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Lebesgue measure

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Concept of area in any dimension

Inmeasure theory, a branch ofmathematics, theLebesgue measure, named afterFrench mathematicianHenri Lebesgue, is the standard way of assigning ameasure tosubsets ofhigher dimensionalEuclideann-spaces. For lower dimensionsn=1,2,or 3{\displaystyle n=1,2,{\text{or }}3}, it coincides with the standard measure oflength,area, orvolume. In general, it is also calledn-dimensional volume,n-volume,hypervolume, or simplyvolume.[1] It is used throughoutreal analysis, in particular to defineLebesgue integration. Sets that can be assigned a Lebesgue measure are calledLebesgue-measurable; the measure of the Lebesgue-measurable setA{\displaystyle A} is here denoted byλ(A){\displaystyle \lambda (A)}.

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 dissertationIntégrale, Longueur, Aire in 1902.[2]

Definition

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For anyintervalI=[a,b]{\displaystyle I=[a,b]}, orI=(a,b){\displaystyle I=(a,b)}, in the setR{\displaystyle \mathbb {R} } of real numbers, letℓ(I)=b−a{\displaystyle \ell (I)=b-a} denote its length. For any subsetE⊆R{\displaystyle E\subseteq \mathbb {R} }, theLebesgueouter measure[3]λ∗(E){\displaystyle \lambda ^{\!*\!}(E)} is defined as aninfimum

λ∗(E)=inf{∑k=1∞ℓ(Ik):(Ik)k∈N is a sequence of open intervals with E⊂⋃k=1∞Ik}.{\displaystyle \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\}.}

The above definition can be generalised to higher dimensions as follows.[4] For anyrectangular cuboidC{\displaystyle C} which is aCartesian productC=I1×⋯×In{\displaystyle C=I_{1}\times \cdots \times I_{n}} of open intervals, letvol⁡(C)=ℓ(I1)×⋯×ℓ(In){\displaystyle \operatorname {vol} (C)=\ell (I_{1})\times \cdots \times \ell (I_{n})} (a real number product) denote its volume. For any subsetE⊆Rn{\displaystyle E\subseteq \mathbb {R^{n}} },

λ∗(E)=inf{∑k=1∞vol⁡(Ck):(Ck)k∈N is a sequence of products of open intervals with E⊂⋃k=1∞Ck}.{\displaystyle \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\}.}

A setE{\displaystyle E} satisfies theCarathéodory criterion whenever, for everyA⊆Rn{\displaystyle A\subseteq \mathbb {R^{n}} }, we have:

λ∗(A)=λ∗(A∩E)+λ∗(A∩E∁).{\displaystyle \lambda ^{\!*\!}(A)=\lambda ^{\!*\!}(A\cap E)+\lambda ^{\!*\!}(A\cap E^{\complement }).}

Here,E∁{\displaystyle E^{\complement }} is the complement ofE{\displaystyle E}. SetsE{\displaystyle E} satisfying the Carathéodory criterion are said to beLebesgue-measurable. The set of all suchE{\displaystyle E} forms aσ-algebra.

TheLebesgue measure of such a set is defined as its Lebesgue outer measure:

λ(E)=λ∗(E){\displaystyle \lambda (E)=\lambda ^{\!*\!}(E)}.

ZFC proves thatnon-measurable sets do exist; examples are theVitali sets.

Intuition

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The first part of the definition states that the subsetE{\displaystyle E} of the real numbers is reduced to its outer measure by coverage by sets of open intervals. Each of these sets of intervalsI{\displaystyle I} coversE{\displaystyle E} in a sense, since the union of these intervals containsE{\displaystyle E}. The total length of any covering interval set may overestimate the measure ofE,{\displaystyle E,} becauseE{\displaystyle E} is a subset of the union of the intervals, and so the intervals may include points which are not inE{\displaystyle E}. The Lebesgue outer measure emerges as thegreatest lower bound (infimum) of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fitE{\displaystyle E} 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 subsetsA{\displaystyle A} of the real numbers usingE{\displaystyle E} as an instrument to splitA{\displaystyle A} into two partitions: the part ofA{\displaystyle A} which intersects withE{\displaystyle E} and the remaining part ofA{\displaystyle A} which is not inE{\displaystyle E}: the set difference ofA{\displaystyle A} andE{\displaystyle E}. These partitions ofA{\displaystyle A} are subject to the outer measure. If for all possible such subsetsA{\displaystyle A} of the real numbers, the partitions ofA{\displaystyle A} cut apart byE{\displaystyle E} have outer measures whose sum is the outer measure ofA{\displaystyle A}, then the outer Lebesgue measure ofE{\displaystyle E} gives its Lebesgue measure. Intuitively, this condition means that the setE{\displaystyle E} must not have some curious properties which causes a discrepancy in the measure of another set whenE{\displaystyle E} 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

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Properties

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Translation invariance: The Lebesgue measure ofA{\displaystyle A} andA+t{\displaystyle A+t} are the same.

The Lebesgue measure onRn{\displaystyle \mathbb {R} ^{n}} has the following properties:

  1. IfA{\textstyle A} is acartesian product ofintervalsI1×I2×...×In{\displaystyle I_{1}\times I_{2}\times ...\times I_{n}}, thenA is Lebesgue-measurable andλ(A)=|I1|⋅|I2|⋅…⋅|In|.{\displaystyle \lambda (A)=|I_{1}|\cdot |I_{2}|\cdot _{\;\dots }\cdot |I_{n}|.}
  2. IfA{\textstyle A} is a union ofcountably many pairwise disjoint Lebesgue-measurable sets, thenA{\textstyle A} is itself Lebesgue-measurable andλ(A){\textstyle \lambda (A)} is equal to the sum (orinfinite series) of the measures of the involved measurable sets.
  3. IfA{\textstyle A} is Lebesgue-measurable, then so is itscomplement.
  4. λ(A)≥0{\textstyle \lambda (A)\geq 0} for every Lebesgue-measurable setA{\textstyle A}.
  5. IfA{\textstyle A} andB{\textstyle B} are Lebesgue-measurable andA{\textstyle A} is a subset ofB{\textstyle B}, thenλ(A)≤λ(B){\textstyle \lambda (A)\leq \lambda (B)}. (A consequence of 2.)
  6. Countableunions andintersections 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:{∅,{1,2,3,4},{1,2},{3,4},{1,3},{2,4}}{\displaystyle \{\emptyset ,\{1,2,3,4\},\{1,2\},\{3,4\},\{1,3\},\{2,4\}\}}.)
  7. IfA{\textstyle A} is anopen orclosed subset ofRn{\displaystyle \mathbb {R} ^{n}} (or evenBorel set, seemetric space), thenA{\textstyle A} is Lebesgue-measurable.
  8. IfA{\textstyle A} is a Lebesgue-measurable set, then it is "approximately open" and "approximately closed" in the sense of Lebesgue measure.
  9. 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,E⊂R{\displaystyle E\subset \mathbb {R} } is Lebesgue-measurable if and only if for everyε>0{\displaystyle \varepsilon >0} there exist an open setG{\displaystyle G} and a closed setF{\displaystyle F} such thatF⊂E⊂G{\displaystyle F\subset E\subset G} andλ(G∖F)<ε{\displaystyle \lambda (G\setminus F)<\varepsilon }.[8]
  10. A Lebesgue-measurable set can be "squeezed" between a containingGδ set and a containedFσ. I.e., ifA{\textstyle A} is Lebesgue-measurable then there exist aGδ setG{\textstyle G} and anFσF{\textstyle F} such thatF⊆A⊆G{\textstyle F\subseteq A\subseteq G} andλ(G∖A)=λ(A∖F)=0{\textstyle \lambda (G\setminus A)=\lambda (A\setminus F)=0}.
  11. Lebesgue measure is bothlocally finite andinner regular, and so it is aRadon measure.
  12. Lebesgue measure isstrictly positive on non-empty open sets, and so itssupport is the whole ofRn{\displaystyle \mathbb {R} ^{n}}.
  13. IfA{\textstyle A} is a Lebesgue-measurable set withλ(A)=0{\textstyle \lambda (A)=0}(anull set),then every subset ofA{\textstyle A} is also a null set.A fortiori, every subset ofA{\displaystyle A} is measurable.
  14. IfA{\textstyle A} is Lebesgue-measurable andx is an element ofRn{\displaystyle \mathbb {R} ^{n}}, then thetranslation ofA{\textstyle A}byx{\textstyle x}, defined byA+x:={a+x:a∈A}{\displaystyle A+x:=\{a+x:a\in A\}}, is also Lebesgue-measurable and has the same measure asA{\textstyle A}.
  15. IfA{\textstyle A} is Lebesgue-measurable andδ>0{\displaystyle \delta >0}, then thedilation ofA{\displaystyle A} byδ{\displaystyle \delta } defined byδA={δx:x∈A}{\displaystyle \delta A=\{\delta x:x\in A\}} is also Lebesgue-measurable and has measureδnλ(A).{\displaystyle \delta ^{n}\lambda \,(A).}
  16. More generally, ifT{\textstyle T} is alinear transformation andA{\textstyle A} is a measurable subset ofRn{\displaystyle \mathbb {R} ^{n}}, thenT(A){\textstyle T(A)} is also Lebesgue-measurable and has the measure|det(T)|λ(A){\displaystyle \left|\det(T)\right|\lambda (A)}.

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

The Lebesgue-measurable sets form aσ-algebra containing all products of intervals, andλ{\displaystyle \lambda } is the uniquecompletetranslation-invariantmeasure on thatσ-algebra withλ([0,1]×[0,1]×⋯×[0,1])=1.{\displaystyle \lambda ([0,1]\times [0,1]\times \cdots \times [0,1])=1.}

The Lebesgue measure also has the property of beingσ-finite.

Null sets

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Main article:Null set

A subset ofRn{\displaystyle \mathbb {R} ^{n}} is anull set if, for everyε>0{\displaystyle \varepsilon >0}, it can be covered with countably many products ofn intervals whose total volume is at mostε{\displaystyle \varepsilon }. Allcountable sets are null sets.

If a subset ofRn{\displaystyle \mathbb {R} ^{n}} hasHausdorff dimension less thann then it is a null set with respect ton-dimensional Lebesgue measure. Here Hausdorff dimension is relative to theEuclidean metric onRn{\displaystyle \mathbb {R} ^{n}} (or any metricLipschitz equivalent to it). On the other hand, a set may havetopological dimension less thann and have positiven-dimensional Lebesgue measure. An example of this is theSmith–Volterra–Cantor set which has topological dimension 0 yet has positive 1-dimensional Lebesgue measure.

In order to show that a given setA{\textstyle A} is Lebesgue-measurable, one usually tries to find a "nicer" setB{\textstyle B} which differs fromA{\textstyle A} only by a null set (in the sense that thesymmetric difference(A∖B)∪(B∖A){\textstyle (A\setminus B)\cup (B\setminus A)} is a null set) and then show thatB{\textstyle B} can be generated using countable unions and intersections from open or closed sets.

Construction of the Lebesgue measure

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The modern construction of the Lebesgue measure is an application ofCarathéodory's extension theorem. It proceeds as follows.

Fixn∈N{\displaystyle n\in \mathbb {N} }. Abox inRn{\displaystyle \mathbb {R} ^{n}} is a set of the formB=∏i=1n[ai,bi],{\displaystyle B=\prod _{i=1}^{n}[a_{i},b_{i}]\,,}wherebi≥ai{\displaystyle b_{i}\geq a_{i}}, and the product symbol here represents a Cartesian product. The volume of this box is defined to bevol⁡(B)=∏i=1n(bi−ai).{\displaystyle \operatorname {vol} (B)=\prod _{i=1}^{n}(b_{i}-a_{i})\,.}Forany subsetA{\displaystyle A} ofRn{\displaystyle \mathbb {R} ^{n}}, we can define itsouter measureλ∗(A){\displaystyle \lambda ^{\!*\!}(A)} by:λ∗(A)=inf{∑B∈Cvol⁡(B):C is a countable collection of boxes whose union covers A}.{\displaystyle \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\}.}We then define the setA{\displaystyle A} to be Lebesgue-measurable if for every subsetS{\displaystyle S} ofRn{\displaystyle \mathbb {R} ^{n}},λ∗(S)=λ∗(S∩A)+λ∗(S∖A).{\displaystyle \lambda ^{*}(S)=\lambda ^{*}(S\cap A)+\lambda ^{*}(S\setminus A)\,.}These Lebesgue-measurable sets form aσ-algebra, and the Lebesgue measure is defined byλ(A)=λ∗(A){\displaystyle \lambda (A)=\lambda ^{\!*\!}(A)} for any Lebesgue-measurable setA{\displaystyle A}.

The existence of sets that are not Lebesgue-measurable is a consequence of the set-theoreticalaxiom of choice, which is independent from many of the conventional systems of axioms forset theory. TheVitali theorem, which follows from the axiom, states that there exist subsets ofR{\displaystyle \mathbb {R} } that are not Lebesgue-measurable. Assuming the axiom of choice,non-measurable sets with many surprising properties have been demonstrated, such as those of theBanach–Tarski paradox.

In 1970,Robert M. Solovay showed that the existence of sets that are not Lebesgue-measurable is not provable within the framework ofZermelo–Fraenkel set theory in the absence of the axiom of choice (seeSolovay's model).[9]

Relation to other measures

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TheBorel 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 onRn{\displaystyle \mathbb {R} ^{n}} is automatically alocally finite Borel measure, not every locally finite Borel measure onRn{\displaystyle \mathbb {R} ^{n}} is necessarily a Lebesgue measure. The Borel measure is translation-invariant, but notcomplete.

TheHaar measure can be defined on anylocally compact group and is a generalization of the Lebesgue measure (Rn{\displaystyle \mathbb {R} ^{n}} with addition is a locally compact group).

TheHausdorff measure is a generalization of the Lebesgue measure that is useful for measuring the subsets ofRn{\displaystyle \mathbb {R} ^{n}} of lower dimensions thann, likesubmanifolds, for example, surfaces or curves inR3{\displaystyle \mathbb {R} ^{3}} andfractal sets. The Hausdorff measure is not to be confused with the notion ofHausdorff dimension.

It can be shown thatthere is no infinite-dimensional analogue of Lebesgue measure.

See also

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References

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  1. ^The termvolume is also used, more strictly, as asynonym of 3-dimensional volume
  2. ^Lebesgue, H. (1902)."Intégrale, Longueur, Aire".Annali di Matematica Pura ed Applicata.7:231–359.doi:10.1007/BF02420592.S2CID 121256884.
  3. ^Royden, H. L. (1988).Real Analysis (3rd ed.). New York: Macmillan. p. 56.ISBN 0-02-404151-3.
  4. ^"Lebesgue-Maß". 29 August 2022. Retrieved9 March 2023 – via Wikipedia.
  5. ^Asaf Karagila."What sets are Lebesgue-measurable?". math stack exchange. Retrieved26 September 2015.
  6. ^Asaf Karagila."Is there a sigma-algebra on R strictly between the Borel and Lebesgue algebras?". math stack exchange. Retrieved26 September 2015.
  7. ^Osgood, William F. (January 1903)."A Jordan Curve of Positive Area".Transactions of the American Mathematical Society.4 (1). American Mathematical Society:107–112.doi:10.2307/1986455.ISSN 0002-9947.JSTOR 1986455.
  8. ^Carothers, N. L. (2000).Real Analysis. Cambridge: Cambridge University Press. pp. 293.ISBN 9780521497565.
  9. ^Solovay, Robert M. (1970). "A model of set-theory in which every set of reals is Lebesgue-measurable".Annals of Mathematics. Second Series.92 (1):1–56.doi:10.2307/1970696.JSTOR 1970696.
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