In measure theory, a branch of mathematics, the **Lebesgue measure**, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets 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**,

- Definition
- Intuition
- Examples
- Properties
- Null sets
- Construction of the Lebesgue measure
- Relation to other measures
- See also
- References

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.^{ [2] }

The Lebesgue measure is often denoted by *dx*, but this should not be confused with the distinct notion of a volume form.

For any interval (or ) in the set of real numbers, let denote its length. For any subset , the Lebesgue outer measure ^{ [3] } is defined as an infimum

Some sets satisfy the Carathéodory criterion, which requires that for every ,

The set of all such forms a *σ*-algebra. For any such , its Lebesgue measure is defined to be its Lebesgue outer measure: .

A set that does not satisfy the Carathéodory criterion is not Lebesgue-measurable. Non-measurable sets do exist; an example is the Vitali sets.

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

- Any closed interval [
*a*,*b*] of real numbers is Lebesgue-measurable, and its Lebesgue measure is the length*b*−*a*. The open interval (*a*,*b*) has the same measure, since the difference between the two sets consists only of the end points*a*and*b*and has measure zero. - 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. - Moreover, every Borel set is Lebesgue-measurable. However, there are Lebesgue-measurable sets which are not Borel sets.
^{ [4] }^{ [5] } - Any countable set of real numbers has Lebesgue measure 0. In particular, the Lebesgue measure of the set of algebraic numbers is 0, even though the set is dense in
**R**. - The Cantor set and the set of Liouville numbers are examples of uncountable sets that have Lebesgue measure 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 sets are examples of sets that are not measurable with respect to the Lebesgue measure. Their existence relies on the axiom of choice.
- Osgood curves are simple plane curves with positive Lebesgue measure
^{ [6] }(it can be obtained by small variation of the Peano curve construction). The dragon curve is another unusual example. - Any line in , for , has a zero Lebesgue measure. In general, every proper hyperplane has a zero Lebesgue measure in its ambient space.

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 Here, |*I*| denotes the length of the interval*I*. - If
*A*is a disjoint union of 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. - 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 does not need to be closed under countable unions: .)
- If
*A*is an open or closed subset of**R**^{n}(or even Borel set, see metric 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 the regularity theorem for 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, is Lebesgue-measurable if and only if for every there exist an open set and a closed set such that and .
^{ [7] } - A Lebesgue-measurable set can be "squeezed" between a containing G
_{δ}set and a contained F_{σ}. I.e, if*A*is Lebesgue-measurable then there exist a G_{δ}set*G*and an F_{σ}*F*such that*G*⊇*A*⊇*F*and*λ*(*G*\*A*) =*λ*(*A*\*F*) = 0. - Lebesgue measure is both locally finite and inner regular, and so it is a Radon 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 (a null 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 , then the*dilation of by*defined by is also Lebesgue-measurable and has measure - More generally, if
*T*is a linear transformation and*A*is a measurable subset of**R**^{n}, then*T*(*A*) is also Lebesgue-measurable and has the measure .

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 a
*σ*-algebra containing all products of intervals, and*λ*is the unique complete translation-invariant measure on that σ-algebra with

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

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. All countable sets are null sets.

If a subset of **R**^{n} 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**^{n} (or any metric Lipschitz 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.

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*) ∪ (*B* − *A*) is a null set) and then show that *B* can be generated using countable unions and intersections from open or closed sets.

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

Fix *n*∈**N**. A **box** in **R**^{n} is a set of the form

where *b _{i}*≥

For *any* subset *A* of **R**^{n}, we can define its outer measure *λ**(*A*) by:

We then define the set *A* to be Lebesgue-measurable if for every subset *S* of **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*.

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 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 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).^{ [8] }

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 any locally compact group and is a generalization of the Lebesgue measure (**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 **R**^{n} of lower dimensions than *n*, like submanifolds, 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.

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

In mathematics, specifically in measure theory, a **Borel measure** on a topological space is a measure that is defined on all open sets. Some authors require additional restrictions on the measure, as described below.

In mathematics, a **measure** on a set is a systematic way to assign a number to subsets of a set, intuitively interpreted as the size of the subset. Those sets which can be associated with such a number, we call **measurable sets**. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space. This assigns the usual length, area, or volume to certain subsets of the given Euclidean space. For instance, the Lebesgue measure of an interval of real numbers is its usual length, but also assigns numbers to other kinds of sets in a way that is consistent with the lengths of intervals.

In mathematical analysis, a **null set** is a set that has **measure zero**. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.

In mathematical analysis and in probability theory, a **σ-algebra** on a set *X* is a collection Σ of subsets of *X* that includes *X* itself, is closed under complement, and is closed under countable unions.

In mathematics, a **Borel set** is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel.

In mathematics and in particular measure theory, a **measurable function** is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable.

In mathematics, a **complete measure** is a measure space in which every subset of every null set is measurable. More formally, a measure space (*X*, Σ, *μ*) is complete if and only if

In calculus, **absolute continuity** is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus—differentiation and integration. This relationship is commonly characterized in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: *absolute continuity of functions* and *absolute continuity of measures.* These two notions are generalized in different directions. The usual derivative of a function is related to the *Radon–Nikodym derivative*, or *density*, of a measure.

In mathematics, the **Radon–Nikodym theorem** is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A *measure* is a set function that assigns a consistent magnitude to the measurable subsets of a measurable space. Examples of a measure include area and volume, where the subsets are sets of points; or the probability of an event, which is a subset of possible outcomes within a wider probability space.

In mathematics, a **Vitali set** is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali in 1905. The **Vitali theorem** is the existence theorem that there are such sets. There are uncountably many Vitali sets, and their existence depends on the axiom of choice. In 1970, Robert Solovay constructed a model of Zermelo–Fraenkel set theory without the axiom of choice where all sets of real numbers are Lebesgue measurable, assuming the existence of an inaccessible cardinal.

In the mathematical field of measure theory, an **outer measure** or **exterior measure** is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer measures was first introduced by Constantin Carathéodory to provide an abstract basis for the theory of measurable sets and countably additive measures. Carathéodory's work on outer measures found many applications in measure-theoretic set theory, and was used in an essential way by Hausdorff to define a dimension-like metric invariant now called Hausdorff dimension. Outer measures are commonly used in the field of geometric measure theory.

In mathematics, **Hausdorff measure** is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in or, more generally, in any metric space.

In mathematics, a **non-measurable set** is a set which cannot be assigned a meaningful "volume". The mathematical existence of such sets is construed to provide information about the notions of length, area and volume in formal set theory. In Zermelo-Fraenkel set theory, the axiom of choice entails that non-measurable subsets of exist.

In mathematics, a **regular measure** on a topological space is a measure for which every measurable set can be approximated from above by open measurable sets and from below by compact measurable sets.

In mathematics, a positive measure *μ* defined on a *σ*-algebra Σ of subsets of a set *X* is called a finite measure if *μ*(*X*) is a finite real number, and a set *A* in Σ is of finite measure if *μ*(*A*) < ∞*.* The measure *μ* is called **σ-finite** if *X* is the countable union of measurable sets with finite measure. A set in a measure space is said to have ** σ-finite measure** if it is a countable union of measurable sets with finite measure. A measure being σ-finite is a weaker condition than being finite, i.e. all finite measures are σ-finite but there are (many) σ-finite measures that are not finite.

In mathematics, a function *f* on the interval [*a*, *b*] has the **Luzin N property**, named after Nikolai Luzin if for all such that , there holds: , where stands for the Lebesgue measure.

In mathematics, **Gaussian measure** is a Borel measure on finite-dimensional Euclidean space **R**^{n}, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named after the German mathematician Carl Friedrich Gauss. One reason why Gaussian measures are so ubiquitous in probability theory is the central limit theorem. Loosely speaking, it states that if a random variable *X* is obtained by summing a large number *N* of independent random variables of order 1, then *X* is of order and its law is approximately Gaussian.

In mathematics, the **support** of a measure *μ* on a measurable topological space is a precise notion of where in the space *X* the measure "lives". It is defined to be the largest (closed) subset of *X* for which every open neighbourhood of every point of the set has positive measure.

In mathematics, a **content** is a set function that is like a measure, but a content must only be finitely additive, whereas a measure must be countably additive. A content is a real function defined on a collection of subsets such that

In mathematics, the notions of **prevalence and shyness** are notions of "almost everywhere" and "measure zero" that are well-suited to the study of infinite-dimensional spaces and make use of the translation-invariant Lebesgue measure on finite-dimensional real spaces. The term "shy" was suggested by the American mathematician John Milnor.

- ↑ The term
*volume*is also used, more strictly, as a synonym of 3-dimensional volume - ↑ Henri Lebesgue (1902). "Intégrale, longueur, aire". Université de Paris.Cite journal requires
`|journal=`

(help) - ↑ Royden, H. L. (1988).
*Real Analysis*(3rd ed.). New York: Macmillan. p. 56. ISBN 0-02-404151-3. - ↑ Asaf Karagila. "What sets are Lebesgue-measurable?". math stack exchange. Retrieved 26 September 2015.
- ↑ Asaf Karagila. "Is there a sigma-algebra on R strictly between the Borel and Lebesgue algebras?". math stack exchange. Retrieved 26 September 2015.
- ↑ Osgood, William F. (January 1903). "A Jordan Curve of Positive Area".
*Transactions of the American Mathematical Society*. American Mathematical Society.**4**(1): 107–112. doi: 10.2307/1986455 . ISSN 0002-9947. JSTOR 1986455. - ↑ Carothers, N. L. (2000).
*Real Analysis*. Cambridge: Cambridge University Press. pp. 293. ISBN 9780521497565. - ↑ 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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