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| In [[calculus]], '''absolute continuity''' is a smoothness property of [[function (mathematics)|function]]s that is stronger than [[continuous function|continuity]] and [[uniform continuity]]. The notion of absolute continuity allows one to obtain generalisations of the relationship between the two central operations of [[calculus]], [[derivative|differentiation]] and [[integral|integration]], expressed by the [[fundamental theorem of calculus]] in the framework of [[Riemann integration]]. Such generalisations are often formulated in terms of [[Lebesgue integration]]. For real-valued functions on the [[real line]] two interrelated notions appear, ''absolute continuity of functions'' and ''absolute continuity of measures.'' These two notions are generalized in different directions. The usual derivative of a function is related to the ''[[Radon–Nikodym derivative]]'', or ''density'', of a measure.
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| ==Absolute continuity of functions==
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| It may happen that a continuous function ''f'' is [[Differentiable function|differentiable]] almost everywhere on [0,1], its derivative ''f'' ′ is [[Lebesgue integration|Lebesgue integrable]], and nevertheless the integral of ''f'' ′ differs from the increment of ''f''. For example, this happens for the [[Cantor function]], which means that this function is not absolutely continuous.
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| ===Definition===
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| Let <math>I</math> be an [[interval (mathematics)|interval]] in the [[real line]] '''R'''. A function <math>f:I \to R</math> is '''absolutely continuous''' on <math>I</math> if for every positive number <math>\epsilon</math>, there is a positive number <math>\delta</math> such that whenever a finite sequence of [[pairwise disjoint]] sub-intervals <math>(x_k, y_k)</math> of <math>I</math> satisfies<ref>{{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.</ref>
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| :<math>\sum_{k} \left| y_k - x_k \right| < \delta</math>
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| then
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| :<math>\displaystyle \sum_{k} | f(y_k) - f(x_k) | < \epsilon.</math>
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| The collection of all absolutely continuous functions on ''I'' is denoted AC(''I'').
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| ===Equivalent definitions===
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| The following conditions on a real-valued function ''f'' on a compact interval [''a'',''b''] are equivalent:<ref>{{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}}.</ref>
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| :(1) ''f'' is absolutely continuous;
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| :(2) ''f'' has a derivative ''f'' ′ [[almost everywhere]], the derivative is Lebesgue integrable, and
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| :: <math> f(x) = f(a) + \int_a^x f'(t) \, dt </math> | |
| :for all ''x'' on [''a'',''b''];
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| :(3) there exists a Lebesgue integrable function ''g'' on [''a'',''b''] such that
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| :: <math> f(x) = f(a) + \int_a^x g(t) \, dt </math>
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| :for all ''x'' on [''a'',''b''].
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| If these equivalent conditions are satisfied then necessarily ''g'' = ''f'' ′ almost everywhere.
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| Equivalence between (1) and (3) is known as the '''fundamental theorem of Lebesgue integral calculus''', due to [[Lebesgue]].<ref>{{harvnb|Athreya|Lahiri|2006|loc=before Theorem 4.4.1 on page 129}}.</ref>
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| For an equivalent definition in terms of measures see the section [[#Relation between the two notions of absolute continuity|Relation between the two notions of absolute continuity]].
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| ===Properties===
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| * 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.<ref>{{harvnb|Royden|1988|loc=Problem 5.14(a,b) on page 111}}.</ref>
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| * If an absolutely continuous function is defined on a bounded closed interval and is nowhere zero then its reciprocal is absolutely continuous.<ref>{{harvnb|Royden|1988|loc=Problem 5.14(c) on page 111}}.</ref>
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| * Every absolutely continuous function is [[uniform continuity|uniformly continuous]] and, therefore, [[Continuous function|continuous]]. Every [[Lipschitz continuity|Lipschitz-continuous]] [[function (mathematics)|function]] is absolutely continuous.<ref>{{harvnb|Royden|1988|loc=Problem 5.20(a) on page 112}}.</ref>
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| * If ''f'': [''a'',''b''] → '''R''' is absolutely continuous, then it is of [[bounded variation]] on [''a'',''b''].<ref>{{harvnb|Royden|1988|loc=Lemma 5.11 on page 108}}.</ref>
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| * If ''f'': [''a'',''b''] → '''R''' is absolutely continuous, then it has the [[Luzin N property|Luzin ''N'' property]] (that is, for any <math>L \subseteq [a,b]</math> such that <math>\lambda(L)=0</math>, it holds that <math>\lambda(f(L))=0</math>, where <math>\lambda</math> stands for the [[Lebesgue measure]] on '''R''').
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| * ''f'': ''I'' → '''R''' is absolutely continuous if and only if it is continuous, is of bounded variation and has the Luzin ''N'' property.
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| ===Examples===
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| The following functions are continuous everywhere but not absolutely continuous:
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| * the [[Cantor function]];
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| * the function
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| ::<math>f(x) = \begin{cases} 0, & \mbox{if }x =0 \\ x \sin(1/x), & \mbox{if } x \neq 0 \end{cases} </math>
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| : on a finite interval containing the origin;
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| * the function ''f''(''x'') = ''x''<sup> 2</sup> on an unbounded interval.
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| ===Generalizations===
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| Let (''X'', ''d'') be a [[metric space]] and let ''I'' be an [[interval (mathematics)|interval]] in the [[real line]] '''R'''. A function ''f'': ''I'' → ''X'' is '''absolutely continuous''' on ''I'' if for every positive number <math>\epsilon</math>, there is a positive number <math>\delta</math> such that whenever a finite sequence of [[pairwise disjoint]] sub-intervals [''x''<sub>''k''</sub>, ''y''<sub>''k''</sub>] of ''I'' satisfies
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| :<math>\sum_{k} \left| y_k - x_k \right| < \delta</math>
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| then
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| :<math>\sum_{k} d \left( f(y_k), f(x_k) \right) < \epsilon.</math>
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| The collection of all absolutely continuous functions from ''I'' into ''X'' is denoted AC(''I''; ''X''). | |
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| A further generalization is the space AC<sup>''p''</sup>(''I''; ''X'') of curves ''f'': ''I'' → ''X'' such that<ref>{{harvnb|Ambrosio|Gigli|Savaré|2005|loc=Definition 1.1.1 on page 23}}</ref>
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| :<math>d \left( f(s), f(t) \right) \leq \int_{s}^{t} m(\tau) \, \mathrm{d} \tau \mbox{ for all } [s, t] \subseteq I</math>
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| for some ''m'' in the [[Lp space|''L''<sup>''p''</sup> space]] ''L''<sup>''p''</sup>(I).
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| ===Properties of these generalizations===
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| * Every absolutely continuous function is [[uniform continuity|uniformly continuous]] and, therefore, [[Continuous function|continuous]]. Every [[Lipschitz continuity|Lipschitz-continuous]] [[function (mathematics)|function]] is absolutely continuous.
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| * If ''f'': [''a'',''b''] → ''X'' is absolutely continuous, then it is of [[bounded variation]] on [''a'',''b''].
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| * For ''f'' ∈ AC<sup>''p''</sup>(''I''; ''X''), the [[metric derivative]] of ''f'' exists for ''λ''-[[almost all]] times in ''I'', and the metric derivative is the smallest ''m'' ∈ ''L''<sup>''p''</sup>(''I''; '''R''') such that<ref>{{harvnb|Ambrosio|Gigli|Savaré|2005|loc=Theorem 1.1.2 on page 24}}</ref>
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| ::<math>d \left( f(s), f(t) \right) \leq \int_{s}^{t} m(\tau) \, \mathrm{d} \tau \mbox{ for all } [s, t] \subseteq I.</math>
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| ==Absolute continuity of measures==
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| ===Definition===
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| A [[measure (mathematics)|measure]] <math>\mu</math> on [[Borel set|Borel subsets]] of the real line is absolutely continuous with respect to [[Lebesgue measure]] <math>\lambda</math> (in other words, dominated by <math>\lambda</math>) if for every measurable set <math>A</math>, <math>\lambda(A) = 0</math> implies <math>\mu(A)=0</math> . This is written as <math>\mu \ll \lambda</math>.
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| 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 Lebesgue measure is meant.
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| The same holds for <math>\mathbb{R}^n, n=1,2,3,\dots</math>
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| ===Equivalent definitions===
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| The following conditions on a finite measure ''μ'' on Borel subsets of the real line are equivalent:<ref>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).</ref>
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| :(1) ''μ'' is absolutely continuous;
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| :(2) for every positive number ''ε'' there is a positive number ''δ'' such that {{nowrap|''μ''(''A'') < ''ε''}} for all Borel sets ''A'' of Lebesgue measure less than ''δ'';
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| :(3) there exists a Lebesgue integrable function ''g'' on the real line such that
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| :: <math> \mu(A) = \int_A g \, \mathrm{d} \lambda</math>
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| :for all Borel subsets ''A'' of the real line.
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| For an equivalent definition in terms of functions see the section [[#Relation between the two notions of absolute continuity|Relation between the two notions of absolute continuity]].
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| 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 ''μ''.
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| Equivalence between (1), (2) and (3) holds also in '''R'''<sup>''n''</sup> for all ''n''=1,2,3,...
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| Thus, the absolutely continuous measures on '''R'''<sup>''n''</sup> are precisely those that have densities; as a special case, the absolutely continuous probability measures are precisely the ones that have [[probability density function]]s.
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| ===Generalizations===
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| If ''μ'' and ''ν'' are two [[measure (mathematics)|measure]]s on the same [[measurable space]] then ''μ'' is said to be '''absolutely continuous with respect to ''ν''''', or '''dominated by ''ν''''' if ''μ''(''A'') = 0 for every set ''A'' for which ''ν''(''A'') = 0.<ref>{{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}}.</ref> This is written as “''μ'' <math>\ll</math> ''ν''”. In symbols:
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| :<math>\mu \ll \nu \iff \left( \nu(A) = 0\ \Rightarrow\ \mu (A) = 0 \right).</math>
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| Absolute continuity of measures is [[reflexive relation|reflexive]] and [[transitive relation|transitive]], but is not [[Antisymmetric relation|antisymmetric]], so it is a [[preorder]] rather than a [[partial order]]. Instead, if ''μ'' <math>\ll</math> ''ν'' and ''ν'' <math>\ll</math> ''μ'', the measures ''μ'' and ''ν'' are said to be [[Equivalence (measure theory)|equivalent]]. Thus absolute continuity induces a partial ordering of such [[equivalence class]]es.
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| If ''μ'' is a [[signed measure|signed]] or [[complex measure]], it is said that ''μ'' is absolutely continuous with respect to ''ν'' if its variation |''μ''| satisfies |''μ''| ≪ ν; equivalently, if every set ''A'' for which ''ν''(''A'') = 0 is ''μ''-[[null set|null]].
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| The [[Radon–Nikodym theorem]]<ref>{{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}}.</ref> states that if ''μ'' is absolutely continuous with respect to ''ν'', and both measures are [[σ-finite]], then ''μ'' has a density, or "Radon-Nikodym derivative", with respect to ''ν'', which means that there exists a ''ν''-measurable function ''f'' taking values in [0, +∞), denoted by ''f'' = d''μ''/d''ν'', such that for any ''ν''-measurable set ''A'' we have
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| :<math>\mu(A) = \int_A f \, \mathrm{d} \nu.</math>
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| ===Singular measures===
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| Via [[Lebesgue's decomposition theorem]],<ref>{{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}}.</ref> every measure can be decomposed into the sum of an absolutely continuous measure and a singular measure. See [[singular measure]] for examples of measures that are not absolutely continuous.
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| ==Relation between the two notions of absolute continuity==
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| A finite measure ''μ'' on [[Borel set|Borel subsets]] of the real line is absolutely continuous with respect to [[Lebesgue measure]] if and only if the point function
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| :<math>F(x)=\mu((-\infty,x])</math>
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| is locally an absolutely continuous real function.
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| In other words, a function is locally absolutely continuous if and only if its [[Distribution (mathematics)|distributional derivative]] is a measure that is absolutely continuous with respect to the Lebesgue measure.
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| If the absolute continuity holds then the Radon-Nikodym derivative of ''μ'' is equal almost everywhere to the derivative of ''F''.<ref>{{harvnb|Royden|1988|loc=Problem 12.17(b) on page 303}}.</ref>
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| More generally, the measure ''μ'' is assumed to be locally finite (rather than finite) and ''F''(''x'') is defined as ''μ''((0,''x'']) for ''x''>0, 0 for ''x''=0, and -''μ''((''x'',0]) for ''x''<0. In this case ''μ'' is the [[Lebesgue–Stieltjes integration|Lebesgue-Stieltjes measure]] generated by ''F''.<ref>{{harvnb|Athreya|Lahiri|2006|loc=Sect. 1.3.2, page 26}}.</ref>
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| The relation between the two notions of absolute continuity still holds.<ref>{{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}}.</ref>
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| ==Notes==
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| {{reflist|29em}}
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| ==References==
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| * {{citation | last1=Ambrosio | first1=Luigi | last2=Gigli | first2=Nicola | last3=Savaré | first3=Giuseppe | title=Gradient Flows in Metric Spaces and in the Space of Probability Measures | publisher=ETH Zürich, Birkhäuser Verlag, Basel | year=2005 | isbn=3-7643-2428-7 }}
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| * {{citation | last1=Athreya | first1=Krishna B. | last2=Lahiri | first2=Soumendra N. | title = Measure theory and probability theory | publisher = Springer | year = 2006 | isbn=0-387-32903-X }}
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| * {{citation | last=Nielsen | first=Ole A. | title = An introduction to integration and measure theory | publisher = Wiley-Interscience | year = 1997 | isbn=0-471-59518-7 }}
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| * {{citation | last=Royden | first=H.L. | title = Real Analysis | publisher = Collier Macmillan | edition=third| year = 1988 | isbn=0-02-404151-3 }}
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| ==External links==
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| * [http://www.encyclopediaofmath.org/index.php/Absolute_continuity Absolute continuity] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| [[Category:Continuous mappings]]
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| [[Category:Real analysis]]
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| [[Category:Measure theory]]
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