As the theory of Lebesgue integration developed, it was assumed that any set of zero measure would be a set of uniqueness — in one dimension the locality principle for Fourier series shows that any set of positive measure is a set of multiplicity ( in higher dimensions this is still an open question ).

Lebesgue presents six conditions which it is desirable that the integral should satisfy, the last of which is " If the sequence f < sub > n </ sub >( x ) increases to the limit f ( x ), the integral of f < sub > n </ sub >( x ) tends to the integral of f ( x ).

In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions.

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.

* Lambda denotes the Lebesgue measure in mathematical set theory.

In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space.

Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon and Maurice Fréchet, among others.

The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory.

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.

Measure theory provides the foundation for the modern notion of integral, the Lebesgue integral.

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

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 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.

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.

For functions on the real line, the Henstock integral is an even more general notion of integral ( based on Riemann's theory rather than Lebesgue's ) that subsumes both Lebesgue integration and improper Riemann integration.

* Dominated convergence theorem, a central mathematical theorem in the theory of integration first proposed by Henri Lebesgue

* it made possible the development of modern measure theory by Lebesgue and the rudiments of functional analysis by Hilbert ;

The Lebesgue theory does not see this as a deficiency: from the point of view of measure theory, and cannot be defined satisfactorily.

* For the Henstock – Kurzweil integral, improper integration is not necessary, and this is seen as a strength of the theory: it encompasses all Lebesgue integrable and improper Riemann integrable functions.

* Henri Lebesgue introduces the theory of Lebesgue integration.

** The Lebesgue measure of a countable disjoint union of measurable sets is equal to the sum of the measures of the individual sets.

* 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.

In analysis, the area of a subset of the plane is defined using Lebesgue measure, though not every subset is measurable.

When speaking about the reals, sometimes it means " all reals but a set of Lebesgue measure zero " ( formally, almost everywhere ).

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 ).

To clarify, when one says that the Lebesgue measure is an extension of the Borel measure, it means that every Borel measurable set E is also a Lebesgue measurable set, and the Borel measure and the Lebesgue measure coincide on the Borel sets ( i. e., for every Borel measurable set ).

Similarly, although a subset of the real numbers that is not Lebesgue measurable can be proven to exist using the axiom of choice, it is consistent that no such set is definable.

** The Vitali theorem on the existence of non-measurable sets which states that there is a subset of the real numbers that is not Lebesgue measurable.

It is also consistent with ZF + DC that every set of reals is Lebesgue measurable ; however, this consistency result, due to Robert M. Solovay, cannot be proved in ZFC itself, but requires a mild large cardinal assumption ( the existence of an inaccessible cardinal ).

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 ).

Sets that can be assigned a Lebesgue measure are called Lebesgue measurable ; the measure of the Lebesgue measurable set A is denoted by λ ( A ).

* Any closed interval b of real numbers is Lebesgue measurable, and its Lebesgue measure is the length b − a.

* Any Cartesian product of intervals b and d is Lebesgue measurable, and its Lebesgue measure is ( b − a )( d − c ), the area of the corresponding rectangle.

In contrast, functions that are not Lebesgue measurable are generally considered pathological, at least in the field of analysis.

* A Lebesgue measurable function is a measurable function, where is the sigma algebra of Lebesgue measurable sets, and is the Borel algebra on the complex numbers C. Lebesgue measurable functions are of interest in mathematical analysis because they can be integrated.

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

He also proves that the Riemann – Lebesgue lemma is a best possible result for continuous functions, and gives some treatment to Lebesgue constants.

Lebesgue integration has the property that every bounded function defined over a bounded interval with a Riemann integral also has a Lebesgue integral, and for those functions the two integrals agree.

But there are many functions with a Lebesgue integral that have no Riemann integral.

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 Lebesgue integral integrates many of these functions ( always reproducing the same answer when it did ), but not all of them.

For Lebesgue measurable functions, the theorem can be stated in the following form:

Independently of Riesz, he discovered the representation theorem in the space of Lebesgue square integrable functions.

In mathematics, Fatou's lemma establishes an inequality relating the integral ( in the sense of Lebesgue ) of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions.

* The Lebesgue integral deals differently with unbounded domains and unbounded functions, so that often an integral which only exists as an improper Riemann integral will exist as a ( proper ) Lebesgue integral, such as.