[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki3\/discrete-measure-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki3\/discrete-measure-wikipedia\/","headline":"Discrete measure – Wikipedia","name":"Discrete measure – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia Schematic representation of the Dirac measure by a line surmounted by an arrow. The","datePublished":"2015-05-26","dateModified":"2015-05-26","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki3\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki3\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/4\/48\/Dirac_distribution_PDF.svg\/325px-Dirac_distribution_PDF.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/4\/48\/Dirac_distribution_PDF.svg\/325px-Dirac_distribution_PDF.svg.png","height":"244","width":"325"},"url":"https:\/\/wiki.edu.vn\/en\/wiki3\/discrete-measure-wikipedia\/","wordCount":5256,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia  Schematic representation of the Dirac measure by a line surmounted by an arrow. The Dirac measure is a discrete measure whose support is the point 0. The Dirac measure of any set containing 0 is 1, and the measure of any set not containing 0 is 0.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In mathematics, more precisely in measure theory, a  measure  on the real line is called a discrete measure (in respect to the Lebesgue measure) if it is concentrated on an at most countable set. 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.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsDefinition and properties[edit]Extensions[edit]References[edit]External links[edit]Definition and properties[edit]A measure \u03bc{displaystyle mu } defined on the Lebesgue measurable sets of the real line with values in        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4[0,\u221e]{displaystyle [0,infty ]} is  said to be discrete if there exists a (possibly finite) sequence of numberss1,s2,\u2026{displaystyle s_{1},s_{2},dots ,}such that\u03bc(R\u2216{s1,s2,\u2026})=0.{displaystyle mu (mathbb {R} backslash {s_{1},s_{2},dots })=0.}The simplest example of a discrete measure on the real line is the Dirac delta function \u03b4.{displaystyle delta .} One has \u03b4(R\u2216{0})=0{displaystyle delta (mathbb {R} backslash {0})=0} and  \u03b4({0})=1.{displaystyle delta ({0})=1.}More generally, if s1,s2,\u2026{displaystyle s_{1},s_{2},dots } is a (possibly finite) sequence of real numbers, a1,a2,\u2026{displaystyle a_{1},a_{2},dots } is a sequence of numbers in [0,\u221e]{displaystyle [0,infty ]} of the same length, one can consider the Dirac measures \u03b4si{displaystyle delta _{s_{i}}} defined by\u03b4si(X)={1\u00a0if\u00a0si\u2208X0\u00a0if\u00a0si\u2209X{displaystyle delta _{s_{i}}(X)={begin{cases}1&{mbox{ if }}s_{i}in X\\0&{mbox{ if }}s_{i}not in X\\end{cases}}}for any Lebesgue measurable set X.{displaystyle X.} Then, the measure\u03bc=\u2211iai\u03b4si{displaystyle mu =sum _{i}a_{i}delta _{s_{i}}}is a discrete measure. In fact, one may prove that any discrete measure on the real line has this form for appropriately chosen sequences s1,s2,\u2026{displaystyle s_{1},s_{2},dots } and a1,a2,\u2026{displaystyle a_{1},a_{2},dots }Extensions[edit]One may extend the notion of discrete measures to more general measure spaces. Given a measurable space (X,\u03a3),{displaystyle (X,Sigma ),} and two measures \u03bc{displaystyle mu } and \u03bd{displaystyle nu } on it, \u03bc{displaystyle mu } is said to be discrete in respect to \u03bd{displaystyle nu } if there exists an at most countable subset S{displaystyle S} of  X{displaystyle X} such thatAll singletons {s}{displaystyle {s}} with s\u2208S{displaystyle sin S} are measurable  (which implies that any subset of S{displaystyle S} is measurable)\u03bd(S)=0{displaystyle nu (S)=0,}\u03bc(X\u2216S)=0.{displaystyle mu (Xbackslash S)=0.,}Notice that the first two requirements are always satisfied for an at most countable subset of the real line if \u03bd{displaystyle nu } 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 \u03bc{displaystyle mu } on (X,\u03a3){displaystyle (X,Sigma )} is discrete in respect to another measure \u03bd{displaystyle nu } on the same space if and only if \u03bc{displaystyle mu } has the form\u03bc=\u2211iai\u03b4si{displaystyle mu =sum _{i}a_{i}delta _{s_{i}}}where S={s1,s2,\u2026},{displaystyle S={s_{1},s_{2},dots },} the singletons {si}{displaystyle {s_{i}}} are in \u03a3,{displaystyle Sigma ,} and their \u03bd{displaystyle nu } measure is 0.One can also define the concept of discreteness for signed measures. Then, instead of conditions 2 and 3 above one should ask that \u03bd{displaystyle nu } be zero on all measurable subsets of S{displaystyle S} and \u03bc{displaystyle mu } be zero on measurable subsets of X\u2216S.{displaystyle Xbackslash S.}References[edit]Kurbatov, V. G. (1999). Functional differential operators and equations. Kluwer Academic Publishers. ISBN\u00a00-7923-5624-1.External links[edit]     (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki3\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki3\/discrete-measure-wikipedia\/#breadcrumbitem","name":"Discrete measure – Wikipedia"}}]}]