- Lebesgue measure
In

mathematics , the**Lebesgue measure**, named afterHenri Lebesgue , is the standard way of assigning alength ,area orvolume tosubset s ofEuclidean space . It is used throughoutreal analysis , in particular to defineLebesgue integration . Sets which can be assigned a volume are called**Lebesgue measurable**; the volume or measure of the Lebesgue measurable set "A" is denoted by λ("A"). A Lebesgue measure of ∞ is possible, but even so, assuming theaxiom of choice , not all subsets of**R**^{"n"}are Lebesgue measurable. The "strange" behavior ofnon-measurable set s gives rise to such statements as the Banach-Tarski paradox, a consequence of the axiom of choice.Lebesgue measure is often denoted $,dx$, but this should not be confused with the distinct notion of a

volume form .**Examples*** If "A" is a closed interval ["a", "b"] , then its Lebesgue measure is the length "b"−"a". The open interval ("a", "b") has the same measure, since the difference between the two sets has measure zero.

* If "A" is theCartesian product of intervals ["a", "b"] and ["c", "d"] , then it is a rectangle and its Lebesgue measure is the area ("b"−"a")("d"−"c").

* TheCantor set is an example of anuncountable set that has Lebesgue measure zero.**Properties**The Lebesgue measure on

**R**^{"n"}has the following properties:# If "A" is a

cartesian product of intervals "I"_{1}× "I"_{2}× ... × "I"_{"n"}, then "A" is Lebesgue measurable and $lambda\; (A)=|I\_1|cdot\; |I\_2|cdots\; |I\_n|.$ Here, |"I"| denotes the length of the interval "I".

# If "A" is adisjoint union of finitely many or countably many disjoint Lebesgue measurable sets, then "A" is itself Lebesgue measurable and λ("A") is equal to the sum (orinfinite series ) of the measures of the involved measurable sets.

# If "A" is Lebesgue measurable, then so is its complement.

# λ("A") ≥ 0 for every Lebesgue measurable set "A".

# 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.)

# Countable unions and 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: $\{emptyset,\; \{1,2,3,4\},\; \{1,2\},\; \{3,4\},\; \{1,3\},\; \{2,4\}\}$.)

# If "A" is an open or closed subset of**R**^{"n"}(or evenBorel set , seemetric space ), then "A" is Lebesgue measurable.

# If "A" is a Lebesgue measurable set, then it is "approximately open" and "approximately closed" in the sense of Lebesgue measure (see theregularity theorem for Lebesgue measure ).

# Lebesgue measure is both locally finite and inner regular, and so it is aRadon measure .

# Lebesgue measure is strictly positive on non-empty open sets, and so its support is the whole of**R**^{"n"}.

# If "A" is a Lebesgue measurable set with λ("A") = 0 (anull set ), then every subset of "A" is also a null set.A fortiori , every subset of "A" is measurable.

# If "A" is Lebesgue measurable and "x" is an element of**R**^{"n"}, 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".

# If "A" is Lebesgue measurable and $delta>0$, then the "dilation of $A$ by $delta$" defined by $delta\; A=\{delta\; x:xin\; A\}$ is also Lebesgue measurable and has measure $delta^\{n\}lambda,(A)$.

# More generally, if "T" is alinear transformation and "A" is a measurable subset of**R**^{"n"}, then "T"("A") is also Lebesgue measurable and has the measure $|det(T)|,\; lambda,(A)$.All the above may be succinctly summarized as follows:

: The Lebesgue measurable sets form a σ-algebra containing all products of intervals, and λ is the unique complete translation-invariant measure on that σ-algebra with $lambda(\; [0,1]\; imes\; [0,\; 1]\; imes\; cdots\; imes\; [0,\; 1]\; )=1.$

The Lebesgue measure also has the property of being σ-finite.

**Null sets**A subset of

**R**^{"n"}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. Allcountable sets are null sets.If a subset of

**R**^{"n"}hasHausdorff dimension less than "n" then it is a null set with respect to "n"-dimensional Lebesgue measure. Here Hausdorff dimension is relative to theEuclidean metric on**R**^{"n"}(or any metricLipschitz equivalent to it). On the other hand a set may havetopological dimension less than "n" and have positive "n"-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 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") $cup$("B" − "A") is a null set) and then show that "B" 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, based on

outer measure s, is due to Carathéodory. It proceeds as follows.Fix $ninmathbb\; N$. A

**box**in $R^n$ is a set of the form $B=prod\_\{i=1\}^n\; [a\_i,b\_i]$, where $b\_ige\; a\_i$. The volume $operatorname\{vol\}(B)$ of this box is defined to be $prod\_\{i=1\}^n\; (b\_i-a\_i).$For "any" subset "A" of

**R**^{"n"}, we can define its outer measure $lambda^*(A)$ by::$lambda^*(A)\; =\; inf\; Bigl\{sum\_\{jin\; J\}operatorname\{vol\}(B\_j)\; :\; \{B\_j:jin\; J\}\; ext\{\; is\; a\; countable\; collection\; of\; boxes\; whose\; union\; covers\; \}ABigr\}\; .$

We then define the set "A" to be Lebesgue measurable if

:$lambda^*(S)\; =\; lambda^*(A\; cap\; S)\; +\; lambda^*(S\; -\; A)$

for all sets $Ssubset\; R^n$. These Lebesgue measurable sets form a

σ-algebra , and the Lebesgue measure is defined by λ("A") = λ^{*}("A") for any Lebesgue measurable set "A".According to the Vitali theorem there exists a subset of the real numbers

**R**that is not Lebesgue measurable. Much more is true: if "A" is any subset of $R^n$ of positive measure, then "A" has subsets which are not Lebesgue measurable.**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. The Borel measure is translation-invariant, but not complete.The

Haar measure can be defined on anylocally compact group and is a generalization of the Lebesgue measure (**R**^{"n"}with addition is a locally compact group).The Hausdorff measure (see

Hausdorff dimension ) is a generalization of the Lebesgue measure that is useful for measuring the subsets of**R**^{"n"}of lower dimensions than "n", likesubmanifold s, for example, surfaces or curves in**R**³ andfractal 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 analogue of Lebesgue measure.

**History**Henri Lebesgue described his measure in 1901, followed the next year by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902.**ee also***

Lebesgue's density theorem

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