Absolute continuity

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 [0, π/2), x² 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 almost everywhere and its derivative f ′ may be Lebesgue integrable, but the integral of f ′ differs from the increment of f (how much f changes over an interval). This happens for example with the Cantor function.

Let ${\displaystyle I}$  be an interval in the real line ${\displaystyle \mathbb {R} }$ . A function ${\displaystyle f\colon I\to \mathbb {R} }$  is absolutely continuous on ${\displaystyle I}$  if for every positive number ${\displaystyle \varepsilon }$ , there is a positive number ${\displaystyle \delta }$  such that whenever a finite sequence of pairwise disjoint sub-intervals ${\displaystyle (x_{k},y_{k})}$  of ${\displaystyle I}$  with ${\displaystyle x_{k}<y_{k}\in I}$  satisfies^[1]

${\displaystyle \sum _{k=1}^{N}(y_{k}-x_{k})<\delta }$ 

then

${\displaystyle \sum _{k=1}^{N}|f(y_{k})-f(x_{k})|<\varepsilon .}$ 

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

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

1. f is absolutely continuous;
2. f has a derivative f ′ almost everywhere, the derivative is Lebesgue integrable, and $${\displaystyle f(x)=f(a)+\int _{a}^{x}f'(t)\,dt}$$  for all x on [a,b];
3. there exists a Lebesgue integrable function g on [a,b] such that $${\displaystyle f(x)=f(a)+\int _{a}^{x}g(t)\,dt}$$  for all x in [a,b].

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

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

For an equivalent definition in terms of measures see the section Relation 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 function is defined on a bounded closed interval and is nowhere zero then its reciprocal is absolutely continuous.^[5]
• Every absolutely continuous function (over a compact interval) is uniformly continuous and, therefore, continuous. Every (globally) Lipschitz-continuous function is absolutely continuous.^[6]
• If f: [a,b] → R is absolutely continuous, then it is of bounded variation on [a,b].^[7]
• 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 (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 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.^[8]
• If f: I → R is absolutely continuous and g: R → R is globally 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.^[9]

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:$${\displaystyle f(x)={\begin{cases}0,&{\text{if }}x=0\\x\sin(1/x),&{\text{if }}x\neq 0\end{cases}}}$$  on a finite interval containing the origin.

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

• The function f(x) = x^β on [0, c], for any 0 < β < α < 1

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

• The function f(x) = √x on [0, c], for α ≤ 1/2.

Let (X, d) be a metric space and let I be an interval in the real line R. A function f: I → X is absolutely continuous on I if for every positive number ${\displaystyle \epsilon }$ , there is a positive number ${\displaystyle \delta }$  such that whenever a finite sequence of pairwise disjoint sub-intervals [x_k, y_k] of I satisfies:

${\displaystyle \sum _{k}\left|y_{k}-x_{k}\right|<\delta }$ 

then:

${\displaystyle \sum _{k}d\left(f(y_{k}),f(x_{k})\right)<\epsilon .}$ 

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

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

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

for some m in the L^p space L^p(I).

• Every absolutely continuous function (over a compact interval) is uniformly continuous and, therefore, continuous. Every Lipschitz-continuous function is absolutely continuous.
• If f: [a,b] → X is absolutely continuous, then it is of bounded variation on [a,b].
• For f ∈ AC^p(I; X), the metric derivative of f exists for λ-almost all times in I, and the metric derivative is the smallest m ∈ L^p(I; R) such that:^[11]$${\displaystyle d\left(f(s),f(t)\right)\leq \int _{s}^{t}m(\tau )\,d\tau {\text{ for all }}[s,t]\subseteq I.}$$ 

A measure ${\displaystyle \mu }$  on Borel subsets of the real line is absolutely continuous with respect to the Lebesgue 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 ${\displaystyle \mu }$  is dominated 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 ${\displaystyle \mathbb {R} ^{n},n\geq 2.}$ 

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 ${\displaystyle \varepsilon }$  there is a positive number ${\displaystyle \delta >0}$  such that ${\displaystyle \mu (A)<\varepsilon }$  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: $${\displaystyle \mu (A)=\int _{A}g\,d\lambda }$$  for all Borel subsets ${\displaystyle A}$  of the real line.

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

Any other function satisfying (3) is equal to ${\displaystyle g}$  almost everywhere. Such a function is called Radon–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 ${\displaystyle n=1,2,3,\ldots .}$ 

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 have probability density functions.

If ${\displaystyle \mu }$  and ${\displaystyle \nu }$  are two measures on the same measurable space ${\displaystyle (X,{\mathcal {A}}),}$  ${\displaystyle \mu }$  is said to be absolutely 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: $${\displaystyle \mu \ll \nu \qquad {\text{ if and only if }}\qquad {\text{ for all }}A\in {\mathcal {A}},\quad (\nu (A)=0\ {\text{ implies }}\ \mu (A)=0).}$$ 

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

Absolute continuity of measures is reflexive and transitive, but is not antisymmetric, so it is a preorder rather than a partial order. Instead, if ${\displaystyle \mu \ll \nu }$  and ${\displaystyle \nu \ll \mu ,}$  the measures ${\displaystyle \mu }$  and ${\displaystyle \nu }$  are said to be equivalent. Thus absolute continuity induces a partial ordering of such equivalence classes.

If ${\displaystyle \mu }$  is a signed or complex 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.

The Radon–Nikodym theorem^[14] states that if ${\displaystyle \mu }$  is absolutely continuous with respect to ${\displaystyle \nu ,}$  and both measures are σ-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 ${\displaystyle [0,+\infty ),}$  denoted by ${\displaystyle f=d\mu /d\nu ,}$  such that for any ${\displaystyle \nu }$ -measurable set ${\displaystyle A}$  we have: $${\displaystyle \mu (A)=\int _{A}f\,d\nu .}$$ 

Via Lebesgue'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. See singular measure for examples of measures that are not absolutely continuous.

A finite measure μ on Borel subsets of the real line is absolutely continuous with respect to Lebesgue measure if and only if the point function:

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

is an absolutely continuous real function. More generally, a function is locally (meaning on every bounded interval) absolutely continuous if and only if 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.^[16]

More generally, the measure μ is assumed to be locally finite (rather than finite) and F(x) is defined as μ((0,x]) for x > 0, 0 for x = 0, and −μ((x,0]) for x < 0. In this case μ is the Lebesgue–Stieltjes measure generated by F.^[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+607 ISBN 978-0-8218-4768-8, MR2527916, 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 at Encyclopedia of Mathematics
• Topics in Real and Functional Analysis by Gerald Teschl