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Topic: Countably additive measure

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	Measure (mathematics)
		Formally, a measure μ is a function which assigns to every element S of a given sigma algebra X a value μ(S), a non-negative real number or ∞.
		The Haar measure for a locally compact topological group is a generalization of the Lebesgue measure and has a similar uniqueness property.
		For instance, a countably additive set function with values in the (signed) real numbers is called a signed measure, while such a function with values in the complex numbers is called a complex measure.
	www.ebroadcast.com.au /lookup/encyclopedia/me/Measure_(mathematics).html (520 words)

	Measure (mathematics) - Wikipedia, the free encyclopedia
		Measure theory is that branch of real analysis which investigates sigma algebras, measures, measurable functions and integrals.
		It is a trivial matter to extend a measure to a complete one; simply consider the σ-algebra of subsets S' which differ by a null set from a measurable set S, that is such that the symmetric difference of S and S' is null.
		For instance, a countably additive set function with values in the (signed) real numbers is called a signed measure, while such a function with values in the complex numbers is called a complex measure.
	en.wikipedia.org /wiki/Measure_(mathematics) (835 words)

	Outer measure - Wikipedia, the free encyclopedia
		In mathematics, in particular in measure theory, an outer measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions.
		A general theory of outer measures was developed by Carathéodory to provide a basis for the theory of measurable sets and countably additive measures.
		The purpose of constructing an outer measure on all subsets of X is to suitably pick out a class of subsets (to be called measurable) in such a way that fulfils the countably additivity property.
	en.wikipedia.org /wiki/Outer_measure (637 words)

	Measure (mathematics) Summary
		Measurement theory is as ancient as formalized mathematics.
		Measurement theory itself is broadly described as the set of rules related to the quantitative measurement of properties of physical entities (e.g., length) or the measurement of the duration between phenomena (e.g., time).
		A measure can be extended to a complete one by considering the σ-algebra of subsets Y which differ by a negligible set from a measurable set X, that is, such that the symmetric difference of X and Y is contained in a null set.
	www.bookrags.com /Measure_(mathematics) (1692 words)

	PlanetMath: Lebesgue outer measure
		The outer measure satisfies all the axioms of a measure except (countable) additivity.
		However, it is countably additive when one restricts to at least the Borel sets, as this is the usual construction of Borel measure.
		This is version 9 of Lebesgue outer measure, born on 2001-10-18, modified 2006-09-11.
	planetmath.org /encyclopedia/OuterMeasure.html (185 words)

	PlanetMath: measure
		Note the slight abuse of terminology: a finitely additive measure is not necessarily a measure.
		Cross-references: Lebesgue measure, unions, finite, algebra, countable additivity, property, disjoint, sequence, equality, extended real numbers, function, measurable space
		This is version 13 of measure, born on 2001-11-11, modified 2006-04-21.
	planetmath.org /encyclopedia/Measure.html (145 words)

	Outer measure (Site not responding. Last check: 2007-10-23)
		In mathematics, in particular in measure theory, an outer measure is a function defined on all subsets of a given set with values in the extended real numbers.
		The measuring function φ is a non-negative extended real-valued function defined for all subsets of R.
		The purpose of constructing an outer measure is to define which sets are measurable, and fulfil the countably additivity axiom.
	www.encyclopedia-1.com /o/ou/outer_measure.html (551 words)

	measure (mathematics) - Article and Reference from OnPedia.com
		Formally, a countably additive measure μ is a function defined on a sigma algebra
		Consider the closed intervals [ k, k+1 for all integers k ; there are countably many such intervals, each has measure 1, and their union is the entire real line.
		This measure is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line.
	www.onpedia.com /encyclopedia/measure-(mathematics) (831 words)

	Measure (mathematics) (Site not responding. Last check: 2007-10-23)
		Formally, a measure μ is a function which assigns to every element S of a given sigma algebra X a value μ(S), a non-negative real number or ∞.
		If μ is a measure on the sigma algebra X, then the members of X are called the μ-measurable sets, or the measurable sets for short.
		The Lebesgue measure is the unique complete translation-invariant measure on a sigma algebra containing the intervalss in R such that μ([0,1]) = 1.
	www.sciencedaily.com /encyclopedia/measure__mathematics_ (818 words)

	Re: Frequentist probability confusion
		By > a countably additive measure we mean that we have the useful theorem > that a countable union of sets of measure zero has measure zero.
		So > a countably additive measure must have a measure of zero for the entire > space and it fails to be a probability space.
		It is possible to have a finitely additive probability measure on a countable set where the probability of each singleton is zero but the probability of the entire set is one.
	www.lns.cornell.edu /spr/2004-04/msg0060414.html (514 words)

	Non-measurable set
		In measure theory, a non-measurable set is one which does not belong to the algebra of measurable sets of some measurable space.
		There is no countably additive measure on all subsets of R which is translation invariant and is finite and non vanishing on [0,1].
		It follows immediately from this theorem, that non-measurable sets exist for any countably additive translation invariant measure which is finite and non vanishing on [0,1].
	www.brainyencyclopedia.com /encyclopedia/n/no/non_measurable_set.html (298 words)

	Haar measure - Wikipedia, the free encyclopedia
		In mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups.
		A frequently used technique for proving the existence of a Haar measure on a locally compact group G is showing the existence of a left invariant Radon measure on G.
		Unless G is a discrete group, it is impossible to define a countably-additive right invariant measure on all subsets of G, assuming the axiom of choice.
	en.wikipedia.org /wiki/Haar_measure (785 words)

	Сибирский Математический Журнал, Том 44 (2003), Номер 3, с. 587-605 (Site not responding. Last check: 2007-10-23)
		We define the gamma-compactification of an arbitrary measurable space and study its structure and properties in the general and topological cases.
		We introduce and study the notion of gamma-extension of a singleton in a topological space.
		We consider the procedure of extension of finitely additive measures from the original space to regular countably additive measures on the gamma-compactification of the space.
	www.emis.de /journals/SMZ/2003/03/587.htm (70 words)

	Re: Frequentist probability confusion
		>>By a countably additive measure we mean that we have the useful >>theorem that a countable union of sets of measure zero has measure >>zero.
		So a countably additive measure must have a measure >>of zero for the entire space and it fails to be a probability space.
		If one only wants some of the sets to be measurable, nothing is needed; consider the field of sets which are periodic from some point on, and give it the limiting frequency.
	www.lns.cornell.edu /spr/2004-04/msg0060448.html (490 words)

	physics - Quantum logic
		Putnam attributes the idea that anomalies associated to quantum measurements originate with anomalies in the logic of physics itself to the physicist David Finkelstein.
		This is the content of the spectral as stated in terms of spectral measures.
		The result of a cluster of measurements intended to determine the truth value of proposition E, is just as in the classical case, a probability distribution of truth values T and F.
	www.physicsdaily.com /physics/Quantum_logic (2618 words)

	Riesz representation theorem
		A non-negative countably additive Borel measure μ on a locally compact Hausdorff space X is regular iff
		If μ is a complex-valued countably additive Borel measure, μ is regular iff the non-negative countably additive measure μ
		Finally, ψ is positive iff the measure μ is non-negative.
	www.brainyencyclopedia.com /encyclopedia/r/ri/riesz_representation_theorem.html (697 words)

	Springer Online Reference Works
		Vector measures of bounded variation are strongly additive, and strongly-additive vector measures are bounded.
		There are also versions for strongly-additive vector measures of the well-known decomposition theorems of Yosida–Hewitt and of Lebesgue (see [a3]).
		Below these developments are given briefly (see also [a1] and [a4]); see [a5] for the role of vector measures in control theory.
	eom.springer.de /v/v096490.htm (501 words)

	Annotated Bibliography on the Range of Vector Measures (Site not responding. Last check: 2007-10-23)
		Brook and Graves (1980) gave a generalization of results of Knowles (1975) and of Tweddle (1972) to strongly countably additive map Phi defined on an algebra of subsets of a non-empty set X with values in a locally convex Hausdorff topological space over the scalar field of complex numbers.
		Obha (1978) proved that the closure of the range of a measure of bounded variation on a sigma-algebra, with values in a Fréchet space with the Radom-Nikodym property, is compact and, if the measure has no atoms, also convex.
		He also showed that every additive measure with values in a locally convex space is convexly bounded (i.e., the range is contained in a convex bounded set).
	www.math.gatech.edu /~hill/publications/annotated.html (5995 words)

	Hausdorff dimension Summary
		To carry this construction of this measure, we use a theory of measure which is appropriate for metric spaces.
		Define a family of metric outer measures on X using the Method II construction of outer measures due to Munroe and described in the article outer measure.
		From the perspective of assigning measure and dimension to sets with unusual metric properties such as fractals, however, this is not a restriction.
	www.bookrags.com /Hausdorff_dimension (2287 words)

	Haar measure -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-23)
		The Haar measures are used in (Analysis of a periodic function into a sum of simple sinusoidal components) harmonic analysis on arbitrary locally compact groups, see (Click link for more info and facts about Pontryagin duality) Pontryagin duality.
		A frequently used technique for proving the existence of a Haar measure on a locally compact group G is showing the existence of a left invariant (Click link for more info and facts about Radon measure) Radon measure on G.
		Note that, unless G is a discrete group, it is impossible to define a countably-additive right invariant measure on all subsets of G, assuming the (Click link for more info and facts about axiom of choice) axiom of choice.
	www.absoluteastronomy.com /encyclopedia/H/Ha/Haar_measure.htm (983 words)

	SeminarI.html
		THEOREM [Ulam] Assume the axiom that there is a (countably additive) nonatomic measure on some set X such that every subset of X is measurable and some subset of X has positive real measure.
		Then we will explore applications of set theory to measure theory, including the existence of nonmeasurable subsets of the real line and various drastic efforts to overcome this ``deficiency.'' In particular, the replacement of countable additivity of measure by finite additivity runs afoul of the Banach-Tarski paradox in 3-dimensional Euclidean space.
		One is equivalent to the statement that there is a (countably additive) nonatomic measure on some set X such that every subset of X is measurable and some subset of X has positive real measure.
	www.math.sc.edu /~nyikos/SeminarI.html (1675 words)

	Haar measure (Site not responding. Last check: 2007-10-23)
		It turns out that there is, up to a positive multiplicative constant, only one left-translation-invariant countably additive regular Borel measure μ such that μ(U) > 0 for any open Borel set U.
		A frequently used technique for showing existence of Haar measure on a locally compact group G is showing the existence of a left invariant Radon measure on 'G''.
		Note that it is impossible to define a countably additive right invariant measure on all subsets of G for all but discrete subgroups, assuming that is the axiom of choice.
	www.sciencedaily.com /encyclopedia/haar_measure (750 words)

	Outer measure (Site not responding. Last check: 2007-10-23)
		Outer measures are used to define measurable set s and countably additive measures.
		This method is known as the Carathéodory construction and is one way of arriving at the concept of Lebesgue measure that is so important for measure theory and the theory of integral s.
		The Borel set s of X are the elements of the smallest σ-algebra generated by the open sets.
	www.worldhistory.com /wiki/O/Outer-measure.htm (654 words)

	KLUEDO - Functions of bounded semivariation and countably additive vector measures
		In the Banach space co there exists a continuous function of bounded semivariation which does not correspond to a countably additive vector measure.
		This result is in contrast to the scalar case, and it has consequences for the characterization of scalar-type operators.
		Besides this negative result we introduce the notion of functions of unconditionally bounded variation which are exactly the generators of countably additive vector measures.
	kluedo.ub.uni-kl.de /volltexte/2000/852 (114 words)

	SPb. Math. Soc.: A.G.Pinsker
		The following result was also obtained in this study: any regular Boolean algebra with a strictly positive finitely additive measure has a strictly positive countably additive measure; so this algebra is normable.
		The first A.G. Pinsker's paper published in 1938 contains the analytical representation of the functionals on the space of measurable functions that are additive on functions with disjoint supports (there are two papers on this subject written jointly with L.V. Kantorovich).
		In the theory of ordered spaces, the operators additive on disjoint elements are called disjointly additive operators.
	www.mathsoc.spb.ru /pers/pinsker/veksl-e.html (1305 words)

	Chair of Math. Anal.: D.Vladmrov
		Besides the general theory of semi-ordered spaces, the research interests of Vladimirov included spaces of measurable functions, invariants of measurable functions with respect to metric isomorphisms of their domains, properties of integral operators, the theory of Boolean algebras and the measure theory as well as their applications to general topology and the probability theory.
		It was devoted to the concept (introduced by him) of strongly compact linear operator in the space of measurable functions.
		In a joint paper with A.A.Samorodnitsky Vladimirov pointed out a class of measurable spaces in which the distribution function was not only one of invariants of the metric type, but completely defined it.
	www.math.spbu.ru /user/analysis/pers/Vlad_e.html (1421 words)

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