Difference between revisions of "Absolute continuity"

Latest revision as of 11:50, 4 February 2021

2020 Mathematics Subject Classification: Primary: 28A33 [MSN][ZBL] (Absolute continuity of measures)

2020 Mathematics Subject Classification: Primary: 26A46 [MSN][ZBL] (Absolute continuity of functions)

Absolute continuity of the Lebesgue integral

Describes a property of absolutely Lebesgue integrable functions. Consider the Lebesgue measure $\lambda$ on the $n$-dimensional euclidean space and let $f\in L^1 (\mathbb R^n, \lambda)$. Then for every $\varepsilon>0$ there is a $\delta>0$ such that \[ \left|\int_A f (x) \rd\lambda (x)\right| < \varepsilon \qquad \mbox{for every measurable set}\, A \mbox{ with } \lambda (A)< \delta\, . \] This property can be generalized to measures $\mu$ on a $\sigma$-algebra $\mathcal{B}$ of subsets of a space $X$ and to functions $f\in L^1 (X, \mu)$ (cp. with Theorem 12.34 of [HS]).

Absolute continuity of measures

A concept in measure theory (see also Absolutely continuous measures). If $\mu$ and $\nu$ are two measures on a σ-algebra $\mathcal{B}$ of subsets of $X$, we say that $\nu$ is absolutely continuous with respect to $\mu$ if $\nu (A) =0$ for any $A\in\mathcal{B}$ such that $\mu (A) =0$ (cp. with Defininition 2.11 of [Ma]). This definition can be generalized to signed measures $\nu$ and even to vector-valued measures $\nu$. Some authors generalize it further to vector-valued $\mu$'s: in that case the absolute continuity of $\nu$ with respect to $\mu$ amounts to the requirement that $\nu (A) = 0$ for any $A\in\mathcal{B}$ such that $|\mu| (A)=0$, where $|\mu|$ is the total variation of $\mu$ (see for instance Section 30 of [Ha]).

The Radon-Nikodym theorem (see Theorem B, Section 31 of [Ha]) characterizes the absolute continuity of $\nu$ with respect to $\mu$ with the existence of a function $f\in L^1 (\mu)$ such that $\nu = f \mu$, i.e. such that \[ \nu (A) = \int_A f\, \rd\mu \qquad \text{for every } A\in\mathcal{B}. \] A corollary of the Radon-Nikodym, the Jordan decomposition Theorem, characterizes signed measures as differences of nonnegative measures. We refer to Signed measure for more on this topic (see also Hahn decomposition).

Absolute continuity of a function

A function $f:I\to \mathbb R$, where $I$ is an interval of the real line, is said absolutely continuous if for every $\varepsilon> 0$ there is $\delta> 0$ such that, for every finite collection of pairwise disjoint intervals $(a_1,b_1), (a_2,b_2), \ldots , (a_n,b_n) \subset I$ with $\sum_i (b_i-a_i) <\delta$, we have \[ \sum_i |f(b_i)-f (a_i)| <\varepsilon \] (see Section 4 in Chapter 5 of [Ro]).

An absolutely continuous function is always continuous. Indeed, if the interval of definition is open, then the absolutely continuous function has a continuous extension to its closure, which is itself absolutely continuous. A continuous function might not be absolutely continuous, even if the interval $I$ is compact. Take for instance the function $f:[0,1]\to \mathbb R$ such that $f(0)=0$ and $f(x) = x \sin x^{-1}$ for $x>0$. The space of absolutely continuous (real-valued) functions is a vector space.

A characterization of absolutely continuous functions on an interval might be given in terms of Sobolev spaces: a continuous function $f:I\to \mathbb R$ is absolutely continuous if and only its distributional derivative is an $L^1$ function, cp. with Theorem 1 in Section 4.9 of [EG] (if $I$ is bounded, this is equivalent to require $f\in W^{1,1} (I)$). Vice versa, for any function with $L^1$ distributional derivative there is an absolutely continuous representative, i.e. an absolutely continuous $\tilde{f}$ such that $\tilde{f} = f$ a.e. (cp. again with [EG]). The latter statement can be proved using the absolute continuity of the Lebesgue integral.

An absolutely continuous function is differentiable almost everywhere and its pointwise derivative coincides with the generalized one. The fundamental theorem of calculus holds for absolutely continuous functions, i.e. if we denote by $f'$ its pointwise derivative, we then have \begin{equation}\label{e:fundamental} f (b)-f(a) = \int_a^b f' (x)\rd x \qquad \forall a<b\in I. \end{equation} In fact this is yet another characterization of absolutely continuous functions (see Theorem 13 and Corollary 11 of Section 4 in Chapter 5 of [Ro]).

The differentiability almost everywhere does not imply the absolute continuity: a notable example is the Cantor ternary function or "Devil's staircase" (see Problem 46 in Chapter 2 of [Ro]). Though such function is differentiable almost everywhere, it fails to satisfy \ref{e:fundamental} since the derivative vanishes almost everywhere but the function is not constant, cp. with Problems 11 and 12 of Chapter 5 in [Ro] (indeed the generalized derivative of the Cantor ternary function is a measure which is not absolutely continuous with respect to the Lebesgue measure, see [AFP]).

It follows from \ref{e:fundamental} that an absolutely continuous function maps a set of (Lebesgue) measure zero into a set of measure zero (i.e. it has the Luzin-N-property), and a (Lebesgue) measurable set into a measurable set. Any continuous function of bounded variation which maps each set of measure zero into a set of measure zero is absolutely continuous (this follows, for instance, from the Radon-Nikodym theorem). Any absolutely continuous function can be represented as the difference of two absolutely continuous non-decreasing functions.

Metric setting

This notion can be easily generalized when the target of the function is a metric space $(X,d)$. In that case the function $f:I\to X$ is absolutely continuous if for every positive $\varepsilon$ there is a positive $\delta$ such that for any $a_1<b_1\leq a_2<b_2 \leq \ldots \leq a_n<b_n \in I$ with $\sum_i |a_i -b_i| <\delta$, we have \[ \sum_i d (f (b_i), f(a_i)) <\varepsilon\, . \] The absolute continuity guarantees the uniform continuity. As for real valued functions, there is a characterization through an appropriate notion of derivative.

Theorem 1 A continuous function $f$ is absolutely continuous if and only if there is a function $g\in L^1_{loc} (I, \mathbb R)$ such that \begin{equation}\label{e:metric} d (f(b), f(a))\leq \int_a^b g(t)\, dt \qquad \forall a<b\in I\, \end{equation} (cp. with [AGS]). This theorem motivates the following

Definition 2 If $f:I\to X$ is absolutely continuous and $I$ is a closed interval, the metric derivative of $f$ is the function $g\in L^1$ with the smallest $L^1$ norm such that \ref{e:metric} holds (cp. with [AGS]).

The definition can be easily generalized to more general domains of definition. Observe also that, if $X$ is the standard Euclidean space $\mathbb R^k$, then the metric derivative of $f$ is the norm of the classical derivative.

References

[AFP]	L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292 Zbl 0957.49001
[AGS]	L. Ambrosio, N. Gigli, G. Savaré, "Gradient flows in metric spaces and in the space of probability measures". Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005. MR2129498 Zbl 1090.35002
[Bi]	P. Billingsley, "Convergence of probability measures" , Wiley (1968) MR0233396 Zbl 0172.21201
[Bo]	N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001
[DS]	N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523 Zbl 0635.47001
[EG]	L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Ha]	P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[HS]	E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202
[KF]	A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961). MR0085462 MR0118796 Zbl 0103.08801
[Ma]	P. Mattila, "Geometry of sets and measures in euclidean spaces". Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005
[Ro]	H.L. Royden, "Real analysis" , Macmillan (1969) MR0151555 Zbl 0197.03501
[Ru]	W. Rudin, "Principles of mathematical analysis", Third edition, McGraw-Hill (1976) MR038502 Zbl 0346.2600
[Ta]	A.E. Taylor, "General theory of functions and integration" , Blaisdell (1965) MRMR0178100 Zbl 0135.11301

How to Cite This Entry:
Absolute continuity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absolute_continuity&oldid=27231

This article was adapted from an original article by A.P. Terekhin, V.F. Emel'yanov, L.D. Kudryavtsev, V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article

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-{{MSC|28A33}}
+{{MSC|28A33}} (Absolute continuity of measures)
+{{MSC|26A46}} (Absolute continuity of functions)
 [[Category:Classical measure theory]]
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 {{TEX|done}}
-====Absolute continuity of the Lebesgue integral====
+===Absolute continuity of the Lebesgue integral===
-Describes a property of absolutely Lebesgue integrable functions. Consider the Lebesgue measure $\mathcal{L}$ on the $n$-dimensional
+Describes a property of absolutely Lebesgue integrable functions. Consider the Lebesgue measure $\lambda$ on the $n$-dimensional
-euclidean space and let $f\in L^1 (\mathbb R^n, \mathcal{L})$. Then for every $\varepsilon>0$ there is a $\delta>0$ such that
+euclidean space and let $f\in L^1 (\mathbb R^n, \lambda)$. Then for every $\varepsilon>0$ there is a $\delta>0$ such that
 \[
-\left|\int_E f (x) d\mathcal{L} (x)\right| < \varepsilon \qquad \mbox{for every measurable set $A$ with $\mathcal{L} (A)< \delta$}.
+\left|\int_A f (x) \rd\lambda (x)\right| < \varepsilon \qquad \mbox{for every measurable set}\, A \mbox{ with } \lambda (A)< \delta\, .
 \]
 This property can be generalized to measures $\mu$ on a $\sigma$-algebra $\mathcal{B}$ of subsets of a space $X$ and
-to functions $f\in L^1 (X, \mu)$.
+to functions $f\in L^1 (X, \mu)$ (cp. with Theorem 12.34 of {{Cite|HS}}).
-====Absolute continuity of measures====
+===Absolute continuity of measures===
-A concept in measure theory. If $\mu$ and $\nu$ are two measures on a $\sigma$-algebra $\mathcal{B}$ of
+A concept in measure theory (see also [[Absolutely continuous measures]]). If $\mu$ and $\nu$ are two measures on a [[Algebra of sets|σ-algebra]] $\mathcal{B}$ of
 subsets of $X$, we say that $\nu$ is absolutely continuous with respect to $\mu$ if $\nu (A) =0$ for
-any $A\in\mathcal{B}$ such that $\mu (A) =0$. This definition can be generalized to [[Signed measure|signed measures]] $\nu$
+any $A\in\mathcal{B}$ such that $\mu (A) =0$ (cp. with Defininition 2.11 of {{Cite|Ma}}). This definition can be generalized to [[Signed measure|signed measures]] $\nu$
-and even to vector-valued measure $\nu$. Some authors generalize it further to vector-valued $\mu$'s: in
+and even to vector-valued measures $\nu$. Some authors generalize it further to vector-valued $\mu$'s: in that case the absolute continuity of $\nu$ with respect to $\mu$ amounts to the requirement that $\nu (A) = 0$ for any $A\in\mathcal{B}$ such that $|\mu| (A)=0$, where $|\mu|$ is the [[Signed measure|total variation]] of $\mu$ (see for instance Section 30 of {{Cite|Ha}}).
-that case the absolute continuity of $\nu$ with respect to $\mu$ amounts to the requirement that
-$\nu (A) = 0$ for any $A\in\mathcal{B}$ such that $|\mu| (A)=0$, where $|\mu|$ is the total variation of $\mu$
-(see [[Signed measure]] for the relevant definition).
-The [[Radon-Nikodym theorem]] characterizes the absolute continuity of $\nu$ with respect to $\mu$ with
+The [[Radon-Nikodym theorem]] (see Theorem B, Section 31 of {{Cite|Ha}}) characterizes the absolute continuity of $\nu$ with respect to $\mu$ with
 the existence of a function $f\in L^1 (\mu)$ such that $\nu = f \mu$, i.e. such that
 \[
-\nu (A) = \int_A f\, d\mu \qquad \mbox{for every $A\in\mathcal{B}$.}
+\nu (A) = \int_A f\, \rd\mu \qquad \text{for every } A\in\mathcal{B}.
 \]
-A corollary of the Radon-Nikodym, the Hahn decomposition theorem, characterize signed measures
+A corollary of the Radon-Nikodym, the [[Jordan decomposition (of a signed measure)|Jordan decomposition Theorem]], characterizes signed measures
-as differences of nonnegative measures. We refer to [[Radon-Nikodym]] for more on this topic.
+as differences of nonnegative measures. We refer to [[Signed measure]] for more on this topic (see also [[Hahn decomposition]]).
+===Absolute continuity of a function===
+A function $f:I\to \mathbb R$, where $I$ is an interval of the real line,
+is said absolutely continuous if for every $\varepsilon> 0$
+there is $\delta> 0$ such that, for every finite collection of pairwise disjoint intervals $(a_1,b_1), (a_2,b_2), \ldots , (a_n,b_n) \subset I$ with
+$\sum_i (b_i-a_i) <\delta$, we have
+\[
+\sum_i |f(b_i)-f (a_i)| <\varepsilon
+\]
+(see Section 4 in Chapter 5 of {{Cite|Ro}}).
+An absolutely continuous function is always continuous. Indeed, if the interval of definition is open,
+then the absolutely continuous function has a continuous extension to its closure, which is itself
+absolutely continuous.
+A continuous function might not be absolutely continuous, even if the interval $I$ is compact.
+Take for instance the function $f:[0,1]\to \mathbb R$ such that $f(0)=0$ and $f(x) = x \sin x^{-1}$ for
+$x>0$. The space of absolutely continuous (real-valued) functions is a vector space.
+A characterization of absolutely continuous functions on an interval might be
+given in terms of [[Sobolev space|Sobolev spaces]]: a continuous function $f:I\to \mathbb R$ is absolutely continuous
+if and only its [[Generalized derivative|distributional derivative]] is an $L^1$ function, cp. with Theorem 1 in Section 4.9 of {{Cite|EG}} (if $I$ is
+bounded, this is equivalent to require $f\in W^{1,1} (I)$). Vice versa,
+for any function with $L^1$ distributional derivative there is an absolutely continuous representative, i.e.
+an absolutely continuous $\tilde{f}$ such that $\tilde{f} = f$ a.e. (cp. again with {{Cite|EG}}). The latter statement
+can be proved using the absolute continuity of the Lebesgue integral.
-Absolute continuity of a function is a stronger notion than continuity. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030034.png" /> defined on a segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030035.png" /> is said to be absolutely continuous if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030036.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030037.png" /> such that for any finite system of pairwise non-intersecting intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030039.png" />, for which
+An absolutely continuous function is differentiable almost everywhere and its pointwise derivative
+coincides with the generalized one. The fundamental theorem of calculus holds for absolutely continuous
+functions, i.e. if we denote by $f'$ its pointwise derivative, we then have
+\begin{equation}\label{e:fundamental}
+f (b)-f(a) = \int_a^b f' (x)\rd x \qquad \forall a<b\in I.
+\end{equation}
+In fact this is yet another characterization of absolutely continuous functions (see Theorem 13 and Corollary 11 of Section 4 in Chapter 5 of {{Cite|Ro}}).
-<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030040.png" /></td> </tr></table>
+The differentiability almost everywhere does not imply the absolute continuity: a notable
+example is the [[Cantor ternary function]] or "Devil's staircase" (see Problem 46 in Chapter 2 of {{Cite|Ro}}). Though such function is differentiable
+almost everywhere, it fails to satisfy \ref{e:fundamental} since the derivative vanishes almost everywhere but the function is not constant, cp. with Problems 11 and 12 of Chapter 5 in {{Cite|Ro}} (indeed the generalized derivative
+of the Cantor ternary function is a measure which is not absolutely continuous with respect to
+the Lebesgue measure, see {{Cite|AFP}}).
-the inequality
+It follows from \ref{e:fundamental} that an absolutely continuous function maps a set of (Lebesgue) measure zero into a set of measure zero (i.e. it has the [[Luzin-N-property]]), and a (Lebesgue) measurable set into a measurable set. Any continuous [[Function of bounded variation|function of bounded variation]] which maps each set of measure zero into a set of measure zero is absolutely continuous (this follows, for instance, from the [[Radon-Nikodym theorem]]). Any absolutely continuous function can be represented as the difference of two absolutely continuous non-decreasing functions.
-<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030041.png" /></td> </tr></table>
+====Metric setting====
+This notion can be easily generalized when the target of the function is a [[Metric space|metric space]] $(X,d)$. In that case the function $f:I\to X$ is absolutely continuous if for every positive $\varepsilon$ there is a positive $\delta$ such that
-holds. Any absolutely continuous function on a segment is continuous on this segment. The opposite implication is not true: e.g. the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030042.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030044.png" /> is continuous on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030045.png" />, but is not absolutely continuous on it. If, in the definition of an absolutely continuous function, the requirement that the pairwise intersections of intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030046.png" /> are empty be discarded, then the function will satisfy an even stronger condition: A [[Lipschitz condition|Lipschitz condition]] with some constant.
+for any $a_1<b_1\leq a_2<b_2 \leq \ldots \leq a_n<b_n \in I$ with $\sum_i |a_i -b_i| <\delta$, we have
+\[
-If two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030048.png" /> are absolutely continuous, then their sum, difference and product are also absolutely continuous and, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030049.png" /> does not vanish, so is their quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030050.png" />. The superposition of two absolutely continuous functions need not be absolutely continuous. However, if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030051.png" /> is absolutely continuous on a segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030052.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030054.png" />, while the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030055.png" /> satisfies a Lipschitz condition on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030056.png" />, then the composite function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030057.png" /> is absolutely continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030058.png" />. If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030059.png" />, which is absolutely continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030060.png" />, is monotone increasing, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030061.png" /> is absolutely continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030062.png" />, then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030063.png" /> is also absolutely continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030064.png" />.
+\sum_i d (f (b_i), f(a_i)) <\varepsilon\, .
+\]
-An absolutely continuous function maps a set of measure zero into a set of measure zero, and a measurable set into a measurable set. Any continuous function of finite variation which maps each set of measure zero into a set of measure zero is absolutely continuous. Any absolutely continuous function can be represented as the difference of two absolutely continuous non-decreasing functions.
+The  absolute continuity guarantees the uniform continuity. As for real valued functions, there is a characterization through an appropriate  notion of derivative.
-A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030065.png" /> that is absolutely continuous on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030066.png" /> has a finite variation on this segment and has a finite derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030067.png" /> at almost every point. The derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030068.png" /> is summable over this segment, and
-<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030069.png" /></td> </tr></table>
-If the derivative of an absolutely continuous function is almost everywhere equal to zero, then the function itself is constant. On the other hand, for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030070.png" /> that is summable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030071.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030072.png" /> is absolutely continuous on this segment. Accordingly, the class of functions that are absolutely continuous on a given segment coincides with the class of functions that can be represented as an indefinite Lebesgue integral, i.e. as a Lebesgue integral with a variable upper limit of a certain summable function plus a constant.
-If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030073.png" /> is absolutely continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030074.png" />, then its total variation is
-<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030075.png" /></td> </tr></table>
-The concept of absolute continuity can be generalized to include both functions of several variables and set functions (see Subsection 4 below).
-====References====
-<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,   S.V. Fomin,   "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Smirnov,   "A course of higher mathematics" , '''5''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR></table>
-''L.D. Kudryavtsev''
-Absolute continuity of a set function is a concept usually applied to countably-additive functions defined on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030076.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030077.png" /> of subsets of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030078.png" />. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030080.png" /> are two countably-additive functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030081.png" /> having values in the extended real number line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030082.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030083.png" /> is absolutely continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030084.png" /> (in symbols this is written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030085.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030086.png" /> entails <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030087.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030088.png" /> is the total variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030089.png" />:
-<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030090.png" /></td> </tr></table>
-<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030091.png" /></td> </tr></table>
-<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030092.png" /></td> </tr></table>
-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030094.png" /> are measures, known as the positive and negative variations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030095.png" />; according to the Jordan–Hahn theorem, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030096.png" />. It turns out that the relations 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030097.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030098.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030099.png" />; 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300100.png" /> are equivalent. If the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300101.png" /> is finite, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300102.png" /> if and only if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300103.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300104.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300105.png" /> entails <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300106.png" />. According to the [[Radon–Nikodým theorem|Radon–Nikodým theorem]], if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300107.png" /> are (completely) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300108.png" />-finite, (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300109.png" /> and there exists a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300111.png" /> such that
-<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300112.png" /></td> </tr></table>
-and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300113.png" />, then there exists on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300114.png" /> a finite measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300115.png" /> such that
-<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300116.png" /></td> </tr></table>
-Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300117.png" /> is (completely) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300118.png" />-finite and the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300119.png" /> makes sense, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300120.png" /> as a function of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300121.png" /> is absolutely continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300122.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300123.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300124.png" /> are (completely) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300125.png" />-finite measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300126.png" />, there exist uniquely defined (completely) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300127.png" />-finite measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300128.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300129.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300130.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300131.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300132.png" /> is singular with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300133.png" /> (i.e. there exists a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300134.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300135.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300136.png" />) (Lebesgue's theorem). A measure, defined on the Borel sets of a finite-dimensional Euclidean space (or, more generally, of a locally compact group), is called absolutely continuous if it is absolutely continuous with respect to the Lebesgue (Haar) measure. A non-negative measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300137.png" /> on the Borel sets of the real line is absolutely continuous if and only if the corresponding distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a010300138.png" /> is absolutely continuous (as a function of a real variable). The concept of absolute continuity of a set function can also be defined for finitely-additive functions and for functions with vector values.
-====References====
-<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.R. Halmos,   "Measure theory" , v. Nostrand  (1950)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Neveu,   "Bases mathématiques du calcul des probabilités" , Masson  (1970)</TD></TR></table>
-''V.V. Sazonov''
+'''Theorem 1'''
+A continuous function $f$ is absolutely continuous if and only if there is a function $g\in L^1_{loc} (I, \mathbb R)$ such that
+\begin{equation}\label{e:metric}
+d (f(b), f(a))\leq \int_a^b g(t)\, dt \qquad \forall a<b\in I\,
+\end{equation}
+(cp. with {{Cite|AGS}}). This theorem motivates the following
-====Comments====
+'''Definition 2'''
+If  $f:I\to X$ is absolutely continuous and $I$ is a closed interval, the metric  derivative of $f$ is the function $g\in L^1$ with the smallest $L^1$ norm  such that \ref{e:metric} holds (cp. with {{Cite|AGS}}).
+The definition can be easily generalized to more general domains of definition. Observe also that, if $X$ is the standard Euclidean space $\mathbb R^k$, then the metric derivative of $f$ is the ''norm'' of the classical derivative.
-====References====
+===References===
-<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.L. Royden,   "Real analysis" , Macmillan  (1968)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.C. Zaanen,   "Integration" , North-Holland  (1967)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Rudin,   "Principles of mathematical analysis" , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W. Rudin,   "Real and complex analysis" , McGraw-Hill  (1966)  pp. 98</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.E. Taylor,   "General theory of functions and integration" , Blaisdell  (1965)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C.D. Aliprantz,   O. Burleinshaw,   "Principles of real analysis" , North-Holland  (1981)</TD></TR></table>
+{|
+|-
+|valign="top"|{{Ref|AFP}}||    L. Ambrosio, N. Fusco, D.  Pallara, "Functions of bounded variations   and  free discontinuity  problems". Oxford Mathematical Monographs. The    Clarendon Press,  Oxford University Press, New York, 2000.     {{MR|1857292}}{{ZBL|0957.49001}}
+|-
+|valign="top"|{{Ref|AGS}}|| L. Ambrosio, N. Gigli, G. Savaré, "Gradient flows in metric spaces and in the space of probability  measures". Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel,  2005. {{MR|2129498}} {{ZBL|1090.35002}}
+|-
+|valign="top"|{{Ref|Bi}}||    P.  Billingsley, "Convergence of probability measures" , Wiley (1968)     {{MR|0233396}} {{ZBL|0172.21201}}
+|-
+|valign="top"|{{Ref|Bo}}||    N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley    (1975) pp. Chapt.6;7;8 (Translated from French) {{MR|0583191}}    {{ZBL|1116.28002}} {{ZBL|1106.46005}} {{ZBL|1106.46006}}    {{ZBL|1182.28002}} {{ZBL|1182.28001}} {{ZBL|1095.28002}}    {{ZBL|1095.28001}} {{ZBL|0156.06001}}
+|-
+|valign="top"|{{Ref|DS}}||    N. Dunford, J.T. Schwartz, "Linear operators. General theory" ,   '''1'''  , Interscience (1958) {{MR|0117523}} {{ZBL|0635.47001}}
+|-
+|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL,  1992. {{MR|1158660}} {{ZBL|0804.2800}}
+|-
+|valign="top"|{{Ref|Ha}}|| P.R. Halmos,   "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
+|-
+|valign="top"|{{Ref|HS}}||  E. Hewitt,   K.R. Stromberg,   "Real and abstract analysis" , Springer  (1965) {{MR|0188387}} {{ZBL|0137.03202}}
+|-
+|valign="top"|{{Ref|KF}}||  A.N. Kolmogorov,   S.V. Fomin,   "Elements of the theory of functions  and functional analysis" , '''1–2''' , Graylock  (1957–1961). {{MR|0085462}}  {{MR|0118796}}{{ZBL|0103.08801}}
+|-
+|valign="top"|{{Ref|Ma}}||    P. Mattila, "Geometry of sets and measures in euclidean spaces".    Cambridge Studies in Advanced Mathematics, 44. Cambridge University    Press, Cambridge,  1995. {{MR|1333890}} {{ZBL|0911.28005}}
+|-
+|valign="top"|{{Ref|Ro}}|| H.L. Royden,   "Real analysis" , Macmillan  (1969) {{MR|0151555}} {{ZBL|0197.03501}}
+|-
+|valign="top"|{{Ref|Ru}}|| W. Rudin, "Principles of  mathematical analysis", Third edition, McGraw-Hill (1976) {{MR|038502}}  {{ZBL|0346.2600}}
+|-
+|valign="top"|{{Ref|Ta}}||  A.E. Taylor,   "General theory of functions and integration" , Blaisdell  (1965) {{MR|MR0178100}}  {{ZBL|0135.11301}}
+|-
+|}

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