[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki12\/borel-measure-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki12\/borel-measure-wikipedia\/","headline":"Borel measure – Wikipedia","name":"Borel measure – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 Measure defined on all open sets of a topological space In mathematics, specifically","datePublished":"2018-04-01","dateModified":"2018-04-01","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki12\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki12\/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:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/68baa052181f707c662844a465bfeeb135e82bab","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/68baa052181f707c662844a465bfeeb135e82bab","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki12\/borel-measure-wikipedia\/","wordCount":5439,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Measure defined on all open sets of a topological spaceIn mathematics, specifically in measure theory, a Borel measure on  a topological space is a measure that is defined on all open sets (and thus on all Borel sets).[1] Some authors require additional restrictions on the measure, as described below.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsFormal definition[edit]On the real line[edit]Product spaces[edit]Applications[edit]Lebesgue\u2013Stieltjes integral[edit]Laplace transform[edit]Hausdorff dimension and Frostman’s lemma[edit]Cram\u00e9r\u2013Wold theorem[edit]References[edit]Further reading[edit]External links[edit]Formal definition[edit]Let X{displaystyle X} be a locally compact Hausdorff space, and let        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4B(X){displaystyle {mathfrak {B}}(X)} be the smallest \u03c3-algebra that contains the open sets of X{displaystyle X}; this is known as the \u03c3-algebra of Borel sets.  A Borel measure is any measure \u03bc{displaystyle mu } defined on the \u03c3-algebra of Borel sets.[2] A few authors require in addition that \u03bc{displaystyle mu } is locally finite, meaning that \u03bc(C){displaystyle mu } is both inner regular and outer regular, it is called a regular Borel measure. If \u03bc{displaystyle mu } is both inner regular, outer regular, and locally finite, it is called a Radon measure.On the real line[edit]The real line R{displaystyle mathbb {R} } with its usual topology is a locally compact Hausdorff space; hence we can define a Borel measure on it. In this case, B(R){displaystyle {mathfrak {B}}(mathbb {R} )} is the smallest \u03c3-algebra that contains the open intervals of R{displaystyle mathbb {R} }. While there are many Borel measures \u03bc, the choice of Borel measure that assigns \u03bc((a,b])=b\u2212a{displaystyle mu ((a,b])=b-a} for every half-open interval (a,b]{displaystyle (a,b]} is sometimes called “the” Borel measure on R{displaystyle mathbb {R} }. This measure turns out to be the restriction to the Borel \u03c3-algebra of the Lebesgue measure \u03bb{displaystyle lambda }, which is a complete measure and is defined on the Lebesgue \u03c3-algebra. The Lebesgue \u03c3-algebra is actually the completion of the Borel \u03c3-algebra, which means that it is the smallest \u03c3-algebra that contains all the Borel sets and can be equipped with a complete measure. Also, the Borel measure and the Lebesgue measure coincide on the Borel sets (i.e., \u03bb(E)=\u03bc(E){displaystyle lambda (E)=mu (E)} for every Borel measurable set, where \u03bc{displaystyle mu } is the Borel measure described above).Product spaces[edit]If X and Y are second-countable, Hausdorff topological spaces, then the set of Borel subsets B(X\u00d7Y){displaystyle B(Xtimes Y)} of their product coincides with the product of the sets B(X)\u00d7B(Y){displaystyle B(X)times B(Y)} of Borel subsets of X and Y.[3] That is, the Borel functorBor:Top2CHaus\u2192Meas{displaystyle mathbf {Bor} colon mathbf {Top} _{mathrm {2CHaus} }to mathbf {Meas} }from the category of second-countable Hausdorff spaces to the category of measurable spaces preserves finite products.Applications[edit]Lebesgue\u2013Stieltjes integral[edit]The Lebesgue\u2013Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue\u2013Stieltjes measure, which may be associated to any function of bounded variation on the real line.  The Lebesgue\u2013Stieltjes measure is a regular Borel measure, and conversely every regular Borel measure on the real line is of this kind.[4]Laplace transform[edit]One can define the Laplace transform of a finite Borel measure \u03bc on the real line by the Lebesgue integral[5](L\u03bc)(s)=\u222b[0,\u221e)e\u2212std\u03bc(t).{displaystyle ({mathcal {L}}mu )(s)=int _{[0,infty )}e^{-st},dmu (t).}An important special case is where \u03bc is a probability measure or, even more specifically, the Dirac delta function. In operational calculus, the Laplace transform of a measure is often treated as though the measure came from a distribution function f.  In that case, to avoid potential confusion, one often writes(Lf)(s)=\u222b0\u2212\u221ee\u2212stf(t)dt{displaystyle ({mathcal {L}}f)(s)=int _{0^{-}}^{infty }e^{-st}f(t),dt}where the lower limit of 0\u2212 is shorthand notation forlim\u03b5\u21930\u222b\u2212\u03b5\u221e.{displaystyle lim _{varepsilon downarrow 0}int _{-varepsilon }^{infty }.}This limit emphasizes that any point mass located at 0 is entirely captured by the Laplace transform.  Although with the Lebesgue integral, it is not necessary to take such a limit, it does appear more naturally in connection with the Laplace\u2013Stieltjes transform.Hausdorff dimension and Frostman’s lemma[edit]Given a Borel measure \u03bc on a metric space X such that \u03bc(X) > 0 and \u03bc(B(x, r)) \u2264 rs holds for some constant s > 0 and for every ball B(x, r) in X, then the Hausdorff dimension dimHaus(X) \u2265 s. A partial converse is provided by the Frostman lemma:[6]Lemma: Let A be a Borel subset of Rn, and let s\u00a0>\u00a00. Then the following are equivalent:Hs(A)\u00a0>\u00a00, where Hs denotes the s-dimensional Hausdorff measure.There is an (unsigned) Borel measure \u03bc satisfying \u03bc(A)\u00a0>\u00a00, and such that\u03bc(B(x,r))\u2264rs{displaystyle mu (B(x,r))leq r^{s}}holds for all x\u00a0\u2208\u00a0Rn and r\u00a0>\u00a00.Cram\u00e9r\u2013Wold theorem[edit]The Cram\u00e9r\u2013Wold theorem in measure theory states that a Borel probability measure on Rk{displaystyle mathbb {R} ^{k}} is uniquely determined by the totality of its one-dimensional projections.[7] It is used as a method for proving joint convergence results. The theorem is named after Harald Cram\u00e9r and Herman Ole Andreas Wold.References[edit]^ D. H. Fremlin, 2000. Measure Theory Archived 2010-11-01 at the Wayback Machine. Torres Fremlin.^ Alan J. Weir (1974). General integration and measure. Cambridge University Press. pp.\u00a0158\u2013184. ISBN\u00a00-521-29715-X.^ Vladimir I. Bogachev. Measure Theory, Volume 1. Springer Science & Business Media, Jan 15, 2007^ Halmos, Paul R. (1974), Measure Theory, Berlin, New York: Springer-Verlag, ISBN\u00a0978-0-387-90088-9^ Feller 1971, \u00a7XIII.1^ Rogers, C. A. (1998). Hausdorff measures. Cambridge Mathematical Library (Third\u00a0ed.). Cambridge: Cambridge University Press. pp.\u00a0xxx+195. ISBN\u00a00-521-62491-6.^ K. Stromberg, 1994. Probability Theory for Analysts. Chapman and Hall.Further reading[edit]Gaussian measure, a finite-dimensional Borel measureFeller, William (1971), An introduction to probability theory and its applications. Vol. II., Second edition, New York: John Wiley & Sons, MR\u00a00270403.J. D. Pryce (1973). Basic methods of functional analysis. Hutchinson University Library. Hutchinson. p.\u00a0217. ISBN\u00a00-09-113411-0.Ransford, Thomas (1995). Potential theory in the complex plane. London Mathematical Society Student Texts. Vol.\u00a028. Cambridge: Cambridge University Press. pp.\u00a0209\u2013218. ISBN\u00a00-521-46654-7. Zbl\u00a00828.31001.Teschl, Gerald, Topics in Real and Functional Analysis, (lecture notes)Wiener’s lemma   relatedExternal 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\/wiki12\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki12\/borel-measure-wikipedia\/#breadcrumbitem","name":"Borel measure – Wikipedia"}}]}]