Lebesgue measure

 ( Measures ) 

In measure theory, a branch of mathematics, the Lebesgue measure, named after france mathematician Henri Lebesgue, is the standard way of assigning a measure to of Euclidean space. For lower dimensions $n\; =\; 1,\; 2,\; \backslash text\{or\; \}\; 3$, it coincides with the standard measure of length, area, or volume. In general, it is also called -dimensional volume, -volume , hypervolume , or simply volume .The term volume is also used, more strictly, as a synonym of 3-dimensional volume 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 here denoted by $\backslash lambda(A)$.

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 dissertation Intégrale, Longueur, Aire in 1902.

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Definition For any interval $I\; =\; a,b$, or $I\; =\; (a,\; b)$, in the set $\backslash mathbb\{R\}$ of real numbers, let $\backslash ell(I)=\; b\; -\; a$ denote its length. For any subset $E\backslash subseteq\backslash mathbb\{R\}$, the Lebesgue outer measure

(1988). 9780024041517, Macmillan. ISBN 9780024041517

$\backslash lambda^\{\backslash !*\backslash !\}(E)$ is defined as an infimum

$$\backslash lambda^\{\backslash !*\backslash !\}(E)\; =\; \backslash inf\; \backslash left\backslash \{\backslash sum\_\{k=1\}^\backslash infty\; \backslash ell(I\_k)\; :\; \{(I\_k)\_\{k\; \backslash in\; \backslash mathbb\; N\}\}\; \backslash text\{\; is\; a\; sequence\; of\; open\; intervals\; with\; \}\; E\backslash subset\; \backslash bigcup\_\{k=1\}^\backslash infty\; I\_k\backslash right\backslash \}.$$

The above definition can be generalised to higher dimensions as follows. For any rectangular cuboid $C$ which is a Cartesian product $C=I\_1\backslash times\backslash cdots\backslash times\; I\_n$ of open intervals, let $\backslash operatorname\{vol\}(C)=\backslash ell(I\_1)\backslash times\backslash cdots\backslash times\; \backslash ell(I\_n)$ (a real number product) denote its volume. For any subset $E\backslash subseteq\backslash mathbb\{R^n\}$,

$$\backslash lambda^\{\backslash !*\backslash !\}(E)\; =\; \backslash inf\; \backslash left\backslash \{\backslash sum\_\{k=1\}^\backslash infty\; \backslash operatorname\{vol\}(C\_k)\; :\; \{(C\_k)\_\{k\; \backslash in\; \backslash mathbb\; N\}\}\; \backslash text\{\; is\; a\; sequence\; of\; products\; of\; open\; intervals\; with\; \}\; E\backslash subset\; \backslash bigcup\_\{k=1\}^\backslash infty\; C\_k\backslash right\backslash \}.$$

A set $E$ satisfies the Carathéodory criterion whenever, for every $A\backslash subseteq\; \backslash mathbb\{R^n\}$, we have:

$$\backslash lambda^\{\backslash !*\backslash !\}(A)\; =\; \backslash lambda^\{\backslash !*\backslash !\}(A\; \backslash cap\; E)\; +\; \backslash lambda^\{\backslash !*\backslash !\}(A\; \backslash cap\; E^\backslash complement).$$

Here, $E^\backslash complement$ is the complement of $E$. Sets $E$ satisfying the Carathéodory criterion are said to be Lebesgue-measurable. The set of all such $E$ forms a Sigma-algebra.

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

$\backslash lambda(E)\; =\; \backslash lambda^\{\backslash !*\backslash !\}(E)$.

ZFC proves that non-measurable sets do exist; examples are the .

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Intuition The first part of the definition states that the subset $E$ of the real numbers is reduced to its outer measure by coverage by sets of open intervals. Each of these sets of intervals $I$ covers $E$ in a sense, since the union of these intervals contains $E$. The total length of any covering interval set may overestimate the measure of $E,$ because $E$ is a subset of the union of the intervals, and so the intervals may include points which are not in $E$. The Lebesgue outer measure emerges as the 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 $E$ 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 subsets $A$ of the real numbers using $E$ as an instrument to split $A$ into two partitions: the part of $A$ which intersects with $E$ and the remaining part of $A$ which is not in $E$: the set difference of $A$ and $E$. These partitions of $A$ are subject to the outer measure. If for all possible such subsets $A$ of the real numbers, the partitions of $A$ cut apart by $E$ have outer measures whose sum is the outer measure of $A$, then the outer Lebesgue measure of $E$ gives its Lebesgue measure. Intuitively, this condition means that the set $E$ must not have some curious properties which causes a discrepancy in the measure of another set when $E$ 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.)

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Examples

• Any closed interval $a,$ of 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 between the two sets consists only of the end points $a$ and $b$, which each have measure zero.
• Any Cartesian product of intervals $a,$ and $c,$ is Lebesgue-measurable, and its Lebesgue measure is $(b\; -\; a)(c-d)$, the area of the corresponding rectangle.
• Moreover, every Borel set is Lebesgue-measurable. However, there are Lebesgue-measurable sets which are not Borel sets.
• Any countable set of real numbers has Lebesgue measure . In particular, the Lebesgue measure of the set of algebraic numbers is , even though the set is Dense set in $\backslash mathbb\{R\}$.
• The Cantor set and the set of are examples of that have Lebesgue measure .
• If the axiom of determinacy holds then all sets of reals are Lebesgue-measurable. Determinacy is however not compatible with the axiom of choice.
• are examples of sets that are not measurable with respect to the Lebesgue measure. Their existence relies on the axiom of choice.
• are simple plane with positive number Lebesgue measure (it can be obtained by small variation of the Peano curve construction). The dragon curve is another unusual example.
• Any line in $\backslash mathbb\{R\}^n$, for $n\; \backslash geq\; 2$, has a zero Lebesgue measure. In general, every proper hyperplane has a zero Lebesgue measure in its ambient space.
• The volume of an -ball can be calculated in terms of Euler's gamma function.

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Properties The Lebesgue measure on $\backslash mathbb\{R\}^n$ has the following properties:

1. If $A$ is a cartesian product of intervals $I\_1\; \backslash times\; I\_2\; \backslash times\; ...\; \backslash times\; I\_n$, then A is Lebesgue-measurable and $\backslash lambda\; (A)=|I\_1|\backslash cdot\; |I\_2|\backslash cdots\; |I\_n|.$
2. If $A$ is a union of countable pairwise disjoint Lebesgue-measurable sets, then $A$ is itself Lebesgue-measurable and $\backslash lambda(A)$ is equal to the sum (or infinite series) of the measures of the involved measurable sets.
3. If $A$ is Lebesgue-measurable, then so is its complement.
4. $\backslash lambda(A)\; \backslash geq\; 0$ for every Lebesgue-measurable set $A$.
5. If $A$ and $B$ are Lebesgue-measurable and $A$ is a subset of $B$, then $\backslash lambda(A)\; \backslash leq\; \backslash lambda(B)$. (A consequence of 2.)
6. 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 does not need to be closed under countable unions: $\backslash \{\backslash emptyset,\; \backslash \{1,2,3,4\backslash \},\; \backslash \{1,2\backslash \},\; \backslash \{3,4\backslash \},\; \backslash \{1,3\backslash \},\; \backslash \{2,4\backslash \}\backslash \}$.)
7. If $A$ is an open set or closed set subset of $\backslash mathbb\{R\}^n$ (or even Borel set, see metric space), then $A$ is Lebesgue-measurable.
8. If $A$ 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, $E\backslash subset\; \backslash mathbb\{R\}$ is Lebesgue-measurable if and only if for every $\backslash varepsilon>0$ there exist an open set $G$ and a closed set $F$ such that $F\backslash subset\; E\backslash subset\; G$ and .
   N. L. Carothers (2025). 9780521497565, Cambridge University Press. . ISBN 9780521497565
10. A Lebesgue-measurable set can be "squeezed" between a containing set and a contained . I.e., if $A$ is Lebesgue-measurable then there exist a set $G$ and an $F$ such that $F\; \backslash subseteq\; A\; \backslash subseteq\; G$ and $\backslash lambda(G\; \backslash setminus\; A)\; =\; \backslash lambda\; (A\; \backslash setminus\; F)\; =\; 0$.
11. Lebesgue measure is both locally finite and inner regular, and so it is a Radon measure.
12. Lebesgue measure is strictly positive on non-empty open sets, and so its support is the whole of $\backslash mathbb\{R\}^n$.
13. If $A$ is a Lebesgue-measurable set with $\backslash lambda(A)\; =\; 0$ (a null set), then every subset of $A$ is also a null set. A fortiori, every subset of $A$ is measurable.
14. If $A$ is Lebesgue-measurable and x is an element of $\backslash mathbb\{R\}^n$, then the translation of $A$ by $x$, defined by $A\; +\; x\; :=\; \backslash \{a\; +\; x:\; a\; \backslash in\; A\backslash \}$, is also Lebesgue-measurable and has the same measure as $A$.
15. If $A$ is Lebesgue-measurable and $\backslash delta>0$, then the dilation of $A$ by $\backslash delta$ defined by $\backslash delta\; A=\backslash \{\backslash delta\; x:x\backslash in\; A\backslash \}$ is also Lebesgue-measurable and has measure $\backslash delta^\{n\}\backslash lambda\backslash ,(A).$
16. More generally, if $T$ is a linear transformation and $A$ is a measurable subset of $\backslash mathbb\{R\}^n$, then $T(A)$ is also Lebesgue-measurable and has the measure $\backslash left|\backslash det(T)\backslash right|\; \backslash lambda(A)$.

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

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

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Null sets A subset of $\backslash mathbb\{R\}^n$ is a null set if, for every $\backslash varepsilon\; >\; 0$, it can be covered with countably many products of n intervals whose total volume is at most $\backslash varepsilon$. All countable sets are null sets.

If a subset of $\backslash mathbb\{R\}^n$ has Hausdorff dimension less than then it is a null set with respect to -dimensional Lebesgue measure. Here Hausdorff dimension is relative to the Euclidean metric on $\backslash mathbb\{R\}^n$ (or any metric Rudolf Lipschitz equivalent to it). On the other hand, a set may have topological dimension less than and have positive -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 $A$ 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 $(A\; \backslash setminus\; B)\; \backslash cup\; (B\; \backslash setminus\; A)$ 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 The modern construction of the Lebesgue measure is an application of Carathéodory's extension theorem. It proceeds as follows.

Fix $n\; \backslash in\; \backslash mathbb\; N$. A box in $\backslash mathbb\{R\}^n$ is a set of the form$$B=\backslash prod\_\{i=1\}^n\; a\_i,b\_i\; \backslash ,\; ,$$where $b\_i\; \backslash geq\; a\_i$, and the product symbol here represents a Cartesian product. The volume of this box is defined to be$$\backslash operatorname\{vol\}(B)=\backslash prod\_\{i=1\}^n\; (b\_i-a\_i)\; \backslash ,\; .$$For any subset $A$ of $\backslash mathbb\{R\}^n$, we can define its outer measure $\backslash lambda^\{\backslash !*\backslash !\}(A)$ by:$$\backslash lambda^*(A)\; =\; \backslash inf\; \backslash left\backslash \{\backslash sum\_\{B\backslash in\; \backslash mathcal\{C\}\}\backslash operatorname\{vol\}(B)\; :\; \backslash mathcal\{C\}\backslash text\{\; is\; a\; countable\; collection\; of\; boxes\; whose\; union\; covers\; \}A\backslash right\backslash \}\; .$$We then define the set $A$ to be Lebesgue-measurable if for every subset $S$ of $\backslash mathbb\{R\}^n$,$$\backslash lambda^*(S)\; =\; \backslash lambda^*(S\; \backslash cap\; A)\; +\; \backslash lambda^*(S\; \backslash setminus\; A)\; \backslash ,\; .$$These Lebesgue-measurable sets form a σ-algebra, and the Lebesgue measure is defined by $\backslash lambda(A)\; =\; \backslash lambda^\{\backslash !*\backslash !\}(A)$ for any Lebesgue-measurable set $A$.

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, which follows from the axiom, states that there exist subsets of $\backslash mathbb\{R\}$ that are not Lebesgue-measurable. Assuming the axiom of choice, non-measurable sets with many surprising properties have been demonstrated, such as those of 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).

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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 $\backslash mathbb\{R\}^n$ is automatically a locally finite Borel measure, not every locally finite Borel measure on $\backslash mathbb\{R\}^n$ is necessarily a Lebesgue measure. The Borel measure is translation-invariant, but not Complete measure.

The Haar measure can be defined on any locally compact group and is a generalization of the Lebesgue measure ($\backslash mathbb\{R\}^n$ 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 $\backslash mathbb\{R\}^n$ of lower dimensions than , like , for example, surfaces or curves in $\backslash mathbb\{R\}^3$ 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 analogue of Lebesgue measure.

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See also

• 4-volume
• Edison Farah
• Lebesgue's density theorem
• Lebesgue measure of the set of Liouville numbers
• Non-measurable set
• Vitali set
• Peano–Jordan measure

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