Riemann integral

Riemann 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. The Riemann integral is unsuitable for many theoretical purposes...

• Riemann sum
  Riemann sum
  In mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. It mayalso be used to define the integration operation. The method was named after German mathematician Bernhard Riemann....
  
  
• Riemann–Stieltjes integral
• Bounded variation
  Bounded variation
  In mathematical analysis, a function of bounded variation, also known as a BV function, is a real-valued function whose total variation is bounded : the graph of a function having this property is well behaved in a precise sense...
  
  
• Jordan content

Measure theory and the Lebesgue integral

• Measure (mathematics)
  Measure (mathematics)
  In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
  
  
  • Sigma algebra
    • Separable sigma algebra
      Separable sigma algebra
      In mathematics, σ-algebras are usually studied in the context of measure theory. A separable σ-algebra is a sigma algebra that can be generated by a countable collection of sets...
      
      
  • Filtration (abstract algebra)
    Filtration (abstract algebra)
    In mathematics, a filtration is an indexed set Si of subobjects of a given algebraic structure S, with the index i running over some index set I that is a totally ordered set, subject to the condition that if i ≤ j in I then Si ⊆ Sj...
    
    
• Borel algebra
  Borel algebra
  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 measure
• Indicator function
• Lebesgue measure
  Lebesgue measure
  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. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
  
  
• Lebesgue integration
  Lebesgue integration
  In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...
  
  
• Lebesgue's density theorem
  Lebesgue's density theorem
  In mathematics, Lebesgue's density theorem states that for any Lebesgue measurable set A, the "density" of A is 1 at almost every point in A...
  
  
• Counting measure
  Counting measure
  In mathematics, the counting measure is an intuitive way to put a measure on any set: the "size" of a subset is taken to be the number of elements in the subset, if the subset is finite, and ∞ if the subset is infinite....
  
  
• Complete measure
  Complete measure
  In mathematics, a complete measure is a measure space in which every subset of every null set is measurable...
  
  
• Haar measure
  Haar measure
  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....
  
  
• Outer measure
  Outer measure
  In mathematics, in particular in 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. A general theory of outer measures was first introduced by...
  
  
• Borel regular measure
  Borel regular measure
  In mathematics, an outer measure μ on n-dimensional Euclidean space Rn is called Borel regular if the following two conditions hold:...
  
  
• Radon measure
  Radon measure
  In mathematics , a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular.-Motivation:...
  
  
• Measurable function
  Measurable function
  In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
  
  
• Null set
  Null set
  In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set...
  
  , negligible set
  Negligible set
  In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose.As common examples, finite sets can be ignored when studying the limit of a sequence, and null sets can be ignored when studying the integral of a measurable function.Negligible sets define...
  
  
• Almost everywhere
  Almost everywhere
  In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
  
  , conull set
  Conull set
  In measure theory, a conull set is a set whose complement is null, i.e., the measure of the complement is zero. For example, the set of irrational numbers is a conull subset of the real line with Lebesgue measure.See also:*Almost everywhere...
  
  
• Lp space
  Lp space
  In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
  
  
• Borel-Cantelli lemma
  Borel-Cantelli lemma
  In probability theory, the Borel–Cantelli lemma is a theorem about sequences of events. In general, it is a result in measure theory. It is named after Émile Borel and Francesco Paolo Cantelli...
  
  
• Lebesgue's monotone convergence theorem
• Fatou's lemma
  Fatou's lemma
  In mathematics, Fatou's lemma establishes an inequality relating the integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions...
  
  
• Absolutely continuous
• Uniform absolute continuity
• Total variation
  Total variation
  In mathematics, the total variation identifies several slightly different concepts, related to the structure of the codomain of a function or a measure...
  
  
• Radon–Nikodym theorem
  Radon–Nikodym theorem
  In mathematics, the Radon–Nikodym theorem is a result in measure theory that states that, given a measurable space , if a σ-finite measure ν on is absolutely continuous with respect to a σ-finite measure μ on , then there is a measurable function f on X and taking values in [0,∞), such that\nu =...
  
  
• Fubini's theorem
  Fubini's theorem
  In mathematical analysis Fubini's theorem, named after Guido Fubini, is a result which gives conditions under which it is possible to compute a double integral using iterated integrals. As a consequence it allows the order of integration to be changed in iterated integrals.-Theorem...
  
  
  • Double integral
• Vitali set
  Vitali set
  In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by . The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vitali sets, and their existence is proven on the assumption of the axiom of...
  
  , non-measurable set
  Non-measurable set
  In mathematics, a non-measurable set is a set whose structure is so complicated that it cannot be assigned any meaningful measure. Such sets are constructed to shed light on the notions of length, area and volume in formal set theory....
  
  

Extensions

• Henstock-Kurzweil integral
  Henstock-Kurzweil integral
  In mathematics, the Henstock–Kurzweil integral, also known as the Denjoy integral and the Perron integral, is one of a number of definitions of the integral of a function. It is a generalization of the Riemann integral which in some situations is more useful than the Lebesgue integral.This integral...
  
  
• Amenable group
  Amenable group
  In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements...
  
  
• Banach–Tarski paradox
  Banach–Tarski paradox
  The Banach–Tarski paradox is a theorem in set theoretic geometry which states the following: Given a solid ball in 3-dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces , which can then be put back together in a different way to yield two...
  
  
• Hausdorff paradox
  Hausdorff paradox
  In mathematics, the Hausdorff paradox, named after Felix Hausdorff, states that if you remove a certain countable subset of the sphere S2, the remainder can be divided into three disjoint subsets A, B and C such that A, B, C and B ∪ C are all congruent...
  
  

Integral equation

Integral equation

In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...

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• Fredholm equation
  • Fredholm operator
    Fredholm operator
    In mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of integral equations. It is named in honour of Erik Ivar Fredholm....
    
    
• Liouville–Neumann series

Integral transforms

• Kernel (integral operator)
• Convolution
  Convolution
  In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
  
  

• Radon transform
  Radon transform
  thumb|right|Radon transform of the [[indicator function]] of two squares shown in the image below. Lighter regions indicate larger function values. Black indicates zero.thumb|right|Original function is equal to one on the white region and zero on the dark region....
  
  

Integral geometry

Integral geometry

In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant transformations from the space of functions on one geometrical space to the...

• Buffon's needle
  Buffon's needle
  In mathematics, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon:Buffon's needle was the earliest problem in geometric probability to be solved; it can be solved using integral geometry...
  
  
• Hadwiger's theorem
  Hadwiger's theorem
  In integral geometry , Hadwiger's theorem characterises the valuations on convex bodies in Rn. It was proved by Hugo Hadwiger.-Valuations:...
  
  
• mean width
  Mean width
  In geometry, the mean width is a measure of dimension length of the "size" a body; see Hadwiger's theorem for more about the available measures of bodies...
  
  
• intrinsic volumes

Other

• Stokes theorem
• Differentiation under the integral sign
• Contour integration
  • Examples of contour integration

See also

• List of calculus topics
• List of multivariable calculus topics
• List of real analysis topics
• List of integrals
• List of integrals of exponential functions
• List of integrals of hyperbolic functions
• List of integrals of irrational functions
• List of integrals of logarithmic functions
• List of integrals of rational functions
• List of integrals of trigonometric functions
• List of integrals of inverse trigonometric functions