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Encyclopedia > Lebesgue measure

In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. It is used throughout real analysis, in particular to define Lebesgue 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 the axiom of choice, not all subsets of Rn are Lebesgue measurable. The "strange" behavior of non-measurable sets gives rise to such statements as the Banach-Tarski paradox, a consequence of the axiom of choice. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... For other uses of this word, see Length (disambiguation). ... Area is a physical quantity expressing the size of a part of a surface. ... The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... The integral can be interpreted as the area under a curve. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... In mathematics, a non-measurable set is a set whose structure is so complicated it sheds light on the very notion of length, area or volume. ... The Banach–Tarski paradox: A ball can be decomposed and reassembled into two balls the same size as the original. ...

Contents

Examples

  • If A is a closed interval [a, b], then its Lebesgue measure is the length ba. The open interval (a, b) has the same measure, since the difference between the two sets has measure zero.
  • If A is the Cartesian product of intervals [a, b] and [c, d], then it is a rectangle and its Lebesgue measure is the area (ba)(dc).
  • The Cantor set is an example of an uncountable set that has Lebesgue measure zero.

In mathematics, the Cartesian product is a direct product of sets. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ... In mathematics, an uncountable or nondenumerable set is a set which is not countable. ...

Properties

The Lebesgue measure on Rn has the following properties:

  1. If A is a cartesian product of intervals I1 × I2 × ... × In, 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.
  2. If A is a disjoint 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 (or infinite series) of the measures of the involved measurable sets.
  3. If A is Lebesgue measurable, then so is its complement.
  4. λ(A) ≥ 0 for every Lebesgue measurable set A.
  5. 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.)
  6. Countable unions and intersections of Lebesgue measurable sets are Lebesgue measurable. (A consequence of 2 and 3.)
  7. If A is an open or closed subset of Rn (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 (see the regularity theorem for Lebesgue measure).
  9. Lebesgue measure is both locally finite and inner regular, and so it is a Radon measure.
  10. Lebesgue measure is strictly positive on non-empty open sets, and so its support is the whole of Rn.
  11. 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.
  12. If A is Lebesgue measurable and x is an element of Rn, then the translation of A by x, defined by A + x = {a + x : aA}, is also Lebesgue measurable and has the same measure as A.
  13. If A is Lebesgue measurable and δ > 0, then the dilation of A by δ defined by delta A={delta x:xin A} is also Lebsgue measurable and has measure delta^{n}lambda,(A).
  14. More generally, if T is a linear transformation and A is a measurable subset of Rn, then T(A) is also Lebesgue measurable and has the measure |det(T)|, lambda,(A).
  15. If A is a Lebesgue measurable subset of Rn and f is an injective continuous function from A to Rn then f(A) is also a measurable set.

All the above may be succinctly summarized as follows: In mathematics, the Cartesian product is a direct product of sets. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. ... In mathematics the term countable set is used to describe the size of a set, e. ... In mathematics, a series is a sum of a sequence of terms. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space is either of two σ-algebras on a topological space X: The minimal σ-algebra containing the open sets. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics, the regularity theorem for Lebesgue measure is a result that, informally speaking, shows that every Lebesgue-measurable subset of the real line is approximately open and approximately closed. Lebesgue measure is a regular measure. ... In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure. ... In mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets. ... In mathematics, a Radon measure on a Hausdorff topological space X is a measure on the σ-algebra of Borel sets of X that is locally finite and inner regular. ... In mathematics, strict positivity is a concept in measure theory. ... In mathematics, the support of a measure μ on a measurable topological space (X, Borel(X)) 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... In measure theory, a null set is a set that is negligible for the purposes of the measure in question. ... This page includes English translations of several Latin phrases and abbreviations such as . ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... An injective function. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

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]times [0, 1]times cdots times [0, 1])=1.

The Lebesgue measure also has the property of being σ-finite. In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... In measure theory, a complete measure is a measure in which every subset of every null set is measurable (having measure 0). ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics, a positive (or signed) measure μ defined on a σ-algebra Σ of subsets of a set X is called finite, if μ(X) is a finite real number (rather than ∞). The measure μ is called σ-finite, if X is the countable union of measurable sets of finite measure. ...


Null sets

Main article: Null set

A subset of Rn 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, and so are sets in Rn whose dimension is smaller than n, for instance straight lines or circles in R2. In measure theory, a null set is a set that is negligible for the purposes of the measure in question. ... In mathematics the term countable set is used to describe the size of a set, e. ... 2-dimensional renderings (ie. ...


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 (AB) cup(BA) is a null set) and then show that B can be generated using countable unions and intersections from open or closed sets. In mathematics, the symmetric difference of two sets is the set of elements which are in one of either set, but not in both. ...


Construction of the Lebesgue measure

The modern construction of the Lebesgue measure, based on outer measures, is due to Carathéodory. It proceeds as follows: In mathematics, in particular in measure theory, an outer measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. ... This article or section does not cite its references or sources. ...


For any subset B of Rn, we can define an outer measure λ * by:

lambda^*(B) = inf {operatorname{vol}(M) : M supseteq B }, and M is a countable union of products of intervals .

Here, vol(M) is sum of the product of the lengths of the involved intervals. We then define the set A to be Lebesgue measurable if

lambda^*(B) = lambda^*(A cap B) + lambda^*(B - A)

for all sets B. 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. In mathematics, the Vitali set is an elementary example of a set of real numbers that is 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. In mathematics, the Borel algebra is the smallest σ-algebra on the real numbers R containing the intervals, and the Borel measure is the measure on this σ-algebra which gives to the interval [a, b] the measure b − a (where a < b). ... In measure theory, a complete measure is a measure in which every subset of every null set is measurable (having measure 0). ...


The Haar measure can be defined on any locally compact group and is a generalization of the Lebesgue measure (Rn with addition is a locally compact group). In mathematical analysis, the Haar measure is a way to assign an invariant volume to subsets of locally compact topological groups and subsequently define an integral for functions on those groups. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ...


The Hausdorff measure (see Hausdorff dimension) is a generalization of the Lebesgue measure that is useful for measuring the subsets of Rn of lower dimensions than n, like submanifolds, for example, surfaces or curves in R3 and fractal sets.[1] In mathematics, the Hausdorff dimension is an extended non-negative real number associated to any metric space. ... This is a glossary of terms specific to differential geometry and differential topology. ... The boundary of the Mandelbrot set is a famous example of a fractal. ...


It can be shown that there is no infinite-dimensional analogue of Lebesgue measure. This article or section does not cite any references or sources. ...


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. Henri Lebesgue Henri Léon Lebesgue (June 28, 1875, Beauvais – July 26, 1941, Paris) was a French mathematician, most famous for his theory of integration. ... 1901 (MCMI) was a common year starting on Tuesday (link will display calendar) of the Gregorian calendar (or a common year starting on Wednesday of the 13-day-slower Julian calendar). ... The integral can be interpreted as the area under a curve. ... 1902 (MCMII) was a common year starting on Wednesday (see link for calendar). ...


See also


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