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Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure Illustration (Vector)
Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Measure of the National Assembly for Wales, primary legislation in Wales * Assembly Measure of the Northern Ireland Assembly (1973) Science and mathematics * Measure (data warehouse), a property on which calculations can be made * Measure (mathematics), a systematic way to assign a number to each suitable subset of that set * Measure (physics), a way to integrate over all possible histories of a system in quantum field theory * Measure (termination), in computer program termination analysis * Measuring coalgebra, a coalgebra constructed from two algebras * Measure (Apple), an iOS augmented reality app Other uses * ''Measure'' (album), by Matt Pond PA, 2000, and its title track * Measure (bartending) or jigger, a bartending tool used to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Considered the greatest mathematician of ancient history, and one of the greatest of all time,* * * * * * * * * * Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems. These include the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral. Heath, Thomas L. 1897. ''Works of Archimedes''. Archimedes' other mathematical achievements include deriving an approximation of pi, defining and in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements. In more technical terms, assuming the axiom of countable choice, a set is ''countable'' if its cardinality (its number of elements) is not greater than that of the natural numbers. A countable set that is not finite is said countably infinite. The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers. A note on terminology Although the terms "countable" and "countably infinite" as defined here are quite comm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sigma Additivity
In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of ''k'' disjoint sets (where ''k'' is a finite number) equals the sum of its values on the sets. Therefore, an additive set function is also called a finitely-additive set function (the terms are equivalent). However, a finitely-additive set function might not have the additivity property for a union of an ''infinite'' number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is, \mu\left(\bigcup_^\infty A_n\right) = \sum_^\infty \mu(A_n). Additivity and sigma-additivity are particularly important properties of measures. They are abstrac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extended Real Number Line
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on infinities and the various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affinely extended real number system is denoted \overline or \infty, +\infty/math> or It is the Dedekind–MacNeille completion of the real numbers. When the meaning is clear from context, the symbol +\infty is often written simply as Motivation Limits It is often useful to describe the behavior of a function f, as either the argument x or the function value f gets "infinitely large" in some sense. For example, consider the function f defined by :f(x) = \frac. The graph of this function has a horizontal asymptote at y = 0. Geometrically, when moving increasingly farther to the right along t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Function
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R and \pm \infty. A set function generally aims to subsets in some way. Measures are typical examples of "measuring" set functions. Therefore, the term "set function" is often used for avoiding confusion between the mathematical meaning of "measure" and its common language meaning. Definitions If \mathcal is a family of sets over \Omega (meaning that \mathcal \subseteq \wp(\Omega) where \wp(\Omega) denotes the powerset) then a is a function \mu with domain \mathcal and codomain \infty, \infty/math> or, sometimes, the codomain is instead some vector space, as with vector measures, complex measures, and projection-valued measures. The domain is a set function may have any number properties; the commonly encountered properties and categor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Countable Additivity Of A Measure
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements. In more technical terms, assuming the axiom of countable choice, a set is ''countable'' if its cardinality (its number of elements) is not greater than that of the natural numbers. A countable set that is not finite is said countably infinite. The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers. A note on terminology Although the terms "countable" and "countably infinite" as defined here are quite c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maurice Fréchet
Maurice may refer to: People *Saint Maurice (died 287), Roman legionary and Christian martyr *Maurice (emperor) or Flavius Mauricius Tiberius Augustus (539–602), Byzantine emperor *Maurice (bishop of London) (died 1107), Lord Chancellor and Lord Keeper of England *Maurice of Carnoet (1117–1191), Breton abbot and saint * Maurice, Count of Oldenburg (fl. 1169–1211) *Maurice of Inchaffray (14th century), Scottish cleric who became a bishop *Maurice, Elector of Saxony (1521–1553), German Saxon nobleman *Maurice, Duke of Saxe-Lauenburg (1551–1612) *Maurice of Nassau, Prince of Orange (1567–1625), stadtholder of the Netherlands *Maurice, Landgrave of Hesse-Kassel or Maurice the Learned (1572–1632) *Maurice of Savoy (1593–1657), prince of Savoy and a cardinal *Maurice, Duke of Saxe-Zeitz (1619–1681) *Maurice of the Palatinate (1620–1652), Count Palatine of the Rhine *Maurice of the Netherlands (1843–1850), prince of Orange-Nassau * Maurice Chevalier (1888–1972), F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constantin Carathéodory
Constantin Carathéodory ( el, Κωνσταντίνος Καραθεοδωρή, Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He made significant contributions to real and complex analysis, the calculus of variations, and measure theory. He also created an axiomatic formulation of thermodynamics. Carathéodory is considered one of the greatest mathematicians of his era and the most renowned Greek mathematician since antiquity. Origins Constantin Carathéodory was born in 1873 in Berlin to Greek parents and grew up in Brussels. His father Stephanos, a lawyer, served as the Ottoman ambassador to Belgium, St. Petersburg and Berlin. His mother, Despina, née Petrokokkinos, was from the island of Chios. The Carathéodory family, originally from Bosnochori or Vyssa, was well established and respected in Constantinople, and its members held many important governmental positions. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johann Radon
Johann Karl August Radon (; 16 December 1887 – 25 May 1956) was an Austrian mathematician. His doctoral dissertation was on the calculus of variations (in 1910, at the University of Vienna). Life RadonBrigitte Bukovics: ''Biography of Johann Radon'', in: 75 Years of Radon Transform, S. Gindikin and P. Michor, eds., International Press Incorporated (1994), pp. 13–18, was born in Tetschen, Bohemia, Austria-Hungary, now Děčín, Czech Republic. He received his doctoral degree at the University of Vienna in 1910. He spent the winter semester 1910/11 at the University of Göttingen, then he was an assistant at the German Technical University in Brno, and from 1912 to 1919 at the Technical University of Vienna. In 1913/14, he passed his habilitation at the University of Vienna. Due to his near-sightedness, he was exempt from the draft during wartime. In 1919, he was called to become Professor extraordinarius at the newly founded University of Hamburg; in 1922, he became ''Prof ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nikolai Luzin
Nikolai Nikolaevich Luzin (also spelled Lusin; rus, Никола́й Никола́евич Лу́зин, p=nʲɪkɐˈlaj nʲɪkɐˈlaɪvʲɪtɕ ˈluzʲɪn, a=Ru-Nikilai Nikilayevich Luzin.ogg; 9 December 1883 – 28 January 1950) was a Soviet/Russian mathematician known for his work in descriptive set theory and aspects of mathematical analysis with strong connections to point-set topology. He was the eponym of Luzitania, a loose group of young Moscow mathematicians of the first half of the 1920s. They adopted his set-theoretic orientation, and went on to apply it in other areas of mathematics. Life He started studying mathematics in 1901 at Moscow State University, where his advisor was Dimitri Egorov. He graduated in 1905. Luzin underwent great personal turmoil in the years 1905 and 1906, when his materialistic worldview had collapsed and he found himself close to suicide. In 1906 he wrote to Pavel Florensky, a former fellow mathematics student who was now studying theology: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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