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Dirac Measure
In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element ''x'' or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields. Definition A Dirac measure is a measure on a set (with any -algebra of subsets of ) defined for a given and any (measurable) set by :\delta_x (A) = 1_A(x)= \begin 0, & x \not \in A; \\ 1, & x \in A. \end where is the indicator function of . The Dirac measure is a probability measure, and in terms of probability it represents the almost sure outcome in the sample space . We can also say that the measure is a single atom at ; however, treating the Dirac measure as an atomic measure is not correct when we consider the sequential definition of Dirac delta, as the limit of a delta sequence. The Dirac measures are the extreme points of the convex set of probability measures on . The name is a back-formation from the Dirac delta fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hasse Diagram Of Powerset Of 3
Hasse is both a surname and a given name. Notable people with the name include: Surname: * Clara H. Hasse (1880–1926), American botanist * Helmut Hasse (1898–1979), German mathematician * Henry Hasse (1913–1977), US writer of science fiction * Johann Adolph Hasse (1699–1783), German composer * Maria Hasse (1921–2014), German mathematician * Peter Hasse (c. 1585–1640), German organist and composer Given name or nickname: * Hans Alfredson (born 1931), Swedish actor, film director, writer and comedian * Hans Backe (born 1952), Swedish football manager * Hasse Borg (born 1953), Swedish footballer * Hasse Börjes (born 1948), Swedish speed skater * Hasse Ekman (1915-2004), Swedish film director and actor * Hans Wind (1919–1995), Finnish flying ace See also * Hasse bound, on the number of points on an elliptic curve * Hasse diagram, a diagram used in set theory {{given name, type=both ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable Space
In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. It captures and generalises intuitive notions such as length, area, and volume with a set X of 'points' in the space, but ''regions'' of the space are the elements of the σ-algebra, since the intuitive measures are not usually defined for points. The algebra also captures the relationships that might be expected of regions: that a region can be defined as an intersection of other regions, a union of other regions, or the space with the exception of another region. Definition Consider a set X and a σ-algebra \mathcal F on X. Then the tuple (X, \mathcal F) is called a measurable space. The elements of \mathcal F are called measurable sets within the measurable space. Note that in contrast to a measure space, no measure is needed for a measurable space. Example Look at the set: X = \. One possible \sigma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Support (measure Theory)
In mathematics, the support (sometimes topological support or spectrum) of a measure \mu on a measurable topological space (X, \operatorname(X)) is a precise notion of where in the space X the measure "lives". It is defined to be the largest ( closed) subset of X for which every open neighbourhood of every point of the set has positive measure. Motivation A (non-negative) measure \mu on a measurable space (X, \Sigma) is really a function \mu : \Sigma \to , +\infty Therefore, in terms of the usual definition of support, the support of \mu is a subset of the σ-algebra \Sigma: \operatorname (\mu) := \overline, where the overbar denotes set closure. However, this definition is somewhat unsatisfactory: we use the notion of closure, but we do not even have a topology on \Sigma. What we really want to know is where in the space X the measure \mu is non-zero. Consider two examples: # Lebesgue measure \lambda on the real line \Reals. It seems clear that \lambda "lives on" the wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radon Measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the -algebra of Borel sets of a Hausdorff topological space that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These conditions guarantee that the measure is "compatible" with the topology of the space, and most measures used in mathematical analysis and in number theory are indeed Radon measures. Motivation A common problem is to find a good notion of a measure on a topological space that is compatible with the topology in some sense. One way to do this is to define a measure on the Borel sets of the topological space. In general there are several problems with this: for example, such a measure may not have a well defined support. Another approach to measure theory is to restrict to locally compact Hausdorff spaces, and only consider the measures that correspond to positive linear functionals on the space of continu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces. One suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singleton (mathematics)
In mathematics, a singleton (also known as a unit set or one-point set) is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the axiom of regularity guarantees that no set is an element of itself. This implies that a singleton is necessarily distinct from the element it contains, thus 1 and \ are not the same thing, and the empty set is distinct from the set containing only the empty set. A set such as \ is a singleton as it contains a single element (which itself is a set, but not a singleton). A set is a singleton if and only if its cardinality is . In von Neumann's set-theoretic construction of the natural numbers, the number 1 is ''defined'' as the singleton \. In axiomatic set theory, the existence of singletons is a consequence of the axiom of pairing: for any set ''A'', the axiom applied to ''A'' and ''A'' asserts the existence of \, which is the same ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inner Regular Measure
In mathematics, a regular measure on a topological space is a measure for which every measurable set can be approximated from above by open measurable sets and from below by compact measurable sets. Definition Let (''X'', ''T'') be a topological space and let Σ be a σ-algebra on ''X''. Let ''μ'' be a measure on (''X'', Σ). A measurable subset ''A'' of ''X'' is said to be inner regular if :\mu (A) = \sup \ This property is sometimes referred to in words as "approximation from within by compact sets." Some authors use the term tight as a synonym for inner regular. This use of the term is closely related to tightness of a family of measures, since a finite measure ''μ'' is inner regular if and only if, for all ''ε'' > 0, there is some compact subset ''K'' of ''X'' such that ''μ''(''X'' \ ''K'') < ''ε''. This is precisely the condition that the [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Space
In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom. Definitions Points x and y in a topological space X can be '' separated by neighbourhoods'' if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U\cap V=\varnothing). X is a Hausdorff space if any two distinct points in X are separated by neighbourhoods. This condition is the third separation axiom (after T0 and T1), which is why Hausdorff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Finite Measure
In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure. Definition Let (X, T) be a Hausdorff topological space and let \Sigma be a \sigma-algebra on X that contains the topology T (so that every open set is a measurable set, and \Sigma is at least as fine as the Borel \sigma-algebra on X). A measure/signed measure/ complex measure \mu defined on \Sigma is called locally finite if, for every point p of the space X, there is an open neighbourhood N_p of p such that the \mu-measure of N_p is finite. In more condensed notation, \mu is locally finite if and only if \text p \in X, \text N_p \in T \mbox p \in N_p \mbox \left, \mu\left(N_p\right)\ < + \infty. Examples # Any on is locally finite, since it assi ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trivial Topology
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means. Every indiscrete space can be viewed as a pseudometric space in which the distance between any two points is zero. Details The trivial topology is the topology with the least possible number of open sets, namely the empty set and the entire space, since the definition of a topology requires these two sets to be open. Despite its simplicity, a space ''X'' with more than one element and the trivial topology lacks a key desirable property: it is not a T0 space. Other properties of an indiscrete space ''X''—many of which are quite unusual—include: * The only closed sets are the empty set and ''X''. * The only possible basi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strictly Positive Measure
In mathematics, strict positivity is a concept in measure theory. Intuitively, a strictly positive measure is one that is "nowhere zero", or that is zero "only on points". Definition Let (X, T) be a Hausdorff topological space and let \Sigma be a \sigma-algebra on X that contains the topology T (so that every open set is a measurable set, and \Sigma is at least as fine as the Borel \sigma-algebra on X). Then a measure \mu on (X, \Sigma) is called strictly positive if every non-empty open subset of X has strictly positive measure. More concisely, \mu is strictly positive if and only if for all U \in T such that U \neq \varnothing, \mu (U) > 0. Examples * Counting measure on any set X (with any topology) is strictly positive. * Dirac measure is usually not strictly positive unless the topology T is particularly "coarse" (contains "few" sets). For example, \delta_0 on the real line \R with its usual Borel topology and \sigma-algebra is not strictly positive; however, if \R is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
