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| In [[Measure (mathematics)|measure theory]], the '''Lebesgue measure''', named after [[france|French]] mathematician [[Henri Lebesgue]], is the standard way of assigning a [[measure (mathematics)|measure]] to [[subset]]s of ''n''-dimensional [[Euclidean space]]. For ''n'' = 1, 2, or 3, it coincides with the standard measure of [[length]], [[area]], or [[volume]]. In general, it is also called '''''n''-dimensional volume''', '''''n''-volume''', or simply '''volume'''.<ref>The term ''[[volume]]'' is also used, more strictly, as a [[synonym]] of 3-dimensional volume</ref> It is used throughout [[real analysis]], in particular to define [[Lebesgue integration]]. Sets that can be assigned a Lebesgue measure are called '''Lebesgue measurable'''; the measure of the Lebesgue measurable set ''A'' is denoted by λ(''A'').
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| Henri Lebesgue described this measure in the year 1901, followed the next year by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902.<ref>{{cite journal |author=Henri Lebesgue |title=Intégrale, longueur, aire |year=1902 |publisher=Université de Paris }}</ref>
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| The Lebesgue measure is often denoted ''dx'', but this should not be confused with the distinct notion of a [[volume form]].
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| == Definition ==
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| Given a subset <math>E\subset\mathbb{R}</math>, with the length of an (open, closed, semi-open) interval <math>I = [a,b]</math> given by <math>l(I)=b - a</math>, the Lebesgue outer measure <math>\lambda^*(E)</math> is defined as
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| :<math>\lambda^*(E) = \text{inf} \left\{\sum_{k=1}^\infty l(I_k) : {(I_k)_{k \in \mathbb N}} \text{ is a sequence of open intervals with } E\subset \bigcup_{k=1}^\infty I_k\right\}</math>.
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| The Lebesgue measure of E is given by its Lebesgue outer measure <math>\lambda(E)=\lambda^*(E)</math> if, for every <math> A\subset\mathbb{R}</math>,
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| :<math>\lambda^*(A) = \lambda^*(A \cap E) + \lambda^*(A \cap E^c) </math>.
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| == Examples ==
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| * Any [[closed interval]] [''a'', ''b''] of [[real number]]s is Lebesgue measurable, and its Lebesgue measure is the length ''b''−''a''. The [[open interval]] (''a'', ''b'') has the same measure, since the [[set difference|difference]] between the two sets consists only of the end points ''a'' and ''b'' and has [[measure zero]].
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| * Any [[Cartesian product]] of intervals [''a'', ''b''] and [''c'', ''d''] is Lebesgue measurable, and its Lebesgue measure is (''b''−''a'')(''d''−''c''), the area of the corresponding [[rectangle]].
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| * The Lebesgue measure of the set of [[rational numbers]] in an interval of the line is 0, although the set is [[Dense set|dense]] in the interval.
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| * The [[Cantor set]] is an example of an [[uncountable set]] that has Lebesgue measure zero.
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| * [[Vitali set]]s are examples of sets that are [[non-measurable set|not measurable]] with respect to the Lebesgue measure. Their existence relies on the [[axiom of choice]].
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| == Properties ==
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| [[File:Translation of a set.svg|thumb|300px|Translation invariance: The Lebesgue measure of <math>A</math> and <math>A+t</math> are the same.]]
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| The Lebesgue measure on '''R'''<sup>''n''</sup> has the following properties:
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| # If ''A'' is a [[cartesian product]] of [[interval (mathematics)|intervals]] ''I''<sub>1</sub> × ''I''<sub>2</sub> × ... × ''I''<sub>''n''</sub>, then ''A'' is Lebesgue measurable and <math>\lambda (A)=|I_1|\cdot |I_2|\cdots |I_n|.</math> Here, |''I''| denotes the length of the interval ''I''.
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| # If ''A'' is a [[disjoint union]] of [[countable|countably many]] disjoint Lebesgue measurable sets, then ''A'' is itself Lebesgue measurable and λ(''A'') is equal to the sum (or [[infinite series]]) of the measures of the involved measurable sets.
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| # If ''A'' is Lebesgue measurable, then so is its [[Complement (set theory)|complement]].
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| # λ(''A'') ≥ 0 for every Lebesgue measurable set ''A''.
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| # If ''A'' and ''B'' are Lebesgue measurable and ''A'' is a subset of ''B'', then λ(''A'') ≤ λ(''B''). (A consequence of 2, 3 and 4.)
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| # Countable [[Union (set theory)|unions]] and [[Intersection (set theory)|intersections]] of Lebesgue measurable sets are Lebesgue measurable. (Not a consequence of 2 and 3, because a family of sets that is closed under complements and disjoint countable unions need not be closed under countable unions: <math>\{\emptyset, \{1,2,3,4\}, \{1,2\}, \{3,4\}, \{1,3\}, \{2,4\}\}</math>.)
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| # If ''A'' is an [[open set|open]] or [[closed set|closed]] subset of '''R'''<sup>''n''</sup> (or even [[Borel set]], see [[metric space]]), then ''A'' is Lebesgue measurable.
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| # If ''A'' is a Lebesgue measurable set, then it is "approximately open" and "approximately closed" in the sense of Lebesgue measure (see the [[regularity theorem for Lebesgue measure]]).
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| # Lebesgue measure is both [[Locally finite measure|locally finite]] and [[Inner regular measure|inner regular]], and so it is a [[Radon measure]].
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| # Lebesgue measure is [[Strictly positive measure|strictly positive]] on non-empty open sets, and so its [[Support (measure theory)|support]] is the whole of '''R'''<sup>''n''</sup>.
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| # If ''A'' is a Lebesgue measurable set with λ(''A'') = 0 (a [[null set]]), then every subset of ''A'' is also a null set. [[A fortiori]], every subset of ''A'' is measurable.
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| # If ''A'' is Lebesgue measurable and ''x'' is an element of '''R'''<sup>''n''</sup>, then the ''translation of ''A'' by x'', defined by ''A'' + ''x'' = {''a'' + ''x'' : ''a'' ∈ ''A''}, is also Lebesgue measurable and has the same measure as ''A''.
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| # If ''A'' is Lebesgue measurable and <math>\delta>0</math>, then the ''dilation of <math>A</math> by <math>\delta</math>'' defined by <math>\delta A=\{\delta x:x\in A\}</math> is also Lebesgue measurable and has measure <math>\delta^{n}\lambda\,(A).</math>
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| # More generally, if ''T'' is a [[linear transformation]] and ''A'' is a measurable subset of '''R'''<sup>''n''</sup>, then ''T''(''A'') is also Lebesgue measurable and has the measure <math>|\det(T)|\, \lambda\,(A)</math>.
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| All the above may be succinctly summarized as follows:
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| : The Lebesgue measurable sets form a [[sigma-algebra|σ-algebra]] containing all products of intervals, and λ is the unique [[Complete measure|complete]] [[translational invariance|translation-invariant]] [[measure (mathematics)|measure]] on that σ-algebra with <math>\lambda([0,1]\times [0, 1]\times \cdots \times [0, 1])=1.</math>
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| The Lebesgue measure also has the property of being [[Sigma-finite measure|σ-finite]].
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| == Null sets ==
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| {{main|Null set}}
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| A subset of '''R'''<sup>''n''</sup> is a ''null set'' if, for every ε > 0, it can be covered with countably many products of ''n'' intervals whose total volume is at most ε. All [[countable]] sets are null sets.
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| If a subset of '''R'''<sup>''n''</sup> has [[Hausdorff dimension]] less than ''n'' then it is a null set with respect to ''n''-dimensional Lebesgue measure. Here Hausdorff dimension is relative to the [[Euclidean metric]] on '''R'''<sup>''n''</sup> (or any metric [[Lipschitz]]{{dn|date=May 2012}} equivalent to it). On the other hand a set may have [[topological dimension]] less than ''n'' and have positive ''n''-dimensional Lebesgue measure. An example of this is the [[Smith–Volterra–Cantor set]] which has topological dimension 0 yet has positive 1-dimensional Lebesgue measure.
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| In order to show that a given set ''A'' is Lebesgue measurable, one usually tries to find a "nicer" set ''B'' which differs from ''A'' only by a null set (in the sense that the [[symmetric difference]] (''A'' − ''B'') <math>\cup</math>(''B'' − ''A'') is a null set) and then show that ''B'' can be generated using countable unions and intersections from open or closed sets.
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| == Construction of the Lebesgue measure ==
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| The modern construction of the Lebesgue measure is an application of [[Carathéodory's extension theorem]]. It proceeds as follows.
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| Fix {{nowrap|''n'' ∈ '''N'''}}. A '''box''' in '''R'''<sup>''n''</sup> is a set of the form
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| :<math>B=\prod_{i=1}^n [a_i,b_i] \, ,</math>
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| where {{nowrap|''b<sub>i</sub>'' ≥ ''a<sub>i</sub>''}}, and the product symbol here represents a Cartesian product. The volume of this box is defined to be
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| :<math>\operatorname{vol}(B)=\prod_{i=1}^n (b_i-a_i) \, .</math>
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| For ''any'' subset ''A'' of '''R'''<sup>''n''</sup>, we can define its [[outer measure]] ''λ''*(''A'') by:
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| :<math>\lambda^*(A) = \inf \Bigl\{\sum_{B\in \mathcal{C}}\operatorname{vol}(B) : \mathcal{C}\text{ is a countable collection of boxes whose union covers }A\Bigr\} .</math>
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| We then define the set ''A'' to be Lebesgue measurable if for every subset ''S'' of '''R'''<sup>''n''</sup>,
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| :<math>\lambda^*(S) = \lambda^*(S \cap A) + \lambda^*(S - A) \, .</math>
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| These Lebesgue measurable sets form a [[σ-algebra]], and the Lebesgue measure is defined by {{nowrap|''λ''(''A'') {{=}} ''λ''*(''A'')}} for any Lebesgue measurable set ''A''.
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| The existence of sets that are not Lebesgue measurable is a consequence of a certain set-theoretical [[axiom]], the [[axiom of choice]], which is independent from many of the conventional systems of axioms for [[set theory]]. The [[Vitali set|Vitali theorem]], which follows from the axiom, states that there exist subsets of '''R''' that are not Lebesgue measurable. Assuming the axiom of choice, [[non-measurable set]]s with many surprising properties have been demonstrated, such as those of the [[Banach–Tarski paradox]].
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| In 1970, [[Robert M. Solovay]] showed that the existence of sets that are not Lebesgue measurable is not provable within the framework of [[Zermelo–Fraenkel set theory]] in the absence of the axiom of choice (see [[Solovay's model]]).<ref>{{Cite journal |last=Solovay |first=Robert M. |title=A model of set-theory in which every set of reals is Lebesgue measurable |journal=[[Annals of Mathematics]] |jstor=1970696 |series=Second Series |volume=92 |year=1970 |issue=1 |pages=1–56 |doi=10.2307/1970696 }}</ref>
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| == Relation to other measures ==
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| The [[Borel measure]] agrees with the Lebesgue measure on those sets for which it is defined; however, there are many more Lebesgue-measurable sets than there are Borel measurable sets. The Borel measure is translation-invariant, but not [[Complete measure|complete]].
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| The [[Haar measure]] can be defined on any [[locally compact]] [[topological group|group]] and is a generalization of the Lebesgue measure ('''R'''<sup>''n''</sup> with addition is a locally compact group).
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| The [[Hausdorff measure]] is a generalization of the Lebesgue measure that is useful for measuring the subsets of '''R'''<sup>''n''</sup> of lower dimensions than ''n'', like [[submanifold]]s, for example, surfaces or curves in '''R'''³ and [[fractal]] sets. The Hausdorff measure is not to be confused with the notion of [[Hausdorff dimension]].
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| It can be shown that [[There is no infinite-dimensional Lebesgue measure|there is no infinite-dimensional analogue of Lebesgue measure]].
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| == See also ==
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| * [[Lebesgue's density theorem]]
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| * [[Duffin–Schaeffer conjecture]]
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| == References ==
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| {{reflist}}
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| [[Category:Measures (measure theory)]]
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