If ''f'': ''I'' → '''R''' is absolutely continuous and ''g'': '''R''' → '''R''' is globally [[Lipschitz continuity|Lipschitz-continuous]], then the composition ''g ∘ f'' is absolutely continuous. Conversely, for every function ''g'' that is not globally Lipschitz continuous there exists an absolutely continuous function ''f'' such that ''g ∘ f'' is not absolutely continuous.<ref>{{harvnb |Fichtenholz|1923}}.</ref>

===Examples===
The following functions are uniformly continuous but '''not''' absolutely continuous:
* The [[Cantor function]] on [0, 1] (it is of bounded variation but not absolutely continuous);
* The function:<math display="block"> f(x) = \begin{cases}
0, & \text{if }x =0 \\
x \sin(1/x), & \text{if } x \neq 0
\end{cases} </math> on a finite interval containing the origin.

The following functions are absolutely continuous but not α-Hölder continuous:
* The function ''f''(''x'')&nbsp;=&nbsp;''x<sup>β</sup>'' on [0,&nbsp;''c''], for any {{nowrap|0 < ''β'' < ''α'' < 1}}

The following functions are absolutely continuous and [[Hölder condition|α-Hölder continuous]] but not [[Lipschitz continuity|Lipschitz continuous]]:
* The function ''f''(''x'')&nbsp;=&nbsp;{{radic|''x''}} on [0,&nbsp;''c''], for ''α''&nbsp;≤&nbsp;1/2.

===Generalizations===
Let (''X'', ''d'') be a [[metric space]] and let ''I'' be an [[interval (mathematics)|interval]] in the [[real line]] '''R'''. A function ''f'': ''I'' → ''X'' is '''absolutely continuous''' on ''I'' if for every positive number <math>\varepsilon</math>, there is a positive number <math>\delta</math> such that whenever a finite sequence of [[pairwise disjoint]] sub-intervals [''x''<sub>''k''</sub>, ''y''<sub>''k''</sub>] of ''I'' satisfies:

:<math>\sum_{k} \left| y_k - x_k \right| < \delta</math>

then:

:<math>\sum_{k} d \left( f(y_k), f(x_k) \right) < \varepsilon.</math>

The collection of all absolutely continuous functions from ''I'' into ''X'' is denoted AC(''I''; ''X'').

A further generalization is the space AC<sup>''p''</sup>(''I''; ''X'') of curves ''f'': ''I'' → ''X'' such that:<ref>{{harvnb|Ambrosio|Gigli|Savaré|2005|loc=Definition 1.1.1 on page 23}}</ref>

:<math>d \left( f(s), f(t) \right) \leq \int_s^t m(\tau) \,d\tau \text{ for all } [s, t] \subseteq I</math>

for some ''m'' in the [[Lp space|''L''<sup>''p''</sup> space]] ''L''<sup>''p''</sup>(I).

===Properties of these generalizations===
* Every absolutely continuous function (over a compact interval) is [[uniform continuity|uniformly continuous]] and, therefore, [[Continuous function|continuous]]. Every [[Lipschitz continuity|Lipschitz-continuous]] [[function (mathematics)|function]] is absolutely continuous.
* If ''f'': [''a'',''b''] → ''X'' is absolutely continuous, then it is of [[bounded variation]] on [''a'',''b''].
* For ''f'' ∈ AC<sup>''p''</sup>(''I''; ''X''), the [[metric derivative]] of ''f'' exists for ''λ''-[[almost all]] times in ''I'', and the metric derivative is the smallest ''m'' ∈ ''L''<sup>''p''</sup>(''I''; '''R''') such that:<ref>{{harvnb |Ambrosio|Gigli|Savaré|2005|loc=Theorem 1.1.2 on page 24}}</ref><math display="block">d \left( f(s), f(t) \right) \leq \int_s^t m(\tau) \,d\tau \text{ for all } [s, t] \subseteq I.</math>

==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.

==Relation between the two notions of absolute continuity==
A finite measure ''μ'' on [[Borel set|Borel subsets]] of the real line is absolutely continuous with respect to [[Lebesgue measure]] if and only if the point function:
:<math>F(x)=\mu((-\infty,x])</math>
is an absolutely continuous real function. More generally, a function is locally (meaning on every bounded interval) absolutely continuous if and only if its [[distributional derivative]] is a measure that is absolutely continuous with respect to the Lebesgue measure.

If absolute continuity holds then the Radon–Nikodym derivative of ''μ'' is equal almost everywhere to the derivative of ''F''.<ref>{{harvnb|Royden|1988|loc=Problem 12.17(b) on page 303}}.</ref>

More generally, the measure ''μ'' is assumed to be locally finite (rather than finite) and ''F''(''x'') is defined as ''μ''((0,''x'']) for {{nowrap|''x'' > 0}}, 0 for {{nowrap|1=''x'' = 0}}, and −''μ''((''x'',0]) for {{nowrap|''x'' < 0}}. In this case ''μ'' is the [[Lebesgue–Stieltjes integration|Lebesgue–Stieltjes measure]] generated by ''F''.<ref>{{harvnb|Athreya|Lahiri|2006|loc=Sect. 1.3.2, page 26}}.</ref> The relation between the two notions of absolute continuity still holds.<ref>{{harvnb|Nielsen|1997|loc=Proposition 15.7 on page 252}}; {{harvnb|Athreya|Lahiri|2006|loc=Theorem 4.4.3 on page 131}}; {{harvnb|Royden|1988|loc=Problem 12.17(a) on page 303}}.</ref>

==Notes==
{{reflist|29em}}

==References==
* {{citation | last1=Ambrosio | first1=Luigi | last2=Gigli | first2=Nicola | last3=Savaré | first3=Giuseppe | title=Gradient Flows in Metric Spaces and in the Space of Probability Measures | publisher=ETH Zürich, Birkhäuser Verlag, Basel | year=2005 | isbn=3-7643-2428-7 }}
* {{citation | last1=Athreya | first1=Krishna B. | last2=Lahiri | first2=Soumendra N. | title = Measure theory and probability theory | publisher = Springer | year = 2006 | isbn=0-387-32903-X }}
* {{citation | last1=Bruckner | first1=A. M. | last2=Bruckner | first2=J. B. | last3=Thomson | first3=B. S. | title =
Real Analysis | publisher = Prentice Hall | year = 1997 | isbn=0-134-58886-X }}
* {{cite journal |last=Fichtenholz |first=Grigorii |author-link=Grigorii Fichtenholz |title=Note sur les fonctions absolument continues |date=1923 |url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=6853&option_lang=eng |journal=Matematicheskii Sbornik |volume=31 |issue=2 |pages=286–295}}
* Leoni, Giovanni (2009), ''[http://bookstore.ams.org/gsm-105 A First Course in Sobolev Spaces]'',  Graduate Studies in Mathematics, American Mathematical Society, pp. xvi+607 {{ISBN|978-0-8218-4768-8}}, {{MR|2527916}}, {{Zbl|1180.46001}}, [http://old.maa.org/press/maa-reviews/a-first-course-in-sobolev-spaces MAA]
* {{citation | last=Nielsen | first=Ole A. | title = An introduction to integration and measure theory | publisher = Wiley-Interscience | year = 1997 | isbn=0-471-59518-7 }}
* {{citation | last=Royden | first=H.L. | title = Real Analysis | publisher = Collier Macmillan | edition=third| year = 1988 | isbn=0-02-404151-3 }}

==External links==
* [https://www.encyclopediaofmath.org/index.php/Absolute_continuity Absolute continuity] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
* [https://www.mat.univie.ac.at/~gerald/ftp/book-ra/index.html Topics in Real Analysis] by [[Gerald Teschl]]

{{Measure theory}}
{{Functional analysis}}

[[Category:Theory of continuous functions]]
[[Category:Real analysis]]
[[Category:Measure theory]]