Interchange Of Limiting Operations
In mathematics, the study of interchange of limiting operations is one of the major concerns of mathematical analysis, in that two given limiting operations, say ''L'' and ''M'', cannot be ''assumed'' to give the same result when applied in either order. One of the historical sources for this theory is the study of trigonometric series. Formulation In symbols, the assumption :''LM'' = ''ML'', where the left-hand side means that ''M'' is applied first, then ''L'', and ''vice versa'' on the right-hand side, is not a valid equation between mathematical operators, under all circumstances and for all operands. An algebraist would say that the operations do not commute. The approach taken in analysis is somewhat different. Conclusions that assume limiting operations do 'commute' are called ''formal''. The analyst tries to delineate conditions under which such conclusions are valid; in other words mathematical rigour is established by the specification of some set of sufficient cond ...
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Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ...
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Fubini's Theorem
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. \, \iint\limits_ f(x,y)\,\text(x,y) = \int_X\left(\int_Y f(x,y)\,\texty\right)\textx=\int_Y\left(\int_X f(x,y) \, \textx \right) \texty \qquad \text \qquad \iint\limits_ , f(x,y), \,\text(x,y) <+\infty. Fubini's theorem implies that two iterated integrals are equal to the corresponding double integral across its integrands. Tonelli's theorem, introduced by in 1909, is similar, but applies to a non-negative measurable function rather than one integrable over their domains. A related theorem is oft ...
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Fatou's Lemma
In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou. Fatou's lemma can be used to prove the Fatou–Lebesgue theorem and Lebesgue's dominated convergence theorem. Standard statement In what follows, \operatorname_ denotes the \sigma-algebra of Borel sets on ,+\infty/math>. Fatou's lemma remains true if its assumptions hold \mu-almost everywhere. In other words, it is enough that there is a null set N such that the values \ are non-negative for every . To see this, note that the integrals appearing in Fatou's lemma are unchanged if we change each function on N. Proof Fatou's lemma does ''not'' require the monotone convergence theorem, but the latter can be used to provide a quick proof. A proof directly from the definitions of integrals is given further below. In each case, the proof begins by ...
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Cafiero Convergence Theorem
Cafiero is an Italian surname. Notable people with the surname include: * Antonio Cafiero (1922–2014), Argentine politician * Carlo Cafiero (1846–1892), Italian anarchist * Claudio Cafiero (born 1989), Italian footballer * Federico Cafiero (1914–1980), Italian mathematician * James Cafiero (1928–2023), American politician * John Cafiero, American musician and director * Juan Pablo Cafiero (born 1953), Argentine politician * Mario Cafiero (died 2020), Argentine politician * Santiago Cafiero Santiago Andrés Cafiero (born 30 August 1979) is an Argentine political scientist and politician, serving as Minister of Foreign Affairs and Worship in the cabinet of President Alberto Fernández since 2021. Previously, from 2019 to 2021, he wa ... (born 1979), Argentine politician Italian-language surnames {{Surname ...
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Fichera Convergence Theorem
Fichera is a surname. Notable people with the surname include: *Gaetano Fichera (1922–1996), Italian mathematician **Fichera's existence principle * Joseph Fichera, American business executive *Marco Fichera Marco Fichera (born 15 April 1993) is an Italian male épée fencer. Career Fichera took up fencing when he was ten years old under the guidance of maestro Domenico Patti at C.S. Acireale. In 2010 he took a double gold haul at the U17 European C ...

Vitali Convergence Theorem
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of the convergence in ''Lp'' in terms of convergence in measure and a condition related to uniform integrability In mathematics, uniform integrability is an important concept in real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties .... Preliminary definitions Let (X,\mathcal,\mu) be a measure space, i.e. \mu : \mathcal\to ,\infty/math> is a set function such that \mu(\emptyset)=0 and \mu is countably-additive. All functions considered in the sequel will be functions f:X\to \mathbb, where \mathbb=\R or \mathbb. We adopt the following definitions according to Bogachev's terminology. * A set of functions \mathcal \sub ...
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Dominated Convergence Theorem
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 ''L''1 norm. Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann integration. In addition to its frequent appearance in mathematical analysis and partial differential equations, it is widely used in probability theory, since it gives a sufficient condition for the convergence of expected values of random variables. Statement Lebesgue's dominated convergence theorem. Let (f_n) be a sequence of complex-valued measurable functions on a measure space . Suppose that the sequence converges pointwise to a function f and is dominated by some integrable function g in the sense that : , f_n(x), \le g(x) for all numbers ''n'' in the index set of the sequence and all points x\in S. Then ''f'' is integrable (in the Lebesgue sense) and : \lim_ \int_ ...
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Schwarz's Theorem
In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility of interchanging the order of taking partial derivatives of a function :f\left(x_1,\, x_2,\, \ldots,\, x_n\right) of ''n'' variables without changing the result under certain conditions (see below). The symmetry is the assertion that the second-order partial derivatives satisfy the identity :\frac \left( \frac \right) \ = \ \frac \left( \frac \right) so that they form an ''n'' × ''n'' symmetric matrix, known as the function's Hessian matrix. This is sometimes known as Schwarz's theorem, Clairaut's theorem, or Young's theorem. In the context of partial differential equations it is called the Schwarz integrability condition. Formal expressions of symmetry In symbols, the symmetry may be expressed as: :\frac \left( \frac \right) \ = \ \frac \left( \frac \right) \qquad\text\qquad \frac \ =\ \frac . Another nota ...
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Tannery's Theorem
In mathematical analysis, Tannery's theorem gives sufficient conditions for the interchanging of the limit and infinite summation operations. It is named after Jules Tannery. Statement Let S_n = \sum_^\infty a_k(n) and suppose that \lim_ a_k(n) = b_k . If , a_k(n), \le M_k and \sum_^\infty M_k < \infty , then $\backslash lim\_\; S\_n\; =\; \backslash sum\_^\; b\_k$.

Proofs

Tannery's theorem follows directly from Lebesgue's applied to the

Moore-Osgood Theorem
In multivariable calculus, an iterated limit is a limit of a sequence or a limit of a function in the form : \lim_ \lim_ a_ = \lim_ \left( \lim_ a_ \right), : \lim_ \lim_ f(x, y) = \lim_ \left( \lim_ f(x, y) \right), or other similar forms. An iterated limit is only defined for an expression whose value depends on at least two variables. To evaluate such a limit, one takes the limiting process as one of the two variables approaches some number, getting an expression whose value depends only on the other variable, and then one takes the limit as the other variable approaches some number. Types of iterated limits This section introduces definitions of iterated limits in two variables. These may generalize easily to multiple variables. Iterated limit of sequence For each n, m \in \mathbf, let a_ \in \mathbf be a real double sequence. Then there are two forms of iterated limits, namely : \lim_ \lim_ a_ \qquad \text \qquad \lim_ \lim_ a_. For example, let :a_ = \frac. Then : ...
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Federico Cafiero
Federico Cafiero (24 May 1914 – 7 May 1980) was an Italian mathematician known for his contributions in real analysis, measure and integration theory, and in the theory of ordinary differential equations. In particular, generalizing the Vitali convergence theorem, the Fichera convergence theorem and previous results of Vladimir Mikhailovich Dubrovskii, he proved a necessary and sufficient condition for the passage to the limit under the sign of integral: this result is, in some sense, definitive. In the field of ordinary differential equations, he studied existence and uniqueness problems under very general hypotheses for the left member of the given first order equation, developing an important approximation method and proving a fundamental uniqueness theorem. Life and academic career Cafiero was born in Riposto, Province of Catania, on May 24, 1914. He obtained his Laurea in mathematics, cum laude, from the University of Naples Federico II in 1939.See . During the 1939–194 ...
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