There are many modern concepts of integration, among these, the most common is based on the abstract mathematical theory known as Lebesgue integration, developed by Henri Lebesgue.

It is used throughout real analysis, in particular to define Lebesgue integration.

In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space ; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral.

This constructive measure theory provides the basis for computable analogues for Lebesgue integration.

Some of the technical deficiencies in Riemann integration can be remedied by the Riemann – Stieltjes integral, and most of these disappear with the Lebesgue integral.

This concept is important in mathematical analysis as the foundation for Lebesgue integration, and in probability theory, where it is interpreted as the collection of events which can be assigned probabilities.

According to Steinhaus, while he was strolling through the gardens he was surprised to over hear the term " Lebesgue measure " ( Lebesgue integration was at the time still a fairly new idea in mathematics ) and walked over to investigate.

According to Lebesgue integration theory, if ƒ and g are functions such that ƒ = g almost everywhere, then ƒ is integrable if and only if g is integrable and the integrals of ƒ and g are identical.

# REDIRECT Lebesgue integration

Henri Léon Lebesgue ForMemRS (; June 28, 1875 – July 26, 1941 ) was a French mathematician most famous for his theory of integration, which was a generalization of the 17th century concept of integration — summing the area between an axis and the curve of a function defined for that axis.

The first of these, unrelated to his development of Lebesgue integration, dealt with the extension of Baire's theorem to functions of two variables.

The next five dealt with surfaces applicable to a plane, the area of skew polygons, surface integrals of minimum area with a given bound, and the final note gave the definition of Lebesgue integration for some function f ( x ).

Lebesgue presents the problem of integration in its historical context, addressing Cauchy, Dirichlet, and Riemann.

In measure-theoretic analysis and related branches of mathematics, the Lebesgue – Stieltjes integral generalizes Riemann – Stieltjes and Lebesgue integration, preserving the many advantages of the latter in a more general measure-theoretic framework.

This is a non-technical treatment from a historical point of view ; see the article Lebesgue integration for a technical treatment from a mathematical point of view.

Lebesgue invented a new method of integration to solve this problem.

As part of the development of Lebesgue integration, Lebesgue invented the concept of measure, which extends the idea of length from intervals to a very large class of sets, called measurable sets ( so, more precisely, simple functions are functions that take a finite number of values, and each value is taken on a measurable set ).

The much stronger axiom of determinacy, or AD, implies that every set of reals is Lebesgue measurable, has the property of Baire, and has the perfect set property ( all three of these results are refuted by AC itself ).

* Hence, the set of algebraic numbers has Lebesgue measure zero ( as a subset of the complex numbers ), i. e. " almost all " complex numbers are not algebraic.

* The Cantor set is an example of an uncountable set that has Lebesgue measure zero.

The Lebesgue measure on R < sup > n </ sup > has the following properties:

A subset N of R has null Lebesgue measure and is considered to be a null set in R if and only if:

Since every singleton set has one-dimensional Lebesgue measure zero,

* if f is a monotonic function defined on an interval I, then f is differentiable almost everywhere on I, i. e. the set of numbers x in I such that f is not differentiable in x has Lebesgue measure zero.

Since x is now fixed, f ( x ) is a fixed countable set and has Lebesgue measure zero.

* Singular measure, a measure or probability distribution whose support has zero Lebesgue ( or other ) measure

While a general proof can be given that almost all numbers are normal ( in the sense that the set of exceptions has Lebesgue measure zero ), this proof is not constructive and only very few specific numbers have been shown to be normal.

It has Lebesgue measure 0.

More precisely: the set of complex n-by-n matrices that are not diagonalizable over C, considered as a subset of C < sup > n × n </ sup >, has Lebesgue measure zero.

In full generality, the isoperimetric inequality states that for any set S ⊂ R < sup > n </ sup > whose closure has finite Lebesgue measure

the corresponding Wishart distribution has no Lebesgue density.

This distribution has neither a probability density function nor a probability mass function, as it is not absolutely continuous with respect to Lebesgue measure, nor has it any point-masses.

This function has a singularity at 0, and is not Lebesgue integrable.

Any set which has a well-defined Lebesgue measure is said to be " measurable ", but the construction of the Lebesgue measure ( for instance using Carathéodory's extension theorem ) does not make it obvious whether there exist non-measurable sets.

In this case, is the smallest σ-algebra that contains the open intervals of R. While there are many Borel measures μ, the choice of Borel measure which assigns for every interval is sometimes called " the " Borel measure on R. In practice, even " the " Borel measure is not the most useful measure defined on the σ-algebra of Borel sets ; indeed, the Lebesgue measure is an extension of " the " Borel measure which possesses the crucial property that it is a complete measure ( unlike the Borel measure ).

The measure μ is equivalent to the Lebesgue measure on I, but it has the additional property that the transformation T preserves the measure μ.

Analytic sets are always Lebesgue measurable ( indeed, universally measurable ) and have the property of Baire and the perfect set property.

The axiom of determinacy is inconsistent with the axiom of choice ( AC ); the axiom of determinacy implies that all subsets of the real numbers are Lebesgue measurable, have the property of Baire, and the perfect set property.

PD implies that all projective sets are Lebesgue measurable ( in fact, universally measurable ) and have the perfect set property and the property of Baire.

* Every set of reals in L ( R ) is Lebesgue measurable ( in fact, universally measurable ) and has the property of Baire and the perfect set property.

By considering other games, we can show that Π < sup > 1 </ sup >< sub > n </ sub > determinacy implies that every Σ < sup > 1 </ sup >< sub > n + 1 </ sub > set of reals has the property of Baire, is Lebesgue measurable ( in fact universally measurable ) and has the perfect set property.