Riemann integral

Riemann integrableRiemann-integrableintegrableLebesgue integrability conditionRiemannRiemann integrabilityRiemann integrationRiemann sumsRiemann's senseRiemann-integrability
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval.wikipedia
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Bernhard Riemann

RiemannGeorg Friedrich Bernhard RiemannRiemann, Bernhard
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval.
In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series.

Riemann–Stieltjes integral

Stieltjes integralRiemann–Stieltjesintegrator
Some of the technical deficiencies in Riemann integration can be remedied with the Riemann–Stieltjes integral, and most disappear with the Lebesgue integral.
In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes.

Integral

integrationintegral calculusdefinite integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval.
In some cases such integrals may be defined by considering the limit of a sequence of proper Riemann integrals on progressively larger intervals.

Fundamental theorem of calculus

First Fundamental Theorem Of Calculusfundamental theorem of real calculusfundamental theorem of the calculus
For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration.
If f is Riemann integrable on [a,b] then

Darboux integral

Darboux integrabilityDarboux sumDarboux sums
(When f is discontinuous on a subinterval, there may not be a tag that achieves the infimum or supremum on that subinterval.) The Darboux integral, which is similar to the Riemann integral but based on Darboux sums, is equivalent to the Riemann integral.
Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal.

Partition of an interval

meshpartitionpartitions
A partition of an interval
Partitions are used in the theory of the Riemann integral, the Riemann–Stieltjes integral and the regulated integral.

Riemann sum

Rectangle methodRiemann sumsrectangle rule
. The Riemann sum of f with respect to the tagged partition
As the shapes get smaller and smaller, the sum approaches the Riemann integral.

Net (mathematics)

netnetsCauchy net
Both of these mean that eventually, the Riemann sum of f with respect to any partition gets trapped close to s. Since this is true no matter how close we demand the sums be trapped, we say that the Riemann sums converge to s. These definitions are actually a special case of a more general concept, a net.

Linear form

linear functionalcontinuous linear functionalcovector
Because the Riemann integral of a function is a number, this makes the Riemann integral a linear functional on the vector space of Riemann-integrable functions.
A typical example of a linear functional is integration: the linear transformation defined by the Riemann integral

Numerical integration

quadraturenumerical quadratureintegration
For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration.

Continuous function

continuouscontinuitycontinuous map
is Riemann integrable if and only if it is continuous almost everywhere (the set of its points of discontinuity has measure zero, in the sense of Lebesgue measure).
This definition is useful in descriptive set theory to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than ε (hence a G δ set) – and gives a very quick proof of one direction of the Lebesgue integrability condition.

Jordan measure

Peano-Jordan measureareaJordan content
Because C is not Jordan measurable,
The key step is then defining a bounded set to be Jordan measurable if it is "well-approximated" by simple sets, exactly in the same way as a function is Riemann integrable if it is well-approximated by piecewise-constant functions.

Uniform convergence

uniformlyconverges uniformlyuniformly convergent
is a uniformly convergent sequence on
The concept, which was first formalized by Karl Weierstrass, is important because several properties of the functions f_n, such as continuity, Riemann integrability, and, with additional hypotheses, differentiability, are transferred to the limit f if the convergence is uniform, but not necessarily if the convergence is not uniform.

Vector space

vectorvector spacesvectors
Because the Riemann integral of a function is a number, this makes the Riemann integral a linear functional on the vector space of Riemann-integrable functions.
(If one uses the Riemann integral instead, the space is not complete, which may be seen as a justification for Lebesgue's integration theory.

Measure (mathematics)

measuremeasure theorymeasurable
The Riemann integral can be interpreted measure-theoretically as the integral with respect to the Jordan measure.
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.

Improper integral

improper Riemann integralimproperimproper integrals
The simplest possible extension is to define such an integral as a limit, in other words, as an improper integral:
When the definite integral exists (in the sense of either the Riemann integral or the more advanced Lebesgue integral), this ambiguity is resolved as both the proper and improper integral will coincide in value.

Henstock–Kurzweil integral

Denjoy integralGauge integralHenstock-Kurzweil integral
An integral which is in fact a direct generalization of the Riemann integral is the Henstock–Kurzweil integral.
It is a generalization of the Riemann integral, and in some situations is more general than the Lebesgue integral.

Smith–Volterra–Cantor set

Fat Cantor set
For example, let C be the Smith–Volterra–Cantor set, and let

Henri Lebesgue

LebesgueHenri Léon LebesgueHenri Leon Lebesgue
It is due to Lebesgue and uses his measure zero, but makes use of neither Lebesgue's general measure or integral.
In the 19th century, Augustin Cauchy developed epsilon-delta limits, and Bernhard Riemann followed up on this by formalizing what is now called the Riemann integral.

Monotonic function

monotonicitymonotonemonotonic
If a real-valued function is monotone on the interval

Fourier series

Fourier coefficientFourier expansionFourier coefficients
In applications such as Fourier series it is important to be able to approximate the integral of a function using integrals of approximations to the function.

Antiderivative

indefinite integralindefinite integrationantidifferentiation
is an antiderivative of the integrable function

Oscillation (mathematics)

oscillationoscillationsMathematics of oscillation
One direction can be proven using the oscillation definition of continuity: For every positive ε, Let
This definition is useful in descriptive set theory to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than ε (hence a G δ set) – and gives a very quick proof of one direction of the Lebesgue integrability condition.

Lebesgue integration

Lebesgue integralLebesgue integrableintegral
Some of the technical deficiencies in Riemann integration can be remedied with the Riemann–Stieltjes integral, and most disappear with the Lebesgue integral.
The Riemann integral—proposed by Bernhard Riemann (1826–1866)—is a broadly successful attempt to provide such a foundation.

Mathematics

mathematicalmathmathematician
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval.