Integrability may refer to:
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Integration may refer to:
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus, and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others.
In measure theory, a branch of mathematics, 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 n-dimensional volume, n-volume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by λ(A).
Measure is a fundamental concept of mathematics. Measures provide a mathematical abstraction for common notions like mass, distance/length, area, volume, probability of events, and — after some adjustments — electrical charge. These seemingly distinct concepts are innately very similar and may, in many cases, be treated as mathematically indistinguishable. Measures are foundational in probability theory. Far-reaching generalizations of measure are widely used in quantum physics and physics in general.
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. It was presented to the faculty at the University of Göttingen in 1854, but not published in a journal until 1868. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration.
Henri Léon Lebesgue was a French mathematician known 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. His theory was published originally in his dissertation Intégrale, longueur, aire at the University of Nancy during 1902.
DCT may refer to:
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus—differentiation and integration. This relationship is commonly characterized in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the Radon–Nikodym derivative, or density, of a measure.
In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if the double integral yields a finite answer when the integrand is replaced by its absolute value.
In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or positive or negative infinity; or in some instances as both endpoints approach limits. Such an integral is often written symbolically just like a standard definite integral, in some cases with infinity as a limit of integration.
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the L1 norm. Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann integration.
In measure-theoretic analysis and related branches of mathematics, Lebesgue–Stieltjes integration generalizes Riemann–Stieltjes and Lebesgue integration, preserving the many advantages of the former in a more general measure-theoretic framework. The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be associated to any function of bounded variation on the real line. The Lebesgue–Stieltjes measure is a regular Borel measure, and conversely every regular Borel measure on the real line is of this kind.
In mathematics, the Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral, Luzin integral or Perron integral, but not to be confused with the more general wide Denjoy integral – is one of a number of definitions of the integral of a function. It is a generalization of the Riemann integral, and in some situations is more general than the Lebesgue integral. In particular, a function is Lebesgue integrable if and only if the function and its absolute value are Henstock–Kurzweil integrable.
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In mathematics, Lebesgue integrability may refer to:
In mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable function is the limit of infinitesimal averages taken about the point. The theorem is named for Henri Lebesgue.
In measure theory, a discipline within mathematics, a pushforward measure is obtained by transferring a measure from one measurable space to another using a measurable function.
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the x-axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined.
Edward James McShane was an American mathematician noted for his advancements of the calculus of variations, integration theory, stochastic calculus, and exterior ballistics. His name is associated with the McShane–Whitney extension theorem. McShane was professor of mathematics at the University of Virginia, president of the American Mathematical Society, president of the Mathematical Association of America, a member of the National Science Board and a member of the National Academy of Sciences.