Namespaces
Variants
Actions

Difference between revisions of "Absolute continuity"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m
Line 1: Line 1:
Absolute continuity of an integral as a property of the (Lebesgue) integral. Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a0103001.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a0103002.png" />-integrable on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a0103003.png" />. The integral of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a0103004.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a0103005.png" />-measurable subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a0103006.png" /> is an absolutely continuous set function (see Subsection 3 below) with respect to the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a0103007.png" /> if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a0103008.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a0103009.png" /> such that the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030010.png" /> for any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030011.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030012.png" />. In the general case the integral with respect to a finitely-additive set function with scalar or vectorial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030013.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030014.png" /> is an absolutely continuous function.
+
{{MSC|28A33}}
 +
 
 +
[[Category:Classical measure theory]]
 +
 
 +
{{TEX|done}}
 +
 
 +
====Absolute continuity of the Lebesgue integral====
 +
 
 +
Describes a property of absolutely Lebesgue integrable functions. Consider the Lebesgue measure $\mathcal{L}$ 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
 +
\[
 +
\left|\int_E f (x) d\mathcal{L} (x)\right| < \varepsilon \qquad \mbox{for every measurable set $A$ with $\mathcal{L} (A)< \delta$}.
 +
\]
 +
This property can be generalized to measures $\mu$ on a $\sigma$-algebra $\mathcal{B}$ of subsets of a space $X$ and
 +
to functions $f\in L^1 (X, \mu)$.
 +
 
 +
====Absolute continuity of measures====
 +
A concept in measure theory. If $\mu$ and $\nu$ are two measures on a $\sigma$-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]] $\nu$
 +
and even to vector-valued measure $\nu$. 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 [[Signed measure]] for the relevant definition).
 +
 
 +
The [[Radon-Nikodym theorem]] 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
 +
\[
 +
\nu (A) = \int_A f\, d\mu \qquad \mbox{for every $A\in\mathcal{B}$.}
 +
\]
 +
A corollary of the Radon-Nikodym, the Hahn decomposition theorem, characterize signed measures
 +
as differences of nonnegative measures. We refer to [[Radon-Nikodym]] for more on this topic.
  
''A.P. TerekhinV.F. Emel'yanov''
 
  
Absolute continuity of a measure as a concept in the theory of measures. A measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030015.png" /> is absolutely continuous with respect to a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030016.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030017.png" /> is an absolutely continuous set function with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030018.png" />. Thus, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030019.png" /> be a finite measure, given together with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030020.png" /> on some fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030021.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030022.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030023.png" /> will then be absolutely continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030024.png" /> if it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030026.png" />, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030027.png" />. A generalized finite measure (cf. [[Charge|Charge]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030028.png" /> is absolutely continuous with respect to a generalized measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030029.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030030.png" />, provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030032.png" /> is the total variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010300/a01030033.png" />.
 
  
''A.P. Terekhin''
 
  
 
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
 
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

Revision as of 20:16, 29 July 2012

2020 Mathematics Subject Classification: Primary: 28A33 [MSN][ZBL]

Absolute continuity of the Lebesgue integral

Describes a property of absolutely Lebesgue integrable functions. Consider the Lebesgue measure $\mathcal{L}$ 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 \[ \left|\int_E f (x) d\mathcal{L} (x)\right| < \varepsilon \qquad \mbox{for every measurable set '"`UNIQ-MathJax6-QINU`"' with '"`UNIQ-MathJax7-QINU`"'}. \] This property can be generalized to measures $\mu$ on a $\sigma$-algebra $\mathcal{B}$ of subsets of a space $X$ and to functions $f\in L^1 (X, \mu)$.

Absolute continuity of measures

A concept in measure theory. If $\mu$ and $\nu$ are two measures on a $\sigma$-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 measures $\nu$ and even to vector-valued measure $\nu$. 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 Signed measure for the relevant definition).

The Radon-Nikodym theorem 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 \[ \nu (A) = \int_A f\, d\mu \qquad \mbox{for every '"`UNIQ-MathJax37-QINU`"'.} \] A corollary of the Radon-Nikodym, the Hahn decomposition theorem, characterize signed measures as differences of nonnegative measures. We refer to Radon-Nikodym for more on this topic.



Absolute continuity of a function is a stronger notion than continuity. A function defined on a segment is said to be absolutely continuous if for any there exists a such that for any finite system of pairwise non-intersecting intervals , , for which

the inequality

holds. Any absolutely continuous function on a segment is continuous on this segment. The opposite implication is not true: e.g. the function if and is continuous on the segment , but is not absolutely continuous on it. If, in the definition of an absolutely continuous function, the requirement that the pairwise intersections of intervals are empty be discarded, then the function will satisfy an even stronger condition: A Lipschitz condition with some constant.

If two functions and are absolutely continuous, then their sum, difference and product are also absolutely continuous and, if does not vanish, so is their quotient . The superposition of two absolutely continuous functions need not be absolutely continuous. However, if the function is absolutely continuous on a segment and if , , while the function satisfies a Lipschitz condition on the segment , then the composite function is absolutely continuous on . If a function , which is absolutely continuous on , is monotone increasing, while is absolutely continuous on , then the function is also absolutely continuous on .

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.

A function that is absolutely continuous on the segment has a finite variation on this segment and has a finite derivative at almost every point. The derivative is summable over this segment, and

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 that is summable on the function 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 is absolutely continuous on , then its total variation is

The concept of absolute continuity can be generalized to include both functions of several variables and set functions (see Subsection 4 below).

References

[1] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[2] V.I. Smirnov, "A course of higher mathematics" , 5 , Addison-Wesley (1964) (Translated from Russian)

L.D. Kudryavtsev

Absolute continuity of a set function is a concept usually applied to countably-additive functions defined on a -ring of subsets of a set . Thus, if and are two countably-additive functions defined on having values in the extended real number line , then is absolutely continuous with respect to (in symbols this is written as ) if entails . Here is the total variation of :

and are measures, known as the positive and negative variations of ; according to the Jordan–Hahn theorem, . It turns out that the relations 1) ; 2) , ; 3) are equivalent. If the measure is finite, if and only if for any there exists a such that entails . According to the Radon–Nikodým theorem, if are (completely) -finite, (i.e. and there exists a sequence , such that

and if , then there exists on a finite measurable function such that

Conversely, if is (completely) -finite and the integral makes sense, then as a function of the set is absolutely continuous with respect to . If and are (completely) -finite measures on , there exist uniquely defined (completely) -finite measures and such that , and is singular with respect to (i.e. there exists a set such that , ) (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 on the Borel sets of the real line is absolutely continuous if and only if the corresponding distribution function 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

[1] P.R. Halmos, "Measure theory" , v. Nostrand (1950)
[2] J. Neveu, "Bases mathématiques du calcul des probabilités" , Masson (1970)

V.V. Sazonov

Comments

References

[a1] H.L. Royden, "Real analysis" , Macmillan (1968)
[a2] A.C. Zaanen, "Integration" , North-Holland (1967)
[a3] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953)
[a4] W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 98
[a5] A.E. Taylor, "General theory of functions and integration" , Blaisdell (1965)
[a6] C.D. Aliprantz, O. Burleinshaw, "Principles of real analysis" , North-Holland (1981)
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