wiktionary measure tocright Measure may refer to Legal List of Church of England Measures Measure of the Church of England is a law passed by the General Synod and the UK Parliament equivalent of an Act Measure of the National Assembly for Wales , a law specific to Wales passed by the Welsh Assembly between 2007 and 2011 Measure of the Northern Ireland Assembly 1973 Mathematics and science Measure probability , a mathematical construct that models a real world experiment Measure mathematics , a way to assign non negative real numbers to subsets Measure physics , a way to integrate over all possible histories of a system in quantum field theory Units Measurement , the process of establishing the magnitude of some attribute of an object relative to some unit of measurement Bar music or measure, a unit of time in Western music representing a regular grouping of beats Coal measure is a seam of coal Measure termination , a way to describe termination of processes, algorithms, or functions Measure bartending or jigger, a bartending tool used to measure liquor Other Measure album , a 2000 album from Matt Pond PA Measure data warehouse , a property on which calculations can be made using precomputed aggregates Measure typography , in typography refers to the width of a block or column of text from the left margin to the right margin Countermeasure , a system designed to prevent weapons from acquiring and or destroying a target disambig cv de Ma es Medida fr Mesure it Misura he pt Medida hu M rt k ... more details
upright thumb Facsimile of the title page of Measure for Measure from the First Folio , published in 1623 Measure for Measure is a play by William Shakespeare , believed to have been written in 1603 ... Lodowick. Isabella , a novice nun. Claudio , Isabella s brother. Angelo Measure for Measure Angelo ... Image William Hunt Claudio and Isabella Shakespeare Measure for Measure.jpg thumb right upright Claudio ... judge, Angelo Measure for Measure Angelo . Claudio, a young nobleman, is betrothed unofficially married ... and Mariana publicly petition him, and he hears their claims against Angelo Measure for Measure ... Bawcutt N. W. Bawcutt ed. , Measure for Measure Oxford, 1991 , p. 17 ref Shakespeare was familiar ... way you judge others, you will be judged, and with the measure you use, it will be measured to you. Date, text and authorship Measure for Measure is believed to have been written in 1603 or 1604 ... of Measure that survives today is not in its original form, but rather the product of a revision after ... also Shakespeare s Mediterranean Measure for Measure , in Shakespeare and the Mediterranean The Selected ... recorded performance of Measure for Measure took place on St. Stephen s night, 26 December 1604. During the English Restoration Restoration , Measure was one of many Shakespearean plays adapted ... reading of both the characters and the overall message of the play. ref S. Nagarajan, Measure for Measure, New York, Penguin, 1998 pp. 181 183. ref Notable more recent productions of Measure for Measure ... Wagner with the libretto written by the composer based on Measure for Measure The musical Desperate ..., Round Heads and Pointed Heads , was originally written as an adaptation of Measure for Measure ... different parts of the body contains an underwater recital of Measure for Measure Mr.Beavis, in Aldous ..., to reading the last scene of Measure for Measure References reflist External links wikisource wikiquote http ise.uvic.ca Library SLT plays measuresubj.html Measure for Measure http www.shakespeare literature.com ... more details
wiktionary unit of measure Unit of measure may refer to Units of measurement for relevance to weights and measures Unit of account for relevance in economics Unit of Measure album Unit of Measure album , a 2000 album by Tony Rice disambiguation ... more details
There are several Initiatives and referendums in the United States ballot measure s called Measure 36 Oregon Ballot Measure 36 1996 minimum wage Oregon Ballot Measure 36 2004 constitutional amendment to ban same sex marriage disambig ... more details
Measure of a Man may refer to In literature The Measure of a Man , a sociopolitical and economic book by Martin Luther King, Jr. The Measure of a Man A Spiritual Autobiography , an autobiographical work by Sidney Poitier In music Measure of a Man , a song from the 2007 Jack Ingram album This Is It Jack Ingram album This Is It The Measure of a Man , a song from the 1997 4Him album The Message 4Him album The Message Measure of a Man Clay Aiken album Measure of a Man Clay Aiken album , Clay Aiken s debut album Measure of a Man Kevin Sharp album Measure of a Man Kevin Sharp album , the debut album of American country music artist Kevin Sharp In visual media The Measure of a Man 1915 film The Measure of a Man 1915 film , a 1915 silent drama film directed by Joe De Grasse and featuring Lon Chaney, Sr. The Measure of a Man Star Trek The Next Generation The Measure of a Man Star Trek The Next Generation , second season episode of Star Trek The Next Generation disambig ... more details
Unreferenced date March 2007 In mathematics , specifically in measure theory , the trivial measure on any measurable space X , is the measure which assigns zero measure to every measurable set A 0 for all A in . Properties of the trivial measure Let denote the trivial measure on some measurable space X , . A measure &nu is the trivial measure &mu if and only if &nu X 0. is an invariant measure and hence a quasi invariant measure for any measurable function f X &rarr X . Suppose that X is a topological space and that &Sigma is the Borel sigma algebra Borel algebra on X . trivially satisfies the condition to be a regular measure . is never a strictly positive measure , regardless of X , , since every measurable set has zero measure. Since X 0, is always a finite measure, and hence a locally finite measure . If X is a Hausdorff space Hausdorff topological space with its Borel algebra, then trivially satisfies the condition to be an tight measure . Hence, is also a Radon measure . In fact, it is the vertex of the cone mathematics pointed cone of all non negative Radon measures on X . If X is an infinity infinite dimension al Banach space with its Borel algebra, then is the only measure on X , that is locally finite and invariant under all translations of X . See the article There is no infinite dimensional Lebesgue measure . If X is n dimensional Euclidean space R sup n sup with its usual algebra and n dimensional Lebesgue measure sup n sup , is a singular measure with respect to sup n sup simply decompose R sup n sup as A R sup n sup 0 and B 0 and observe that A sup n sup B 0. Category Measures measure theory nl Triviale maat pl Miara trywialna ... more details
Unit measure is an Probability axioms axiom of probability theory that states that the probability of the entire sample space is equal to one 1 number unity that is, P S 1 where S is the sample space. Loosely speaking, it means that S must be chosen so that when the experiment is performed, something happens. The term measure here refers to the measure theory measure theoretic approach to probability. Unreferenced date November 2010 DEFAULTSORT Unit Measure Category Probability theory probability stub ... more details
The measure in quantum physics is the integration measure mathematics measure used for performing a path integral . In quantum field theory , one must sum over all possible histories of a system. When summing over possible histories, which may be very similar to each other, one has to decide when two histories are to be considered different, and when they are to be considered the same, in order not to count the same history twice. This decision is coded within the concept of the measure by an Observer quantum physics observer . In fact, the possible histories can be deformed continuously, and therefore the sum is in fact an integral , known as path integral formulation path integral . br In the limit where the sum is becoming an integral, the concept of the measure described above is replaced by an integration measure mathematics measure . See also Action physics Category physics quantum stub ... more details
Unreferenced date February 2007 In measure theory , a pushforward measure also push forward , push forward or image measure is obtained by transferring pushing forward a measure mathematics measure from ... X sub 1 sub X sub 2 sub and a measure sub 1 sub 0, , the pushforward of is defined to be the measure f sub sub sub 2 sub 0 ... mutatis mutandis for a signed measure signed or complex measure . Examples and applications A natural Lebesgue measure on the unit circle S sup 1 sup here thought of as a subset of the complex plane C may be defined using a push forward construction and Lebesgue measure on the real line R . Let also denote the restriction of Lebesgue measure to the interval 0, 2 and let f 0 ... Lebesgue measure on S sup 1 sup is then the push forward measure f sub sub . The measure f sub sub might also be called arc length measure or angle measure , since the f sub sub measure ... at the centre of the circle. The previous example extends nicely to give a natural Lebesgue measure ... sup T sup 1 sup . This Lebesgue measure on T sup n sup is, up to normalization, the Haar measure for the compact space compact , connected space connected Lie group T sup n sup . Gaussian measure s on infinite dimensional vector spaces are defined using the push forward and the standard Gaussian measure on the real line a Borel measure on a separable space separable Banach space X is called ... to X is a Gaussian measure on R . Consider a measurable function f X X and the Function composition ... of such systems to find a measure &mu on X that the map f leaves unchanged, a so called invariant measure , one for which f sub &lowast sub &mu &mu . One can also consider quasi invariant measure s for such a dynamical system a measure on X is called quasi invariant under f if the push forward of by f is merely equivalence of measures equivalent to the original measure , not necessarily ... more details
In mathematics , specifically in Measure mathematics measure theory , a Borel measure is defined as follows let X be a locally compact Hausdorff space , and let math mathfrak B X math be the Sigma algebra Generated .CF.83 algebra smallest algebra that contains the open sets of X this is known as the algebra of Borel set s. Any measure defined on the algebra of Borel sets is called a Borel measure . Some authors require in addition that C for every compact set C . If a Borel measure is both inner regular and outer regular , it is called a regular Borel measure . If is both inner regular and Locally finite measure locally finite , it is called a Radon measure . Note that a locally finite Borel measure automatically satisfies C for every compact set C . On the real line The real line R with its Real line usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it. In this case, math mathfrak B textbf R math is the smallest algebra that contains the open intervals of R . While there are many Borel measures , the choice of Borel measure which assigns math mu a,b b a math for every interval math a,b math is sometimes called the Borel measure on R . In practice, even the Borel measure is not the most useful measure defined on the algebra of Borel sets indeed, the Lebesgue measure math lambda math is an extension of the Borel measure which possesses the crucial property that it is a complete measure unlike the Borel measure . To clarify, when one says that the Lebesgue measure math lambda math is an extension of the Borel measure math mu math , it means that every Borel measurable set E is also a Lebesgue measurable set, and the Borel measure and the Lebesgue measure coincide on the Borel sets ... General integration and measure publisher Cambridge University Press year 1974 isbn 0 521 29715 X pages 158 184 DEFAULTSORT Borel Measure Category Measures measure theory de Borelma fr mesure de Borel ... more details
A Baire measure is a measure mathematics measure on the algebra of Baire set s of a topological space . In spaces that are not metric space s, the Borel set s and the Baire set s may differ. In this context, Baire measures are used because they connect to the properties of continuous function s more directly. References Leonard Gillman and Meyer Jerison , Rings of Continuous Functions , Springer Verlag 43, 1960 mathanalysis stub Category Measures measure theory nl Baire maat ... more details
In mathematics , the counting measure is an intuitive way to put a measure mathematics measure on any Set mathematics set the size of a subset is taken to be the number of elements in the subset, if the subset is finite, and &infin if the subset is infinite . ref PlanetMath id CountingMeasure title Counting Measure ref Formally, start with a set and consider the sigma algebra on consisting of power set all subsets of &Omega . Define a measure on this sigma algebra by setting A A if A is a finite subset of and A if A is an infinite subset of , where A denotes the cardinality of set A. Then , , is a measure space . Notes Reflist DEFAULTSORT Counting Measure Category Measures measure theory ca Mesura de comptar da T llem let de Z hlma Ma theorie eo Kalkula mezuro fr Mesure de comptage ko nl Telmaat ja pl Miara licz ca ru sv Kardinalitetm tt uk ... more details
In mathematics, a measure algebra is a Boolean algebra structure Boolean algebra with a countably additive positive measure. A probability measure on a measure space gives a measure algebra on the Boolean algebra of measurable sets modulo null set s. Definition A measure algebra is a Boolean algebra B with a measure m , which is a real valued function on B such that m 0 0, m 1 1 m x 0 if x 0 m is countably additive m x sub i sub m x sub i sub if the x sub i sub are a countable set of elements that are disjoint x sub i sub x sub j sub 0 whenever i j . References Citation last1 Jech first1 Thomas author1 link Thomas Jech title Set Theory publisher Springer Verlag location Berlin, New York edition third millennium series Springer Monographs in Mathematics isbn 978 3 540 44085 7 doi 10.1007 3 540 44761 X 22 year 2003 chapter Saturated ideals page 415 Category Measure theory ... more details
In mathematics , Banach measure in measure theory may mean a real valued function on the algebra of all sets for example, in the plane , by means of which a rigid, finitely additive area can be defined for every set, even when a set does not have a true geometric area. That is, this is a theoretical definition getting round the phenomenon of non measurable set s. However, as the Vitali set shows, it cannot be countably additive. The existence of Banach measures proves the impossibility of a Banach Tarski paradox in two dimensions. clarify date October 2011 fact date October 2011 The concept of Banach measure is to be distinguished from the idea of a measure taking values in a Banach space , for example in the theory of spectral measure s. External links http www groups.dcs.st and.ac.uk history Mathematicians Banach.html Stefan Banach bio Category Measures measure theory mathanalysis stub ... more details
In mathematics , a Measure mathematics measure is said to be saturated if every locally measurable set is also measurable . ref Bogachev, Vladmir 2007 . Measure Theory Volume 2 . Springer. ISBN 978 3540345138. ref A set math E math , not necessarily measurable, is said to be locally measurable if for every measurable set math A math of finite measure, math E cap A math is measurable. math sigma math finite measures, and measures arising as the restriction of Outer measure outer measures , are saturated. References reflist Category Measures measure theory mathanalysis stub ... more details
No footnotes date February 2010 expand Italian Misura matematica date December 2011 Image Measure illustration.png right thumb Informally, a measure has the property of being monotone function monotone in the sense that if A is a subset of B , the measure of A is less than or equal to the measure of B . Furthermore, the measure of the empty set is required to be 0. In mathematical analysis , a measure ... interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean ... subsets of the n dimensional Euclidean space R sup n sup . For instance, the Lebesgue measure ... sense of the word, specifically 1. To qualify as a measure see Definition Definition below ... set and also satisfy the other axioms of a measure. This problem was resolved by defining measure only on a sub collection of all subsets the subsets on which the measure is to be defined are called ... measure cannot be defined consistently, are necessarily complicated, in the sense of being ... of choice . Measure theory was developed in successive stages during the late 19th and early 20th ... s axiomatisation of probability theory and in ergodic theory . In integration theory, specifying a measure ..., the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer ... is given by the measure. Ergodic theory considers measures that are invariant under, or arise naturally ... is called a measure if it satisfies the following properties Non negativity math mu E geq 0 math for all ... one set E has finite measure. Then the null set automatically has measure zero because of countable ... has measure zero. anchor Measurable space The pair math X , &Sigma is called a measurable space , the members ... a visible anchor measure space . If only the second and third conditions of the definition of measure ... a signed measure . A probability measure is a measure with total measure one i.e., math &mu X ... more details
Refimprove date October 2010 In mathematics , a complete measure or, more precisely, a complete measure space is a measure mathematics measure space in which every subset of every null set is measurable having measure zero . More formally, X , , is complete if and only if math S subseteq ... constructed Lebesgue measure on the real line denote this measure space by R , B , . We now wish to construct two dimensional Lebesgue measure sup 2 sup on the plane R sup 2 sup as a product measure . Na vely, we would take the sigma algebra &sigma algebra on R sup 2 sup to be B ... A sub 2 sub for A sub i sub B . While this approach does define a measure space , it has a flaw. Since every singleton mathematics singleton set has one dimensional Lebesgue measure ... sup 2 sup measure of 0 × A is not defined, but math 0 times A subseteq 0 times mathbb R , math and this larger set does have sup 2 sup measure zero. So, two dimensional Lebesgue measure as just defined is not complete, and some kind of completion procedure is required. Construction of a complete measure Given a possibly incomplete measure space X , , , there is an extension X , sub 0 sub , sub 0 sub of this measure space that is complete. The smallest such extension i.e. the smallest algebra sub 0 sub is called the completion of the measure space. The completion can be constructed as follows let Z be the set of all subsets of measure ... sub 0 sub , sub 0 sub is a complete measure space, and is the completion of X , , ... Examples Borel measure as defined on the Borel algebra generated by the Open set open Interval mathematics ... to define the complete Lebesgue measure. n dimensional Lebesgue measure is the completion of the n ... measure, as in the one dimensional case. References SpringerEOM title Complete measure id C c023800 first A.P. last Terekhin DEFAULTSORT Complete Measure Category Measures measure theory fr Compl tion ... more details
Measure or sometimes The Measure in typography refers to the width of a block or Column typography column of text from the left Margin typography margin to the right margin. Punctuation should preferably Hanging punctuation hang outside the measure. It is relative to each design , but can be used as a unit of measurement within that design to create unity and harmony. The measure of a text influences legibility. Long lines are hard to read, short lines are more easily read. 45 75 characters per line are regarded as the ideal range for the measure. For multiple column setups, 40 50 characters are often preferred. 66 is sometimes considered ideal for one column setups. Generally, if the measure is wide, the leading of a text should be increased if the measure is short, it is can safely be decreased. Reverse text, i.e. white text on black also requires more leading. ref http webtypography.net Rhythm and Proportion Horizontal Motion 2.1.2 ref ref http www.markboulton.co.uk journal comments five simple steps to better typography ref The measure or the length of a legible line is the length of the alphabet Caps, Lowercase and Numerals of a typeface of a size being used. References Reflist Typography terms DEFAULTSORT Measure Typography Category Page layout ... more details
In mathematics , a regular measure on a topological space is a measure mathematics measure for which every measurable set is approximately open and approximately closed . Definition Let X , T be a topological space and let be a sigma algebra &sigma algebra on X that contains the topology T so that all open set open and closed set s are measurable set s, and is at least as fine as the Borel sigma algebra Borel &sigma algebra on X . Let be a measure on X , . A measurable subset A of X is said to be &mu regular if math mu A sup mu F F subseteq A, F mbox closed math and math mu A inf mu G G supseteq A, G mbox open . math Alternatively, A is a &mu regular set if and only if , for every > 0, there exists a closed set F and an open set G such that math F subseteq A subseteq G math and math mu G setminus F delta. math The two definitions are equivalent if math mu A math is finite otherwise, the second definition is stronger . If every measurable set is regular, then the measure &mu is said to be a regular measure . Some authors require the set F to be compact not just closed . ref harvnb Dudley 1989 loc Sect. 7.1 ref Examples Lebesgue measure on the real line is a regular measure see the regularity theorem for Lebesgue measure . Any Borel probability measure on any metric space is a regular measure. The trivial measure , which assigns measure zero to every measurable subset, is a regular measure. A trivial example of a non regular measure on the real line with its usual topology is the measure where math mu emptyset 0 math , math mu left 1 right 0 , , math , and math mu A infty , , math for any other set math A math . Notes references References cite book last Billingsley first Patrick title Convergence of Probability Measures publisher John Wiley & Sons, Inc. location New York year 1999 isbn 0 471 19745 9 cite book last Parthasarathy first K. R. title Probability ... regular measure Radon measure Regularity theorem for Lebesgue measure Category Measures measure ... more details
In mathematics , in particular in measure theory , an inner measure is a Function mathematics function on the Set mathematics set of all subsets of members of a given algebra with values in the extended real line extended real numbers satisfying some technical conditions. Intuitively, the inner measure of a set is a lower bound of the size of that set. Definition An inner measure is a function math varphi 2 X rightarrow 0, infty , math defined on all subsets of a set X , that satisfies the following conditions Null empty set The empty set has zero inner measure see also measure zero . math varphi varnothing 0 math superadditivity Superadditive For any disjoint sets A and B, math varphi A cup B geq varphi A varphi B . math Limits of decreasing towers For any sequence A sub j sub of sets such that math A j supseteq A j 1 math for each j and math varphi A 1 infty math math varphi left bigcap j 1 infty A j right lim j to infty varphi A j math Infinity must be approached If math varphi A infty math for a set A then for every positive number c, there exists a B which is a subset of A such that, math c leq varphi B leq varphi A math The inner measure induced by a measure Let be a algebra over a set X and be a Measure mathematics measure on . Then the inner measure sub sub induced ... sub gives a lower bound of the size of any set by insuring it is at least as big as the measure of any of its measurable subsets. Even though the set function sub sub is usually not a measure, ..., If E &sube F then sub sub E &le sub sub F . Measure completion Induced inner measures are often used in combination with outer measure s to extend a measure to a larger algebra. If is a measure ... mu T mu T mu T math , for all T math scriptstyle hat Sigma math is a measure on math scriptstyle hat Sigma math known as the completion of . References Halmos, Paul R., Measure Theory , D. Van Nostrand ... Measures measure theory ... more details
In mathematics , a Dirac measure is a measure mathematics measure sub x sub on a set X with any sigma algebra &sigma algebra of subset s of X defined by math delta x A 1 A x begin cases 0, & x not in A 1, & x in A. end cases math for a given math x in X math and any measurable set measurable set A X . The Dirac measure is a probability measure , and in terms of probability it represents the almost sure outcome x in the sample space X . We can also say that the measure is a single Atom measure theory atom at x 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 point s of the convex set of probability measures on X . The name is a back ... as a theorem of Lebesgue integration . Properties of the Dirac measure Let sub x sub denote the Dirac measure centred on some fixed point x in some measurable space X , . sub x sub is a probability measure, and hence a finite measure. Suppose that X , T is a topological space and that ... positive measure if and only if the topology T is such that x lies within every non empty open set, e.g. in the case of the trivial topology , X . Since sub x sub is probability measure, it is also a locally finite measure . If X is a Hausdorff space Hausdorff topological space with its Borel algebra, then sub x sub satisfies the condition to be an inner regular measure , since Singleton ... is also a Radon measure . Assuming that the topology T is fine enough that x is closed, which is the case in most applications, the Support measure theory support of sub x sub is x . Otherwise, supp sub x sub is the closure of x in X , T . Furthermore, sub x sub is the only probability measure ... and n dimensional Lebesgue measure sup n sup , then sub x sub is a singular measure with respect ... publisher CRC Press See also Discrete measure DEFAULTSORT Dirac Measure Category Measures measure ... more details
Unreferenced date December 2009 In mathematics , Gaussian measure is a Borel measure on finite dimensional ... denote the complete measure completion of the Borel sigma algebra Borel &sigma algebra on R sup n sup . Let sup n sup B sub 0 sub R sup n sup 0, denote the usual n dimensional Lebesgue measure . Then the standard Gaussian measure sup n sup B sub 0 sub R sup n sup 0, is defined by math gamma n A frac ... generally, the Gaussian measure with mean R sup n sup and variance sup 2 sup > 0 is given by math ... measures . The Dirac measure sub sub is the weak convergence of measures weak limit of math gamma mu, sigma 2 n math as 0, and is considered to be a degenerate Gaussian measure in contrast, Gaussian ... of Gaussian measure The standard Gaussian measure sup n sup on R sup n sup is a Borel measure ... structure is Equivalence measure theory equivalent to Lebesgue measure math lambda n ll gamma n ll lambda n math , where math ll math stands for absolute continuity of measures is Support measure theory supported on all of Euclidean space supp sup n sup R sup n sup is a probability measure sup n sup R sup n sup 1 , and so it is Locally finite measure locally finite is Strictly positive measure strictly positive every non empty open set has positive measure is Inner regular measure inner ... so Gaussian measure is a Radon measure is not Translation geometry translation Invariant mathematics ... measure push forward of standard Gaussian measure by the translation map T sub h sub R sup n sup &rarr R sup n sup , T sub h sub x x h is the probability measure associated to a normal distribution ... dimensional Lebesgue measure there is no analogue of Lebesgue measure on an infinite ... dimensional spaces, the main example being the abstract Wiener space construction. A Borel measure on a separable space separable Banach space E is said to be a non degenerate centered Gaussian measure if, for every linear functional L E sup sup except L 0, the push forward measure L sub sub is a non ... more details
Convergence in measure can refer to two distinct mathematical concepts which both generalize the concept ... functions on a measure space X ,&Sigma , &mu . The sequence f sub n sub is said to converge globally in measure to f if for every &epsilon 0, math lim n to infty mu x in X f x f n x geq varepsilon 0 math , and to converge locally in measure to f if for every &epsilon 0 and every math F in Sigma ... . Convergence in measure can refer to either global convergence in measure or local convergence in measure ... N are measurable functions X &rarr real numbers R . Global convergence in measure implies local convergence in measure. The converse, however, is false i.e. , local convergence in measure is strictly weaker than global convergence in measure, in general. If, however, math mu X infty math or, more generally, if all the f sub n sub vanish outside some set of finite measure, then the distinction between local and global convergence in measure disappears. If &mu is sigma finite &sigma finite and f sub n sub converges locally or globally to f in measure, there is a subsequence converging to f almost everywhere . If &mu is &sigma finite, f sub n sub converges to f locally in measure if and only ..., if f sub n sub converges to f almost everywhere, then f sub n sub converges to f locally in measure ... convergence is replaced by local or global convergence in measure. Clarify date June 2011 If &mu ... is replaced by local or global convergence in measure. Clarify date June 2011 If X a , b R and &mu is Lebesgue measure , there are sequences g sub n sub of step functions and h sub n sub of continuous functions converging globally in measure to f . If f and f sub n sub n N are in Lp space L sup ... globally in measure. The converse is false. If f sub n sub converges to f in measure and g sub n sub converges to g in measure then f sub n sub g sub n sub converges to f g in measure. Additionally, if the measure space is finite, f sub n sub g sub n sub also converges to fg . Counterexamples Let ... more details
In mathematics , more precisely in measure theory , a measure mathematics measure on the real line is called a discrete measure in respect to the Lebesgue measure if its support measure theory support is at most a countable set . Note that the support need not be a discrete set . Geometrically, a discrete measure on the real line, with respect to Lebesgue measure is a collection of point masses. Definition and properties A measure math mu math defined on the Lebesgue measure Lebesgue measurable set s of the real line with values in math 0, infty math is said to be discrete if there exists a possibly finite sequence of numbers math s 1, s 2, dots , math such that math mu mathbb R backslash s 1, s 2, dots 0. math The simplest example of a discrete measure on the real line is the Dirac delta function math delta. math One has math delta mathbb R backslash 0 0 math and math delta 0 1. math More generally, if math s 1, s 2, dots math is a possibly finite sequence of real numbers, math a 1, a 2 ... measure s math delta s i math defined by math delta s i X begin cases 1 & mbox if s i in X 0 & mbox if s i not in X end cases math for any Lebesgue measurable set math X. math Then, the measure math mu sum i a i delta s i math is a discrete measure. In fact, one may prove that any discrete measure ... measure space s. Given a measure space math X, Sigma , math and two measures math mu math and math nu ... for an at most countable subset of the real line if math nu math is the Lebesgue measure, so they were not necessary in the first definition above. As in the case of measures on the real line, a measure math mu math on math X, Sigma math is discrete in respect to another measure math nu math on the same ..., s 2, dots , math the singletons math s i math are in math Sigma, math and their math nu math measure is 0. One can also define the concept of discreteness for signed measure s. Then, instead of conditions ... measure author A.P. Terekhin Category Measures measure theory it Misura discreta ... more details
Refimprove date December 2009 A conical measure is a type of laboratory glassware which consists of a conical cup with a notch on the top to allow for the easy pouring of liquids, and graduated markings on the side to allow easy and accurate measurement of volumes of liquid. They may be made of plastic , glass , or Pyrex borosilicate glass . The use of the conical measure usually dictates its construction material. Plastic conical measures, commonly referred to as measuring cups are used by patients to measure liquid medicaments for oral administration. Glass and borosilicate conical measures are commonly used when compounding by the pharmacy profession. Conical measures are the most commonly used item of glassware used in the preparation of extemporaneous medicaments. They are not as precise as graduated cylinder s for measuring liquids, but make up for this in terms of easy pouring and ability to mix solutions within the measure itself. History During his experiment s, Ab al Rayh n al B r n 973 1048 invented the conical measure, ref Marshall Clagett 1961 . The Science of Mechanics in the Middle Ages , p. 64. University of Wisconsin Press . ref in order to find the ratio between the weight of a substance in air and the weight of water displaced, and to accurately measure the specific weights of the gemstones and their corresponding metal s, which are very close to modern measurements. fact date September 2011 References reflist Laboratory equipment DEFAULTSORT Conical Measure Category Volumetric instruments Category Laboratory glassware ar es Medida c nica fa simple Conical measure th ... more details
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