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Scheffé’s Lemma
In mathematics, Scheffé's lemma is a proposition in measure theory concerning the convergence of sequences of integrals
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Measure Theory
In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure
Lebesgue measure
on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry
Euclidean geometry
to suitable subsets of the n-dimensional Euclidean space
Euclidean space
Rn
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Convergence (mathematics)
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. Given an infinite sequence ( a 1 ,   a 2 ,   a 3 , … ) displaystyle left(a_ 1 , a_ 2 , a_ 3 ,dots right) , the nth partial sum S n displaystyle S_ n is the sum of the first n terms of the sequence, that is, S n = ∑ k = 1 n a k . displaystyle S_ n =sum _ k=1 ^ n a_ k
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Integral
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse, differentiation, being the other. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral ∫ a b f ( x ) d x displaystyle int _ a ^ b !f(x),dx is defined informally as the signed area of the region in the xy-plane that is bounded by the graph of f, the x-axis and the vertical lines x = a and x = b. The area above the x-axis adds to the total and that below the x-axis subtracts from the total. Roughly speaking, the operation of integration is the reverse of differentiation
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Lebesgue Integration
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 extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined. Long before the advent of the 20th century, mathematicians already understood that for non-negative functions with a smooth enough graph—such as continuous functions on closed bounded intervals—the area under the curve could be defined as the integral, and computed using approximation techniques on the region by polygons
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Measure Space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. Measure spaces contain information about the underlying set, the subsets of said set that are feasible for measuring (the σ displaystyle sigma -algebra) and the method that is used for measuring (the measure)
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Probability Theory
Probability
Probability
theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion. Although it is not possible to perfectly predict random events, much can be said about their behavior
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Probability Density Function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function, whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample.[citation needed] In other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0 (since there are an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would equal one sample compared to the other sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value
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Absolute Continuity
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—expressed by the fundamental theorem of calculus in the framework of Riemann integration. Such generalizations are often formulated in terms of Lebesgue
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
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Random Variable
In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable whose possible values are outcomes of a random phenomenon.[1] As a function, a random variable is required to be measurable, which rules out certain pathological cases where the quantity which the random variable returns is infinitely sensitive to small changes in the outcome. It is common that these outcomes depend on some physical variables that are not well understood. For example, when tossing a fair coin, the final outcome of heads or tails depends on the uncertain physics. Which outcome will be observed is not certain. The coin could get caught in a crack in the floor, but such a possibility is excluded from consideration. The domain of a random variable is the set of possible outcomes. In the case of the coin, there are only two possible outcomes, namely heads or tails
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Convergence In Distribution
In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behaviour that is essentially unchanging when items far enough into the sequence are studied
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Henry Scheffé
Henry Scheffé (New York City, United States, 11 April 1907 – Berkeley, California, USA, 5 July 1977) was an American statistician.[1][2] He is known for the Lehmann–Scheffé theorem and Scheffé's method. Education and career[edit] Scheffé was born in New York City
New York City
on April 11, 1907, the child of German immigrants. The family moved to Islip, New York, where Scheffé went to high school. He graduated in 1924, took night classes at Cooper Union, and a year later entered the Polytechnic Institute of Brooklyn
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Frigyes Riesz
Frigyes Riesz
Frigyes Riesz
(Hungarian: Riesz Frigyes, pronounced [ˈriːs ˈfriɟɛʃ]; 22 January 1880 – 28 February 1956) was a Hungarian[1][2] mathematician who made fundamental contributions to functional analysis.Contents1 Life and career 2 Publications 3 See also 4 References 5 External linksLife and career[edit] He was born into a Jewish family in Győr, Kingdom of Hungary, Austria-Hungary
Austria-Hungary
and died in Budapest, Hungary. Between 1911 and 1919 he was a professor at the Franz Joseph University
Franz Joseph University
in Kolozsvár, Austria-Hungary
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Lp Spaces
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz
Frigyes Riesz
(Riesz 1910). Lp spaces form an important class of Banach
Banach
spaces in functional analysis, and of topological vector spaces
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Digital Object Identifier
In computing, a Digital Object Identifier or DOI is a persistent identifier or handle used to uniquely identify objects, standardized by the International Organization for Standardization
International Organization for Standardization
(ISO).[1] An implementation of the Handle System,[2][3] DOIs are in wide use mainly to identify academic, professional, and government information, such as journal articles, research reports and data sets, and official publications though they also have been used to identify other types of information resources, such as commercial videos. A DOI aims to be "resolvable", usually to some form of access to the information object to which the DOI refers. This is achieved by binding the DOI to metadata about the object, such as a URL, indicating where the object can be found. Thus, by being actionable and interoperable, a DOI differs from identifiers such as ISBNs and ISRCs which aim only to uniquely identify their referents
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Special
Special
Special
or specials may refer to:Contents1 Music 2 Film and television 3 Other uses 4 See alsoMusic[edit] Special
Special
(album), a 1992
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