Editing Lebesgue measure

{{Short description|Concept of area in any dimension}}
In [[Measure (mathematics)|measure theory]], a branch of [[mathematics]], the '''Lebesgue measure''', named after [[france|French]] mathematician [[Henri Lebesgue]], is the standard way of assigning a [[measure (mathematics)|measure]] to [[subset]]s of [[higher dimension]]al [[Euclidean space|Euclidean ''{{Math|n}}''-spaces]]. For lower dimensions {{Math|1=n=1, 2,}} or {{Math|3}}, it coincides with the standard measure of [[length]], [[area]], or [[volume]]. In general, it is also called '''''{{Math|n}}''-dimensional volume''', '''''{{Math|n}}''-volume''', '''hypervolume''', 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 <math>A</math> is here denoted by <math>\lambda(A)</math>.

Henri Lebesgue described this measure in the year 1901 which, a year after, was followed up by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902.<ref>{{cite journal |doi=10.1007/BF02420592|title=Intégrale, Longueur, Aire |year=1902 |last1=Lebesgue |first1=H. |journal=Annali di Matematica Pura ed Applicata |volume=7 |pages=231–359 |s2cid=121256884 |url=https://zenodo.org/record/2313710 }}</ref>

==Definition==

For any [[Interval (mathematics)|interval]] <math>I = [a,b]</math>, or <math>I = (a, b)</math>, in the set <math>\mathbb{R}</math> of real numbers, let <math>\ell(I)= b - a</math> denote its length. For any subset <math>E\subseteq\mathbb{R}</math>, the Lebesgue [[outer measure]]<ref>{{cite book |title=Real Analysis |last1=Royden |first1=H. L. |author-link=Halsey Royden |date=1988 |publisher=Macmillan |isbn=0-02-404151-3 |edition=3rd |location=New York |page=56 }}</ref> <math>\lambda^{\!*\!}(E)</math> is defined as an [[infimum]]

<math display="block">\lambda^{\!*\!}(E) = \inf \left\{\sum_{k=1}^\infty \ell(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>

The above definition can be generalised to higher dimensions as follows.<ref>{{Cite web|url=https://de.wikipedia.org/w/index.php?title=Lebesgue-Ma%C3%9F&oldid=225731376|title=Lebesgue-Maß|date=29 August 2022|accessdate=9 March 2023|via=Wikipedia}}</ref>
For any [[rectangular cuboid]] <math>C</math> which is a [[Cartesian product]] <math>C=I_1\times\cdots\times I_n</math> of open intervals, let <math>\operatorname{vol}(C)=\ell(I_1)\times\cdots\times \ell(I_n)</math> (a real number product) denote its volume.
For any subset <math>E\subseteq\mathbb{R^n}</math>,

<math display="block">\lambda^{\!*\!}(E) = \inf \left\{\sum_{k=1}^\infty \operatorname{vol}(C_k) : {(C_k)_{k \in \mathbb N}} \text{ is a sequence of products of open intervals with } E\subset \bigcup_{k=1}^\infty C_k\right\}.</math>

Some sets <math>E</math> satisfy the [[Carathéodory's criterion|Carathéodory criterion]], which requires that for every <math> A\subseteq \mathbb{R^n}</math>,

<math display="block">\lambda^{\!*\!}(A) = \lambda^{\!*\!}(A \cap E) + \lambda^{\!*\!}(A \cap E^c).</math>

Here <math>E^c</math> denotes the complement set. The sets <math>E</math> that satisfy the Carathéodory criterion are said to be Lebesgue-measurable, with its Lebesgue measure being defined as its Lebesgue outer measure: <math>\lambda(E) = \lambda^{\!*\!}(E)</math>. The set of all such <math>E</math> forms a [[Sigma-algebra|''σ''-algebra]]. 

A set <math>E</math> that does not satisfy the Carathéodory criterion is not Lebesgue-measurable. [[ZFC]] proves that [[non-measurable set]]s do exist; examples are the [[Vitali set]]s.

=== Intuition ===

The first part of the definition states that the subset <math>E</math> of the real numbers is reduced to its outer measure by coverage by sets of open intervals.  Each of these sets of intervals <math>I</math> covers <math>E</math> in a sense, since the union of these intervals contains <math>E</math>. The total length of any covering interval set may overestimate the measure of <math>E,</math> because <math>E</math> is a subset of the union of the intervals, and so the intervals may include points which are not in <math>E</math>. The Lebesgue outer measure emerges as the [[Infimum and supremum|greatest lower bound (infimum)]] of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit <math>E</math> most tightly and do not overlap.

That characterizes the Lebesgue outer measure. Whether this outer measure translates to the Lebesgue measure proper depends on an additional condition. This condition is tested by taking subsets <math>A</math> of the real numbers using <math>E</math> as an instrument to split <math>A</math> into two partitions: the part of <math>A</math> which intersects with <math>E</math> and the remaining part of <math>A</math> which is not in <math>E</math>: the set difference of <math>A</math> and <math>E</math>.  These partitions of <math>A</math> are subject to the outer measure. If for all possible such subsets <math>A</math> of the real numbers, the partitions of <math>A</math> cut apart by <math>E</math> have outer measures whose sum is the outer measure of <math>A</math>, then the outer Lebesgue measure of <math>E</math> gives its Lebesgue measure. Intuitively, this condition means that the set <math>E</math> must not have some curious properties which causes a discrepancy in the measure of another set when <math>E</math> is used as a "mask" to "clip" that set, hinting at the existence of sets for which the Lebesgue outer measure does not give the Lebesgue measure. (Such sets are, in fact, not Lebesgue-measurable.)

== Examples ==

* Any closed [[Interval (mathematics)|interval]] <math display="inline">[a, b]</math> of [[real number]]s is Lebesgue-measurable, and its Lebesgue measure is the length <math display="inline">b - a</math>. The [[open interval]] <math display="inline">(a, b)</math> has the same measure, since the [[set difference|difference]] between the two sets consists only of the end points <math>a</math> and <math>b</math>, which each have [[measure zero]].
* Any [[Cartesian product]] of intervals <math display="inline">[a, b]</math> and <math display="inline">[c, d]</math> is Lebesgue-measurable, and its Lebesgue measure is <math display="inline">(b - a)(c-d)</math>, the area of the corresponding [[rectangle]].
* Moreover, every [[Borel set]] is Lebesgue-measurable. However, there are Lebesgue-measurable sets which are not Borel sets.<ref>{{cite web | url=https://math.stackexchange.com/q/556756 | title=What sets are Lebesgue-measurable? | publisher=math stack exchange | access-date=26 September 2015 | author=Asaf Karagila}}</ref><ref>{{cite web | url=https://math.stackexchange.com/q/142385 | title=Is there a sigma-algebra on R strictly between the Borel and Lebesgue algebras? | publisher=math stack exchange | access-date=26 September 2015 | author=Asaf Karagila}}</ref>
* Any [[countable]] set of real numbers has Lebesgue measure {{Math|0}}. In particular, the Lebesgue measure of the set of [[algebraic numbers]] is {{Math|0}}, even though the set is [[Dense set|dense]] in <math>\mathbb{R}</math>.
* The [[Cantor set]] and the set of [[Liouville number]]s are examples of [[uncountable set]]s that have Lebesgue measure {{Math|0}}.
* If the [[axiom of determinacy]] holds then all sets of reals are Lebesgue-measurable. Determinacy is however not compatible with the [[axiom of choice]].
* [[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]].
* [[Osgood curve]]s are simple plane [[curve]]s with [[positive number|positive]] Lebesgue measure<ref>{{cite journal|last=Osgood|first=William F.|date=January 1903|title=A Jordan Curve of Positive Area|journal=Transactions of the American Mathematical Society|publisher=American Mathematical Society|volume=4|issue=1|pages=107–112|doi=10.2307/1986455|issn=0002-9947|jstor=1986455|author-link1=William Fogg Osgood|doi-access=free}}<!--|access-date=2008-06-04--></ref> (it can be obtained by small variation of the [[Peano curve]] construction). The [[dragon curve]] is another unusual example.
* Any line in <math>\mathbb{R}^n</math>, for <math>n \geq 2</math>, has a zero Lebesgue measure. In general, every proper [[hyperplane]] has a zero Lebesgue measure in its [[ambient space]].
* The [[volume of an n-ball|volume of an ''{{Math|n}}''-ball]] can be calculated in terms of Euler's gamma function.

== Properties ==
[[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.]]
The Lebesgue measure on <math>\mathbb{R}^n</math> has the following properties:

# If <math display="inline">A</math> is a [[cartesian product]] of [[interval (mathematics)|intervals]] <math>I_1 \times I_2 \times ... \times I_n</math>, then ''A'' is Lebesgue-measurable and <math>\lambda (A)=|I_1|\cdot |I_2|\cdots |I_n|.</math>
# If ''<math display="inline">A</math>'' is a union of [[countable|countably many]] pairwise disjoint Lebesgue-measurable sets, then ''<math display="inline">A</math>'' is itself Lebesgue-measurable and ''<math display="inline">\lambda(A)</math>'' is equal to the sum (or [[infinite series]]) of the measures of the involved measurable sets.
# If ''<math display="inline">A</math>'' is Lebesgue-measurable, then so is its [[Complement (set theory)|complement]].
# ''<math display="inline">\lambda(A) \geq 0</math>'' for every Lebesgue-measurable set ''<math display="inline">A</math>''.
# If ''<math display="inline">A</math>'' and''<math display="inline">B</math>'' are Lebesgue-measurable and ''<math display="inline">A</math>'' is a subset of ''<math display="inline">B</math>'', then ''<math display="inline">\lambda(A) \leq \lambda(B)</math>''. (A consequence of 2.)
# 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 does not need to be closed under countable unions: <math>\{\emptyset, \{1,2,3,4\}, \{1,2\}, \{3,4\}, \{1,3\}, \{2,4\}\}</math>.)
# If ''<math display="inline">A</math>'' is an [[open set|open]] or [[closed set|closed]] subset of <math>\mathbb{R}^n</math> (or even [[Borel set]], see [[metric space]]), then ''<math display="inline">A</math>'' is Lebesgue-measurable.
# If ''<math display="inline">A</math>'' is a Lebesgue-measurable set, then it is "approximately open" and "approximately closed" in the sense of Lebesgue measure. 
# A Lebesgue-measurable set can be "squeezed" between a containing open set and a contained closed set.  This property has been used as an alternative definition of Lebesgue measurability.  More precisely, <math>E\subset \mathbb{R}</math> is Lebesgue-measurable if and only if for every <math>\varepsilon>0</math> there exist an open set <math>G</math> and a closed set <math>F</math> such that <math>F\subset E\subset G</math> and <math>\lambda(G\setminus F)<\varepsilon</math>.<ref>{{Cite book|title=Real Analysis|last=Carothers|first=N. L.|publisher=Cambridge University Press|year=2000|isbn=9780521497565|location=Cambridge|pages=[https://archive.org/details/realanalysis0000caro/page/293 293]|url=https://archive.org/details/realanalysis0000caro/page/293}}</ref>
# A Lebesgue-measurable set can be "squeezed" between a containing [[Gδ set|{{Math|G <sub>δ</sub>}} set]] and a contained [[Fσ set|{{Math|F <sub>σ</sub>}}]]. I.e, if ''<math display="inline">A</math>'' is Lebesgue-measurable then there exist a [[Gδ set|{{Math|G <sub>δ</sub>}} set]] ''<math display="inline">G</math>'' and an [[Fσ set|{{Math|F <sub>σ</sub>}}]] ''<math display="inline">F</math>'' such that ''<math display="inline">F \subseteq A \subseteq G</math>'' and ''<math display="inline">\lambda(G \setminus A) = \lambda (A \setminus F) = 0</math>''.
# Lebesgue measure is both [[Locally finite measure|locally finite]] and [[Inner regular measure|inner regular]], and so it is a [[Radon measure]].
# Lebesgue measure is [[Strictly positive measure|strictly positive]] on non-empty open sets, and so its [[Support (measure theory)|support]] is the whole of <math>\mathbb{R}^n</math>.
# If ''<math display="inline">A</math>'' is a Lebesgue-measurable set with ''<math display="inline">\lambda(A) = 0</math>'' ''(a [[null set]]), ''then every subset of ''<math display="inline">A</math>'' is also a null set. [[A fortiori|''A fortiori'']], every subset of A is measurable.
# If ''<math display="inline">A</math>'' is Lebesgue-measurable and ''x'' is an element of <math>\mathbb{R}^n</math>, then the ''translation of <math display="inline">A</math>'' ''by <math display="inline">x</math>'', defined by <math>A + x := \{a + x: a \in A\}</math>, is also Lebesgue-measurable and has the same measure as ''<math display="inline">A</math>''.
# If ''<math display="inline">A</math>'' 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>
# More generally, if ''<math display="inline">T</math>'' is a [[linear transformation]] and ''<math display="inline">A</math>'' is a measurable subset of <math>\mathbb{R}^n</math>, then ''<math display="inline">T(A)</math>'' is also Lebesgue-measurable and has the measure <math>\left|\det(T)\right| \lambda(A)</math>.

All the above may be succinctly summarized as follows (although the last two assertions are non-trivially linked to the following):

{{block indent|The Lebesgue-measurable sets form a [[sigma-algebra|{{mvar|σ}}-algebra]] containing all products of intervals, and <math>\lambda</math> is the unique [[Complete measure|complete]] [[translational invariance|translation-invariant]] [[measure (mathematics)|measure]] on that {{mvar|σ}}-algebra with <math>\lambda([0,1]\times [0, 1]\times \cdots \times [0, 1])=1.</math>}}

The Lebesgue measure also has the property of being [[Σ-finite measure|{{mvar|σ}}-finite]].

== Null sets ==
{{main|Null set}}
A subset of <math>\mathbb{R}^n</math> is a ''null set'' if, for every <math>\varepsilon > 0</math>, it can be covered with countably many products of ''n'' intervals whose total volume is at most <math>\varepsilon</math>. All [[countable]] sets are null sets.

If a subset of <math>\mathbb{R}^n</math> has [[Hausdorff dimension]] less than ''{{Mvar|n}}'' then it is a null set with respect to ''{{Mvar|n}}''-dimensional Lebesgue measure. Here Hausdorff dimension is relative to the [[Euclidean metric]] on <math>\mathbb{R}^n</math> (or any metric [[Rudolf Lipschitz|Lipschitz]] equivalent to it). On the other hand, a set may have [[topological dimension]] less than {{Mvar|n}} and have positive ''{{Mvar|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.

In order to show that a given set ''<math display="inline">A</math>'' is Lebesgue-measurable, one usually tries to find a "nicer" set ''<math display="inline">B</math>'' which differs from ''<math display="inline">A</math>'' only by a null set (in the sense that the [[symmetric difference]] ''<math display="inline">(A \setminus B) \cup (B \setminus A)</math>'' is a null set) and then show that ''<math display="inline">B</math>'' can be generated using countable unions and intersections from open or closed sets.

== Construction of the Lebesgue measure ==
The modern construction of the Lebesgue measure is an application of [[Carathéodory's extension theorem]]. It proceeds as follows.

Fix <math>n \in \mathbb N</math>. A '''box''' in <math>\mathbb{R}^n</math> is a set of the form<math display="block">B=\prod_{i=1}^n [a_i,b_i] \, ,</math>where <math>b_i \geq a_i</math>, and the product symbol here represents a Cartesian product. The volume of this box is defined to be<math display="block">\operatorname{vol}(B)=\prod_{i=1}^n (b_i-a_i) \, .</math>For ''any'' subset ''<math>A</math>'' of <math>\mathbb{R}^n</math>, we can define its [[outer measure]] <math>\lambda^{\!*\!}(A)</math> by:<math display="block">\lambda^*(A) = \inf \left\{\sum_{B\in \mathcal{C}}\operatorname{vol}(B) : \mathcal{C}\text{ is a countable collection of boxes whose union covers }A\right\} .</math>We then define the set ''<math>A</math>'' to be Lebesgue-measurable if for every subset ''<math>S</math>'' of <math>\mathbb{R}^n</math>,<math display="block">\lambda^*(S) = \lambda^*(S \cap A) + \lambda^*(S \setminus A) \, .</math>These Lebesgue-measurable sets form a [[σ-algebra|''σ''-algebra]], and the Lebesgue measure is defined by <math>\lambda(A) = \lambda^{\!*\!}(A)</math> for any Lebesgue-measurable set ''<math>A</math>''.

The existence of sets that are not Lebesgue-measurable is a consequence of the set-theoretical [[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 '''<math>\mathbb{R}</math>''' 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]].

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>

== Relation to other measures ==
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. While the Lebesgue measure on <math>\mathbb{R}^n</math> is automatically a [[Locally finite measure|locally finite]] Borel measure, not every  locally finite Borel measure on <math>\mathbb{R}^n</math> is necessarily a Lebesgue measure. The Borel measure is translation-invariant, but not [[Complete measure|complete]].

The [[Haar measure]] can be defined on any [[locally compact group]] and is a generalization of the Lebesgue measure (<math>\mathbb{R}^n</math> with addition is a locally compact group).

The [[Hausdorff measure]] is a generalization of the Lebesgue measure that is useful for measuring the subsets of <math>\mathbb{R}^n</math> of lower dimensions than ''{{Mvar|n}}'', like [[submanifold]]s, for example, surfaces or curves in <math>\mathbb{R}^3</math> and [[fractal]] sets. The Hausdorff measure is not to be confused with the notion of [[Hausdorff dimension]].

It can be shown that [[There is no infinite-dimensional Lebesgue measure|there is no infinite-dimensional analogue of Lebesgue measure]].

== See also ==
* [[4-volume]]
* [[Edison Farah]]
* [[Lebesgue's density theorem]]
* [[Liouville number#Liouville numbers and measure|Lebesgue measure of the set of Liouville numbers]]
* [[Non-measurable set]]
** [[Vitali set]]
* [[Peano–Jordan measure]]

==References==

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{{Measure theory}}
{{Lp spaces}}

[[Category:Measures (measure theory)]]