Unreferenced date February 2007 In measure theory , a pushforwardmeasure 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 ... or Koopman operator . See also Measure preserving dynamical system DEFAULTSORT PushforwardMeasure ... 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 ... more details
The notion of pushforward in mathematics is dual to the notion of pullback , and can mean a number of different, but closely related things. Pushforward differential the differential of a smooth map between manifold s, and the pushforward operations it defines. Pushforward differential Pushforward of vector fields Direct image sheaf the pushforward of a sheaf mathematics sheaf by a map. Pushforward homology the map induced in Homology mathematics homology by a continuous map between topological space s. Fiberwise integral the direct image of a differential form or Cohomology cohomology class by a smooth map, defined by integration on the fibres . Pushout category theory the categorical dual of Pullback category theory . Pushforwardmeasure the measure mathematics measure induced on the target measure space by a measurable function . The transfer operator is the pushforward on the space of measurable function s its adjoint, the pull back, is the composition or Koopman operator . mathdab zh ... more details
in mathematics , see pushforward . Suppose that &phi M N is a smooth map between smooth manifold ... itself called the pushforward . Motivation Let &phi U &rarr V be a Smooth function ... x . The application of d &phi sub x sub to a tangent vector X is sometimes called the pushforward of X by &phi . The exact definition of this pushforward depends on the definition one uses for tangent ... varphi circ gamma 0 . math Here &gamma is a curve in M with &gamma 0 x . In other words, the pushforward ... is a smooth real valued function on N . By definition, the pushforward of X at a given x in M is in T sub .... If &phi is a local diffeomorphism , then the pushforward at x is invertible and its inverse gives ... bundle section of the vector bundle Hom TM , &phi sup sup TN over M . Pushforward of vector fields ... a pushforward of X by &phi as a vector field on N . For example, if the map &phi is not surjective, there is no natural way to define such a pushforward outside of the image of &phi . Also, if &phi is not injective there may be more than one choice of pushforward at a given point. Nevertheless, one ... , i.e., a section of TM . Then, applying the differential pointwise to X yields the pushforward ... when &phi is a diffeomorphism . In this case, the pushforward defines a vector field Y on N , given ... in &phi sup 1 sup y . This is precisely the condition that guarantees that a pushforward of X , as a vector ... Category Differential geometry Category Smooth functions de Pushforward es Diferencial matem tica ... more details
In algebraic topology , the pushforward of a continuous function math f math math X rightarrow Y math between two topological space s is a homomorphism math f H n left X right rightarrow H n left Y right math between the homology group s for math n geq 0 math . Homology is a functor which converts a topological space math X math into a sequence of homology groups math H n left X right math . Often, the collection of all such groups is referred to using the notation math H left X right math this collection has the structure of a graded group . In any category mathematics category , a functor must induce a corresponding morphism . The pushforward is the morphism corresponding to the homology functor. Definition for singular and simplicial homology We build the pushforward homomorphism as follows for singular or simplicial homology First we have an induced homomorphism between the singular or simplicial chain complex math C n left X right math and math C n left Y right math defined by composing each singular n simplex math sigma X math math Delta n rightarrow X math with math f math to obtain a singular n simplex of math Y math , math f left sigma X right f sigma X math math Delta n rightarrow Y math . Then we extend math f math linearly via math f left sum tn t sigma t right sum tn tf left sigma t right math . The maps math f math math C n left X right rightarrow C n left Y right math satisfy math f partial partial f math where math partial math is the boundary operator between chain groups, so math partial f math defines a chain complex Chain maps chain map . We have a diagram math begin array ccccccccc dots & rightarrow & C n 1 left X right & overset partial rightarrow & C n left X right & overset partial rightarrow & C n 1 left X right & rightarrow & dots & & downarrow scriptstyle f & & downarrow scriptstyle f & & downarrow scriptstyle f dots & rightarrow & C n 1 left Y right & overset partial rightarrow & C n left Y right & overset partial rightarrow & C n 1 left Y ... more details
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
right 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 ... disguised as Friar Lodowick. Isabella , a novice nun. Claudio , Isabella s brother. Angelo Measure for Measure Angelo , rules in the Duke s absence. Escalus , a lord. Lucio , a debauched lord. Mariana ... end Synopsis Image William Hunt Claudio and Isabella Shakespeare Measure for Measure.jpg thumb right ... of a strict judge, Angelo Measure for Measure Angelo . Claudio, a young nobleman, is betrothed unofficially ... Measure for Measure Angelo , which Angelo smoothly denies. As the scene develops, it appears that Friar ... in 1565. ref name Bawcutt N. W. Bawcutt ed. , Measure for Measure Oxford, 1991 , p. 17 ref Shakespeare ... 7 2 For in the same 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 ... of the text of Measure that survives today is not in its original form, but rather the product ... Press, 1993 . See also Shakespeare s Mediterranean Measure for Measure , in Shakespeare and the Mediterranean ... William Topham The earliest 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 ..., Measure for Measure, New York, Penguin, 1998 pp. 181 183. ref Notable more recent productions of Measure for Measure are Charles Laughton as Angelo at the Old Vic Theatre in 1933, Peter Brook ... 1836 by Richard Wagner with the libretto written by the composer based on Measure for Measure The musical ... Brecht play, Round Heads and Pointed Heads , was originally written as an adaptation of Measure for Measure . Thomas Pynchon s early short story, Mortality and Mercy in Vienna , take its title ... How to recognize different parts of the body contains an underwater recital of Measure for Measure ... more details
In mathematics &mdash specifically, in measure theory &mdash a perfect measure or, more accurately, a perfect measure space is one that is &ldquo well behaved &rdquo in some sense. Intuitively, a perfect measure &mu is one for which, if we consider the pushforwardmeasure on the real line R , then every measurable set is &ldquo &mu approximately a Borel set &rdquo . The notion of perfectness is closely related to tightness of measures indeed, in metric space s, tight measures are always perfect. Definition A measure space X , &Sigma , &mu is said to be perfect if, for every &Sigma measurable function f X &rarr R and every A &sube R with f sup &minus 1 sup A &isin &Sigma , there exist Borel subsets A sub 1 sub and A sub 2 sub of R such that math A 1 subseteq A subseteq A 2 mbox and mu big f 1 A 2 setminus A 1 big 0. math Results concerning perfect measures If X is any metric space and &mu is an inner regular measure inner regular or tightness of measures tight measure on X , then X , B sub X sub , &mu is a perfect measure space, where B sub X sub denotes the Borel sigma algebra Borel &sigma algebra on X . References SpringerEOM title Perfect measue id P p072070 first V.V. last Sazonov cite book last Parthasarathy first K. R. title Probability measures on metric spaces publisher AMS Chelsea Publishing, Providence, RI year 2005 chapter Chapter 2, Section 4 isbn 0 8218 3889 X id MathSciNet id 2169627 Category Measures measure theory pl Miara doskona a ... 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 ... hand side is the Radon Nikodym derivative , and T sub h sub sub &lowast sub &gamma sup n sup is the pushforwardmeasure 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 ... more details
In mathematics , an invariant measure is a measure mathematics measure that is preserved by some function mathematics function . Ergodic theory is the study of invariant measures in dynamical systems . The Krylov Bogolyubov theorem proves the existence of invariant measures under certain conditions on the function and space under consideration. Definition Let X , be a measurable space and let f be a measurable function from X to itself. A measure on X , is said to be invariant under f if, for every measurable set A in , math mu left f 1 A right mu A . math In terms of the pushforwardmeasure push forward , this states that f sub sub . The collection of measures usually probability measure s on X that are invariant under f is sometimes denoted M sub f sub X . The collection of ergodic adjective ergodic measures , E sub f sub X , is a subset of M sub f sub X . Moreover, any convex combination of two invariant measures is also invariant, so M sub f sub X is a convex set E sub f sub X consists precisely of the extreme points of M sub f sub X . In the case ... map, a measure on X , is said to be an invariant measure if it is an invariant measure for each ... measure for a sequence of random variable s Z sub t sub sub t 0 sub perhaps a Markov chain or the solution ... T sub a sub R R given by math T a x x a. math Then one dimensional Lebesgue measure &lambda is an invariant measure for T sub a sub . More generally, on n dimensional Euclidean space R sup n sup with its usual Borel algebra, n dimensional Lebesgue measure sup n sup is an invariant measure for any ... measure in the first example is unique up to trivial renormalization with a constant factor ... probability measure math mu boldsymbol rm S rightarrow boldsymbol rm R math is invariant. Note ... measure References Invariant measures, John Von Neumann, AMS Bookstore, 1999, ISBN 9780821809129 DEFAULTSORT Invariant Measure Category Dynamical systems Category Measures measure theory pl Miara niezmiennicza ... more details
measure Product measurePushforwardmeasure Vector measure Volume form Measurable cardinal div References ...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 on a set mathematics set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization ... measure on a Euclidean space , which assigns the conventional length , area , and volume ..., the Lebesgue measure of the Interval mathematics interval 0, 1 in the real line real numbers is its length in the everyday sense of the word, specifically 1. To qualify as a measure see ... size to each subset of a given 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 ... space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated ... consequence of the axiom of choice . Measure theory was developed in successive stages during ... theory, specifying a measure allows one to define integral s on spaces more general than subsets of Euclidean space moreover, the integral with respect to the Lebesgue measure on Euclidean spaces ... whose probability is given by the measure. Ergodic theory considers measures that are invariant ... real number line is called a measure if it satisfies the following properties Non negativity ... One may require that at least one set E has finite measure. Then the null set automatically has measure zero because of countable additivity, because math mu E mu E cup varnothing cup varnothing cup ... if the empty set has measure zero. anchor Measurable space The pair math X , &Sigma is called ... more details
wiktionary unit of measure Unit of measure commonly refers to Units of measurement for relevance to weights and measures. It may also refer to 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
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 cs Aritmetick m ra 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
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 ... already constructed Lebesgue measure on the real line denote this measure space by R , B , . We now wish to construct some 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 ... measure zero, math lambda 2 0 times A lambda 0 cdot lambda A 0 math for any subset A of R . However ... . Then the 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, this 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 ... of the measure space. The completion can be constructed as follows let Z be the set of all subsets of measure zero subsets of X intuitively, those elements of Z that are not already in are the ones ... Then X , sub 0 sub , sub 0 sub is a complete measure space, and is the completion of X ... B mu A . math Examples Borel measure as defined on the Borel algebra generated by the Open set open ... must be used to define the complete Lebesgue measure. n dimensional Lebesgue measure is the completion ... of the Borel 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 ... 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
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