Incalculus andreal analysis,absolute continuity is asmoothness property offunctions that is stronger thancontinuity anduniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations ofcalculus—differentiation andintegration. This relationship is commonly characterized (by thefundamental theorem of calculus) in the framework ofRiemann integration, but with absolute continuity it may be formulated in terms ofLebesgue integration. For real-valued functions on thereal line, two interrelated notions appear:absolute continuity of functions andabsolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to theRadon–Nikodym derivative, ordensity, of a measure. We have the following chains of inclusions for functionsover acompact subset of the real line:

absolutely continuous ⊆uniformly continuous${\displaystyle =}$continuous

and, for a compact interval,

continuously differentiable ⊆Lipschitz continuous ⊆absolutely continuous ⊆bounded variation ⊆differentiablealmost everywhere.

A continuous function fails to be absolutely continuous if it fails to beuniformly continuous, which can happen if the domain of the function is not compact – examples are tan(x) over[0,π/2),x² over the entire real line, and sin(1/x) over (0, 1]. But a continuous functionf can fail to be absolutely continuous even on a compact interval. It may not be "differentiable almost everywhere" (like theWeierstrass function, which is not differentiable anywhere). Or it may bedifferentiable almost everywhere and its derivativef ′ may beLebesgue integrable, but the integral off ′ differs from the increment off (how muchf changes over an interval). This happens for example with theCantor function.

Let${\displaystyle I}$ be aninterval in thereal line${\displaystyle \mathbb{R}}$. A function${\displaystyle f:I\to \mathbb{R}}$ isabsolutely continuous on${\displaystyle I}$ if for every positive number${\displaystyle \epsilon }$, there is a positive number${\displaystyle \delta }$ such that whenever a finite sequence ofpairwise disjoint sub-intervals${\displaystyle (x_k,y_k)}$ of${\displaystyle I}$ with satisfies^[1]

then

The collection of all absolutely continuous functions on${\displaystyle I}$ is denoted${\displaystyle \mathrm{AC}(I)}$.

The following conditions on a real-valued functionf on a compact interval [a,b] are equivalent:^[2]

1. f is absolutely continuous;
2. f has a derivativef ′almost everywhere, the derivative is Lebesgue integrable, and$${\displaystyle f(x)=f(a)+{\int }_a^x{f}^{\prime }(t)dt}$$ for allx on [a,b];
3. there exists a Lebesgue integrable functiong on [a,b] such that$${\displaystyle f(x)=f(a)+{\int }_a^{x}g(t)dt}$$ for allx in [a,b].

If these equivalent conditions are satisfied, then necessarily any functiong as in condition 3. satisfiesg =f ′ almost everywhere.

Equivalence between (1) and (3) is known as thefundamental theorem of Lebesgue integral calculus, due toLebesgue.^[3]

For an equivalent definition in terms of measures see the sectionRelation between the two notions of absolute continuity.

• 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.^[4]
• If an absolutely continuous functionf is defined on a bounded closed interval and is nowhere zero then1/f is absolutely continuous.^[5]
• Every absolutely continuous function (over a compact interval) isuniformly continuous and, therefore,continuous. Every (globally)Lipschitz-continuousfunction is absolutely continuous.^[6]
• Iff: [a,b] →R is absolutely continuous, then it is ofbounded variation on [a,b].^[7]
• Iff: [a,b] →R is absolutely continuous, then it can be written as the difference of two monotonic nondecreasing absolutely continuous functions on [a,b].
• Iff: [a,b] →R is absolutely continuous, then it has theLuzinN property (that is, for any${\displaystyle N\subseteq [a,b]}$ such that${\displaystyle \lambda (N)=0}$, it holds that${\displaystyle \lambda (f(N))=0}$, where${\displaystyle \lambda }$ stands for theLebesgue measure onR).
• f:I →R is absolutely continuous if and only if it is continuous, is of bounded variation and has the LuzinN property. This statement is also known as the Banach-Zareckiǐ theorem.^[8]
• Iff:I →R is absolutely continuous andg:R →R is globallyLipschitz-continuous, then the compositiong${\displaystyle \circ }$ f is absolutely continuous. Conversely, for every functiong that is not globally Lipschitz continuous there exists an absolutely continuous functionf such thatg${\displaystyle \circ }$ f is not absolutely continuous.^[9]

The following functions are uniformly continuous butnot absolutely continuous:

• TheCantor function on [0, 1] (it is of bounded variation but not absolutely continuous);
• The function: on a finite interval containing the origin.

The following functions are absolutely continuous but not α-Hölder continuous:

• The functionf(x) = x^β on [0, c], for any0 <β <α < 1

The following functions are absolutely continuous andα-Hölder continuous but notLipschitz continuous:

• The functionf(x) = √x on [0, c], forα ≤ 1/2.

Let (X,d) be ametric space and letI be aninterval in thereal lineR. A functionf:I →X isabsolutely continuous onI if for every positive number${\displaystyle \epsilon }$, there is a positive number${\displaystyle \delta }$ such that whenever a finite sequence ofpairwise disjoint sub-intervals [x_k,y_k] ofI satisfies:

then:

The collection of all absolutely continuous functions fromI intoX is denoted AC(I;X).

A further generalization is the space AC^p(I;X) of curvesf:I →X such that:^[10]

${\displaystyle d\left(f(s),f(t)\right)\le {\int }_s^{t}m(\tau )d\tau \text{ for all }[s,t]\subseteq I}$

for somem in theL^p spaceL^p(I).

• Every absolutely continuous function (over a compact interval) isuniformly continuous and, therefore,continuous. EveryLipschitz-continuousfunction is absolutely continuous.
• Iff: [a,b] →X is absolutely continuous, then it is ofbounded variation on [a,b].
• Forf ∈ AC^p(I;X), themetric derivative off exists forλ-almost all times inI, and the metric derivative is the smallestm ∈L^p(I;R) such that:^[11]$${\displaystyle d\left(f(s),f(t)\right)\le {\int }_s^{t}m(\tau )d\tau \text{ for all }[s,t]\subseteq I.}$$

Ameasure${\displaystyle \mu }$ onBorel subsets of the real line is absolutely continuous with respect to theLebesgue measure${\displaystyle \lambda }$ if for every${\displaystyle \lambda }$-measurable set${\displaystyle A,}$${\displaystyle \lambda (A)=0}$ implies${\displaystyle \mu (A)=0}$. Equivalently,${\displaystyle \mu (A)>0}$ implies${\displaystyle \lambda (A)>0}$. This condition is written as${\displaystyle \mu \ll \lambda .}$ We say${\displaystyle \mu }$ isdominated by${\displaystyle \lambda .}$

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${\displaystyle {\mathbb{R}}^n,n\ge 2.}$

The following conditions on a finite measure${\displaystyle \mu }$ on Borel subsets of the real line are equivalent:^[12]

1. ${\displaystyle \mu }$ is absolutely continuous;
2. For every positive number${\displaystyle \epsilon }$ there is a positive number${\displaystyle \delta >0}$ such that for all Borel sets${\displaystyle A}$ of Lebesgue measure less than${\displaystyle \delta ;}$
3. There exists a Lebesgue integrable function${\displaystyle g}$ on the real line such that:$${\displaystyle \mu (A)={\int }_{A}gd\lambda }$$ for all Borel subsets${\displaystyle A}$ of the real line.

For an equivalent definition in terms of functions see the sectionRelation between the two notions of absolute continuity.

Any other function satisfying (3) is equal to${\displaystyle g}$ almost everywhere. Such a function is calledRadon–Nikodym derivative, or density, of the absolutely continuous measure${\displaystyle \mu .}$

Equivalence between (1), (2) and (3) holds also in${\displaystyle {\mathbb{R}}^n}$ for all${\displaystyle n=1,2,3,\dots .}$

Thus, the absolutely continuous measures on${\displaystyle {\mathbb{R}}^n}$ are precisely those that have densities; as a special case, the absolutely continuous probability measures are precisely the ones that haveprobability density functions.

If${\displaystyle \mu }$ and${\displaystyle \nu }$ are twomeasures on the samemeasurable space${\displaystyle (X,\mathcal{A}),}$${\displaystyle \mu }$ is said to beabsolutely continuous with respect to${\displaystyle \nu }$ if${\displaystyle \mu (A)=0}$ for every set${\displaystyle A}$ for which${\displaystyle \nu (A)=0.}$^[13] This is written as "${\displaystyle \mu \ll \nu }$". That is:$${\displaystyle \mu \ll \nu \text{ if and only if }\text{ for all }A\in \mathcal{A},(\nu (A)=0\text{ implies }\mu (A)=0).}$$

When${\displaystyle \mu \ll \nu ,}$ then${\displaystyle \nu }$ is said to bedominating${\displaystyle \mu .}$

Absolute continuity of measures isreflexive andtransitive, but is notantisymmetric, so it is apreorder rather than apartial order. Instead, if${\displaystyle \mu \ll \nu }$ and${\displaystyle \nu \ll \mu ,}$ the measures${\displaystyle \mu }$ and${\displaystyle \nu }$ are said to beequivalent. Thus absolute continuity induces a partial ordering of suchequivalence classes.

If${\displaystyle \mu }$ is asigned orcomplex measure, it is said that${\displaystyle \mu }$ is absolutely continuous with respect to${\displaystyle \nu }$ if its variation${\displaystyle |\mu |}$ satisfies${\displaystyle |\mu |\ll \nu ;}$ equivalently, if every set${\displaystyle A}$ for which${\displaystyle \nu (A)=0}$ is${\displaystyle \mu }$-null.

TheRadon–Nikodym theorem^[14] states that if${\displaystyle \mu }$ is absolutely continuous with respect to${\displaystyle \nu ,}$ and both measures areσ-finite, then${\displaystyle \mu }$ has a density, or "Radon-Nikodym derivative", with respect to${\displaystyle \nu ,}$ which means that there exists a${\displaystyle \nu }$-measurable function${\displaystyle f}$ taking values in${\displaystyle [0,+\mathrm{\infty }),}$ denoted by${\displaystyle f=d\mu /d\nu ,}$ such that for any${\displaystyle \nu }$-measurable set${\displaystyle A}$ we have:$${\displaystyle \mu (A)={\int }_{A}fd\nu .}$$

ViaLebesgue's decomposition theorem,^[15] 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. Seesingular measure for examples of measures that are not absolutely continuous.

A finite measureμ onBorel subsets of the real line is absolutely continuous with respect toLebesgue measure if and only if the point function:

${\displaystyle F(x)=\mu ((-\mathrm{\infty },x])}$

is an absolutely continuous real function. More generally, a function is locally (meaning on every bounded interval) absolutely continuous if and only if itsdistributional derivative is a measure that is absolutely continuous with respect to the Lebesgue measure.

If absolute continuity holds then the Radon–Nikodym derivative ofμ is equal almost everywhere to the derivative ofF.^[16]

More generally, the measureμ is assumed to be locally finite (rather than finite) andF(x) is defined asμ((0,x]) forx > 0, 0 forx = 0, and −μ((x,0]) forx < 0. In this caseμ is theLebesgue–Stieltjes measure generated byF.^[17] The relation between the two notions of absolute continuity still holds.^[18]

• Ambrosio, Luigi; Gigli, Nicola; Savaré, Giuseppe (2005),Gradient Flows in Metric Spaces and in the Space of Probability Measures, ETH Zürich, Birkhäuser Verlag, Basel,ISBN 3-7643-2428-7
• Athreya, Krishna B.; Lahiri, Soumendra N. (2006),Measure theory and probability theory, Springer,ISBN 0-387-32903-X
• Bruckner, A. M.; Bruckner, J. B.; Thomson, B. S. (1997),Real Analysis, Prentice Hall,ISBN 0-134-58886-X
• Fichtenholz, Grigorii (1923)."Note sur les fonctions absolument continues".Matematicheskii Sbornik.31 (2):286–295.
• Leoni, Giovanni (2009),A First Course in Sobolev Spaces, Graduate Studies in Mathematics, American Mathematical Society, pp. xvi+607ISBN 978-0-8218-4768-8,MR 2527916,Zbl 1180.46001,MAA
• Nielsen, Ole A. (1997),An introduction to integration and measure theory, Wiley-Interscience,ISBN 0-471-59518-7
• Royden, H.L. (1988),Real Analysis (third ed.), Collier Macmillan,ISBN 0-02-404151-3

• Absolute continuity atEncyclopedia of Mathematics
• Topics in Real Analysis byGerald Teschl

• Theory of continuous functions
• Real analysis
• Measure theory

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