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 {{closed-open|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 {{prime}} may be Lebesgue integrable, but the integral of f {{prime}} differs from the increment of f (how much f changes over an interval). This happens for example with the Cantor function.

Let $I$ be an interval in the real line $\backslash R$. A function $f\backslash colon\; I\; \backslash to\; \backslash R$ is absolutely continuous on $I$ if for every positive number $\backslash varepsilon$, there is a positive number $\backslash delta$ such that whenever a finite sequence of pairwise disjoint sub-intervals $(x\_k,\; y\_k)$ of $I$ with satisfies{{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 $I$ is assumed to be bounded and closed in the former two books but not the latter book.

:

then

:

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

The following conditions on a real-valued function f on a compact interval [a,b] are equivalent:{{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}}.

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

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

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

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.{{harvnb |Royden|1988|loc=Problem 5.14(a,b) on page 111}}.
• If an absolutely continuous function f is defined on a bounded closed interval and is nowhere zero then 1/f is absolutely continuous.{{harvnb |Royden|1988|loc=Problem 5.14(c) on page 111}}.
• Every absolutely continuous function (over a compact interval) is uniformly continuous and, therefore, continuous. Every (globally) Lipschitz-continuous function is absolutely continuous.{{harvnb |Royden|1988|loc=Problem 5.20(a) on page 112}}.
• If f: [a,b] → R is absolutely continuous, then it is of bounded variation on [a,b].{{harvnb|Royden|1988|loc=Lemma 5.11 on page 108}}.
• 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 $N\; \backslash subseteq\; [a,b]$ such that $\backslash lambda(N)\; =\; 0$, it holds that $\backslash lambda(f(N))\; =\; 0$, where $\backslash lambda$ 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.{{harvnb |Bruckner|Bruckner|Thomson|1997|loc=Theorem 7.11}}.
• If f: I → R is absolutely continuous and g: R → R is globally Lipschitz-continuous, then the composition g $\backslash circ$ f is absolutely continuous. Conversely, for every function g that is not globally Lipschitz continuous there exists an absolutely continuous function f such that $\backslash circ$ f'' is not absolutely continuous.{{harvnb |Fichtenholz|1923}}.

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:$$f(x)\; =\; \backslash begin\{cases\}$$

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 {{nowrap|0 < β < α < 1}}

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

• The function f(x) = {{radic|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 $\backslash varepsilon$, there is a positive number $\backslash delta$ such that whenever a finite sequence of pairwise disjoint sub-intervals [x_k, y_k] of I satisfies:

:

then:

:

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:{{harvnb|Ambrosio|Gigli|Savaré|2005|loc=Definition 1.1.1 on page 23}}

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

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:{{harvnb |Ambrosio|Gigli|Savaré|2005|loc=Theorem 1.1.2 on page 24}}$$d\; \backslash left(\; f(s),\; f(t)\; \backslash right)\; \backslash leq\; \backslash int\_s^t\; m(\backslash tau)\; \backslash ,d\backslash tau\; \backslash text\{\; for\; all\; \}\; [s,\; t]\; \backslash subseteq\; I.$$

A measure $\backslash mu$ on Borel subsets of the real line is absolutely continuous with respect to the Lebesgue measure $\backslash lambda$ if for every $\backslash lambda$-measurable set $A,$ $\backslash lambda(A)\; =\; 0$ implies $\backslash mu(A)\; =\; 0$. Equivalently, $\backslash mu(A)\; >\; 0$ implies $\backslash lambda(A)\; >\; 0$. This condition is written as $\backslash mu\; \backslash ll\; \backslash lambda.$ We say $\backslash mu$ is dominated by $\backslash lambda.$

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 $\backslash mathbb\{R\}^n,\; n\; \backslash geq\; 2.$

The following conditions on a finite measure $\backslash mu$ on Borel subsets of the real line are equivalent: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).

1. $\backslash mu$ is absolutely continuous;
2. For every positive number $\backslash varepsilon$ there is a positive number $\backslash delta\; >\; 0$ such that for all Borel sets $A$ of Lebesgue measure less than $\backslash delta;$
3. There exists a Lebesgue integrable function $g$ on the real line such that: $$\backslash mu(A)\; =\; \backslash int\_A\; g\; \backslash ,d\backslash lambda$$ for all Borel subsets $A$ 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 $g$ almost everywhere. Such a function is called Radon–Nikodym derivative, or density, of the absolutely continuous measure $\backslash mu.$

Equivalence between (1), (2) and (3) holds also in $\backslash R^n$ for all $n\; =\; 1,\; 2,\; 3,\; \backslash ldots.$

Thus, the absolutely continuous measures on $\backslash R^n$ 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 $\backslash mu$ and $\backslash nu$ are two measures on the same measurable space $(X,\; \backslash mathcal\{A\}),$ $\backslash mu$ is said to be {{visible anchor|Absolutely continuous measure|text=absolutely continuous}} with respect to $\backslash nu$ if $\backslash mu(A)\; =\; 0$ for every set $A$ for which $\backslash nu(A)\; =\; 0.${{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}}. This is written as "$\backslash mu\backslash ll\backslash nu$". That is:

$$\backslash mu\; \backslash ll\; \backslash nu\; \backslash qquad\; \backslash text\{\; if\; and\; only\; if\; \}\; \backslash qquad\; \backslash text\{\; for\; all\; \}\; A\backslash in\backslash mathcal\{A\},\; \backslash quad\; (\backslash nu(A)\; =\; 0\backslash \; \backslash text\{\; implies\; \}\; \backslash \; \backslash mu\; (A)\; =\; 0).$$

When $\backslash mu\backslash ll\backslash nu,$ then $\backslash nu$ is said to be {{visible anchor|Domination (measure theory)|text=dominating}} $\backslash mu.$

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

If $\backslash mu$ is a signed or complex measure, it is said that $\backslash mu$ is absolutely continuous with respect to $\backslash nu$ if its variation $|\backslash mu|$ satisfies $|\backslash mu|\; \backslash ll\; \backslash nu;$ equivalently, if every set $A$ for which $\backslash nu(A)\; =\; 0$ is $\backslash mu$-null.

The Radon–Nikodym theorem{{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}}. states that if $\backslash mu$ is absolutely continuous with respect to $\backslash nu,$ and both measures are σ-finite, then $\backslash mu$ has a density, or "Radon-Nikodym derivative", with respect to $\backslash nu,$ which means that there exists a $\backslash nu$-measurable function $f$ taking values in $[0,\; +\backslash infty),$ denoted by $f\; =\; d\backslash mu\; /\; d\backslash nu,$ such that for any $\backslash nu$-measurable set $A$ we have:

$$\backslash mu(A)\; =\; \backslash int\_A\; f\; \backslash ,d\backslash nu.$$

Via Lebesgue's decomposition theorem,{{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}}. 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:

:$F(x)=\backslash mu((-\backslash infty,x])$

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.{{harvnb|Royden|1988|loc=Problem 12.17(b) on page 303}}.

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 measure generated by F.{{harvnb|Athreya|Lahiri|2006|loc=Sect. 1.3.2, page 26}}. The relation between the two notions of absolute continuity still holds.{{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}}.

{{reflist|29em}}

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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}}
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• [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

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Category:Theory of continuous functions

Category:Real analysis

Category:Measure theory