View source for Absolute continuity

{{MSC|28A33}}

[[Category:Classical measure theory]]

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====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.




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

<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 inequality

<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>

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.

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" />.

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 <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''

====Comments====


====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>