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Lebesgue Differentiation Theorem
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. Statement For a Lebesgue integrable real or complex-valued function ''f'' on R''n'', the indefinite integral is a set function which maps a measurable set ''A'' to the Lebesgue integral of f \cdot \mathbf_A, where \mathbf_ denotes the characteristic function of the set ''A''. It is usually written A \mapsto \int_ f\ \mathrm\lambda, with ''λ'' the ''n''–dimensional Lebesgue measure. The ''derivative'' of this integral at ''x'' is defined to be \lim_ \frac \int_f \, \mathrm\lambda, where , ''B'', denotes the volume (i.e., the Lebesgue measure) of a ball ''B'' centered at ''x'', and ''B'' → ''x'' means that the diameter of ''B'' tends to 0. The ''Lebesgue differentiation theorem'' states th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Functions
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue's Density Theorem
In mathematics, Lebesgue's density theorem states that for any Lebesgue measurable set A\subset \R^n, the "density" of ''A'' is 0 or 1 at almost every point in \R^n. Additionally, the "density" of ''A'' is 1 at almost every point in ''A''. Intuitively, this means that the "edge" of ''A'', the set of points in ''A'' whose "neighborhood" is partially in ''A'' and partially outside of ''A'', is negligible. Let μ be the Lebesgue measure on the Euclidean space R''n'' and ''A'' be a Lebesgue measurable subset of R''n''. Define the approximate density of ''A'' in a ε-neighborhood of a point ''x'' in R''n'' as : d_\varepsilon(x)=\frac where ''B''ε denotes the closed ball of radius ε centered at ''x''. Lebesgue's density theorem asserts that for almost every point ''x'' of ''A'' the density : d(x)=\lim_ d_(x) exists and is equal to 0 or 1. In other words, for every measurable set ''A'', the density of ''A'' is 0 or 1 almost everywhere in R''n''. However, if μ(''A'') >&nb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doubling Space
In mathematics, a metric space with metric is said to be doubling if there is some doubling constant such that for any and , it is possible to cover the ball with the union of at most balls of radius . The base-2 logarithm of is called the doubling dimension of . Euclidean spaces \mathbb^d equipped with the usual Euclidean metric are examples of doubling spaces where the doubling constant depends on the dimension . For example, in one dimension, ; and in two dimensions, . In general, Euclidean space \mathbb^d has doubling dimension \Theta(d). Assouad's embedding theorem An important question in metric space geometry is to characterize those metric spaces that can be embedded in some Euclidean space by a bi-Lipschitz function. This means that one can essentially think of the metric space as a subset of Euclidean space. Not all metric spaces may be embedded in Euclidean space. Doubling metric spaces, on the other hand, would seem like they have more of a chance, since ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultrametric Space
In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems for ultrametric spaces may seem strange at a first glance, they appear naturally in many applications. Formal definition An ultrametric on a set is a real-valued function :d\colon M \times M \rightarrow \mathbb (where denote the real numbers), such that for all : # ; # (''symmetry''); # ; # if then ; # (strong triangle inequality or ultrametric inequality). An ultrametric space is a pair consisting of a set together with an ultrametric on , which is called the space's associated distance function (also called a metric). If satisfies all of the conditions except possibly condition 4 then is called an ultrapseudometric on . An ultrapseudometric space is a pair consisting of a set and an ultrapseudometric on . In the case ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Compact
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood. In mathematical analysis locally compact spaces that are Hausdorff are of particular interest; they are abbreviated as LCH spaces. Formal definition Let ''X'' be a topological space. Most commonly ''X'' is called locally compact if every point ''x'' of ''X'' has a compact neighbourhood, i.e., there exists an open set ''U'' and a compact set ''K'', such that x\in U\subseteq K. There are other common definitions: They are all equivalent if ''X'' is a Hausdorff space (or preregular). But they are not equivalent in general: :1. every point of ''X'' has a compact neighbourhood. :2. every point of ''X'' has a closed compact neighbourhood. :2′. every point of ''X'' has a relatively compact neighbourhoo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ''g''''p'' on the tangent space ''T''''p''''M'' at each point ''p''. The family ''g''''p'' of inner products is called a metric tensor, Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds. A common convention is to take ''g'' to be Smoothness, smooth, which means that for any smooth coordinate chart on ''M'', the ''n''2 functions :g\left(\frac,\frac\right):U\to\mathbb are smooth functions. These functions are commonly designated as g_. With further restrictions on the g_, one could also consider Lipschitz continuity, Lipschitz Riemannian metrics or Measurable function, measurable Riemannian metrics, among many other possibilities. A Riemannian metric (tensor) makes it possible to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Density Theorem
In mathematics, Lebesgue's density theorem states that for any Lebesgue measurable set A\subset \R^n, the "density" of ''A'' is 0 or 1 at almost every point in \R^n. Additionally, the "density" of ''A'' is 1 at almost every point in ''A''. Intuitively, this means that the "edge" of ''A'', the set of points in ''A'' whose "neighborhood" is partially in ''A'' and partially outside of ''A'', is negligible. Let μ be the Lebesgue measure on the Euclidean space R''n'' and ''A'' be a Lebesgue measurable subset of R''n''. Define the approximate density of ''A'' in a ε-neighborhood of a point ''x'' in R''n'' as : d_\varepsilon(x)=\frac where ''B''ε denotes the closed ball of radius ε centered at ''x''. Lebesgue's density theorem asserts that for almost every point ''x'' of ''A'' the density : d(x)=\lim_ d_(x) exists and is equal to 0 or 1. In other words, for every measurable set ''A'', the density of ''A'' is 0 or 1 almost everywhere in R''n''. However, if μ(''A'') >&nb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henstock–Kurzweil Integral
In mathematics, the Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced ), Luzin integral or Perron integral, but not to be confused with the more general wide Denjoy integral – is one of a number of inequivalent 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. This integral was first defined by Arnaud Denjoy (1912). Denjoy was interested in a definition that would allow one to integrate functions like :f(x)=\frac\sin\left(\frac\right). This function has a singularity at 0, and is not Lebesgue integrable. However, it seems natural to calculate its integral except over the interval and then let . Trying to create a general theory, Denjoy used trans ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Integrable
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. Overview Let be a non-negative real-valued function on the interval , and let be the region of the plane under the graph of the function and above the interval . See the figure on the top right. This region can be expressed in set-builder notation as S = \left \. We are interested in measuring the area of . Once we have measured it, we will denote the area in the usual way by \int_a^b f(x)\,dx. The basic idea of the Riemann integral is to use very simple approximations for the area of . By taking better and bet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vitali Covering Lemma
In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem. The covering theorem is credited to the Italian mathematician Giuseppe Vitali.. The theorem states that it is possible to cover, up to a Lebesgue-negligible set, a given subset ''E'' of R''d'' by a disjoint family extracted from a ''Vitali covering'' of ''E''. Vitali covering lemma There are two basic version of the lemma, a finite version and an infinite version. Both lemmas can be proved in the general setting of a metric space, typically these results are applied to the special case of the Euclidean space \mathbb^d. In both theorems we will use the following notation: if B = B(x,r) is a ball and c \in \mathbb, we will write cB for the ball B(x,cr). Finite version Theorem (Finite Covering Lemma). Let B_, \dots, B_ be any finite coll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Inequality
In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function (mathematics), function of a random variable is greater than or equal to some positive Constant (mathematics), constant. It is named after the Russian mathematician Andrey Markov, although it appeared earlier in the work of Pafnuty Chebyshev (Markov's teacher), and many sources, especially in Mathematical analysis, analysis, refer to it as Chebyshev's inequality (sometimes, calling it the first Chebyshev inequality, while referring to Chebyshev's inequality as the second Chebyshev inequality) or Irénée-Jules Bienaymé, Bienaymé's inequality. Markov's inequality (and other similar inequalities) relate probabilities to expected value, expectations, and provide (frequently loose but still useful) bounds for the cumulative distribution function of a random variable. Statement If is a nonnegative random variable and , then the probability that is at least is at most th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
