Editing Borel measure

{{Use American English|date = February 2019}}
{{Short description|Measure defined on all open sets of a topological space}}
In [[mathematics]], specifically in [[Measure (mathematics)|measure theory]], a '''Borel measure''' on  a [[topological space]] is a [[measure (mathematics)|measure]] that is defined on all [[open set]]s (and thus on all [[Borel sets]]).<ref>D. H. Fremlin, 2000. ''[http://www.essex.ac.uk/maths/people/fremlin/mt.htm Measure Theory] {{Webarchive|url=https://web.archive.org/web/20101101220236/http://www.essex.ac.uk/maths/people/fremlin/mt.htm# |date=2010-11-01 }}''. Torres Fremlin.</ref> Some authors require additional restrictions on the measure, as described below.

==Formal definition==
Let <math>X</math> be a [[locally compact]] [[Hausdorff space]], and let <math>\mathfrak{B}(X)</math> be the [[Sigma-algebra#Generated .CF.83-algebra|smallest σ-algebra]] that contains the [[open set]]s of <math>X</math>; this is known as the σ-algebra of [[Borel set]]s. A Borel measure is any measure <math>\mu</math> defined on the σ-algebra of Borel sets.<ref>{{cite book | author=Alan J. Weir | title=General integration and measure | publisher=[[Cambridge University Press]] | year=1974 | isbn=0-521-29715-X | pages=158–184 }}</ref> A few authors require in addition that <math>\mu</math> is [[Locally finite measure|locally finite]], meaning that every point has an open neighborhood with finite measure.  For Hausdorff spaces, this implies that <math>\mu(C)<\infty</math> for every [[compact set]] <math>C</math>; and for locally compact Hausdorff spaces, the two conditions are equivalent. If a Borel measure <math>\mu</math> is both [[inner regular]] and [[Regular_measure#Definition|outer regular]], it is called a [[Borel regular measure|regular Borel measure]]. If <math>\mu</math> is both inner regular, outer regular, and [[Locally finite measure|locally finite]], it is called a [[Radon measure]]. Alternatively, if a regular Borel measure <math>\mu</math> is [[Tightness_of_measures|tight]], it is a Radon measure.

If <math>X</math> is a [[Separable_space|separable]] [[complete metric space]], then every Borel measure <math>\mu</math> on <math>X</math> is a Radon measure.<ref>{{citation| last=Bogachev | first=Vladimir I.| authorlink=Vladimir I. Bogachev| title=Measure Theory | chapter=Measures on topological spaces | publisher=Springer Berlin Heidelberg | publication-place=Berlin, Heidelberg | date=2007 | isbn=978-3-540-34513-8 | doi=10.1007/978-3-540-34514-5_7|page=70}}
</ref>

==On the real line==
The [[real line]] <math>\mathbb R</math> with its [[Real line#As a topological space|usual topology]] is a locally compact Hausdorff space; hence we can define a Borel measure on it. In this case, <math>\mathfrak{B}(\mathbb R)</math> is the smallest σ-algebra that contains the [[open interval]]s of <math>\mathbb R</math>. While there are many Borel measures ''μ'', the choice of Borel measure that assigns <math>\mu((a,b])=b-a</math> for every half-open interval <math>(a,b]</math> is sometimes called "the" Borel measure on <math>\mathbb R</math>. This measure turns out to be the restriction to the Borel σ-algebra of the [[Lebesgue measure]] <math>\lambda</math>, which is a [[complete measure]] and is defined on the Lebesgue σ-algebra. The Lebesgue σ-algebra is actually the ''completion'' of the Borel σ-algebra, which means that it is the smallest σ-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., <math>\lambda(E)=\mu(E)</math> for every Borel measurable set, where <math>\mu</math> is the Borel measure described above). This idea extends to finite-dimensional spaces <math>\mathbb R^n</math> (the [[Cramér–Wold theorem]], below) but does not hold, in general, for infinite-dimensional spaces. [[Infinite-dimensional Lebesgue measure]]s do not exist.

==Product spaces==
If ''X'' and ''Y'' are [[second-countable]], [[Hausdorff topological space|Hausdorff topological spaces]], then the set of Borel subsets <math>B(X\times Y)</math> of their product coincides with the product of the sets <math>B(X)\times B(Y)</math> of Borel subsets of ''X'' and ''Y''.<ref>[[Vladimir I. Bogachev]]. Measure Theory, Volume 1. Springer Science & Business Media, Jan 15, 2007</ref> That is, the Borel [[functor]] 
: <math>\mathbf{Bor}\colon\mathbf{Top}_\mathrm{2CHaus}\to\mathbf{Meas}</math> 
from the [[category (mathematics)|category]] of second-countable Hausdorff spaces to the category of [[Measurable space|measurable spaces]] preserves finite [[Product (category theory)|products]].

==Applications==

===Lebesgue–Stieltjes integral===
{{Main|Lebesgue–Stieltjes integration}}
The [[Lebesgue–Stieltjes integral]] is the ordinary [[Lebesgue integral]] with respect to a measure known as the Lebesgue–Stieltjes measure, which may be associated to any function of [[bounded variation]] on the real line.  The Lebesgue–Stieltjes measure is a [[Borel regular measure|regular Borel measure]], and conversely every regular Borel measure on the real line is of this kind.<ref>{{Citation|last1=Halmos|first1=Paul R.|author1-link=Paul R. Halmos|title=Measure Theory|publisher=[[Springer-Verlag]]|location=Berlin, New York|isbn=978-0-387-90088-9|year=1974|url-access=registration|url=https://archive.org/details/measuretheory00halm}}</ref>

===Laplace transform===
{{Main|Bernstein's theorem on monotone functions}}
One can define the [[Laplace transform]] of a finite Borel measure ''μ'' on the [[real line]] by the [[Lebesgue integration|Lebesgue integral]]<ref>{{harvnb|Feller|1971|loc=§XIII.1}}</ref>

: <math>(\mathcal{L}\mu)(s) = \int_{[0,\infty)} e^{-st}\,d\mu(t).</math>

An important special case is where ''μ'' 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 [[Cumulative distribution function|distribution function]] ''f''.  In that case, to avoid potential confusion, one often writes

: <math>(\mathcal{L}f)(s) = \int_{0^-}^\infty e^{-st}f(t)\,dt</math>

where the lower limit of 0<sup>−</sup> is shorthand notation for

: <math>\lim_{\varepsilon\downarrow 0}\int_{-\varepsilon}^\infty.</math>

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–Stieltjes transform]].

===Moment problem===
{{Main|Moment problem}}
One can define the [[moment (mathematics)|moments]] of a finite Borel measure ''μ'' on the [[real line]] by the integral

: <math>m_n = \int_a^b x^n\,d\mu(x).</math>

For <math>(a,b)=(-\infty,\infty),\;(0,\infty),\;(0,1)</math> these correspond to the [[Hamburger moment problem]], the [[Stieltjes moment problem]] and the [[Hausdorff moment problem]], respectively. The question or problem to be solved is, given a collection of such moments, is there a corresponding measure? For the Hausdorff moment problem, the corresponding measure is unique. For the other variants, in general, there are an infinite number of distinct measures that give the same moments.

===Hausdorff dimension and Frostman's lemma===
{{Main|Hausdorff dimension|Frostman lemma}}

Given a Borel measure ''μ'' on a metric space ''X'' such that ''μ''(''X'') > 0 and ''μ''(''B''(''x'', ''r'')) ≤ ''r<sup>s</sup>'' holds for some constant ''s'' > 0 and for every ball ''B''(''x'', ''r'') in ''X'', then the [[Hausdorff dimension]] dim<sub>Haus</sub>(''X'') ≥ ''s''. A partial converse is provided by the [[Frostman lemma]]:<ref>{{cite book
|    author = Rogers, C. A.
|     title = Hausdorff measures
|   edition = Third
|    series = Cambridge Mathematical Library
| publisher = Cambridge University Press
|  location = Cambridge
|      year = 1998
|     pages = xxx+195
|        isbn = 0-521-62491-6
}}</ref>

'''Lemma:''' Let ''A'' be a [[Borel measurable|Borel]] subset of '''R'''<sup>''n''</sup>, and let ''s''&nbsp;>&nbsp;0. Then the following are equivalent:
*''H''<sup>''s''</sup>(''A'')&nbsp;>&nbsp;0, where ''H''<sup>''s''</sup> denotes the ''s''-dimensional [[Hausdorff measure]].
*There is an (unsigned) Borel measure ''&mu;'' satisfying ''&mu;''(''A'')&nbsp;>&nbsp;0, and such that 
::<math>\mu(B(x,r))\le r^s</math> 
:holds for all ''x''&nbsp;&isin;&nbsp;'''R'''<sup>''n''</sup> and ''r''&nbsp;>&nbsp;0.

===Cramér–Wold theorem===
{{Main|Cramér–Wold theorem}}
The [[Cramér–Wold theorem]] in [[measure theory]] states that a Borel [[probability measure]] on <math>\mathbb R^k</math> is uniquely determined by the totality of its one-dimensional projections.<ref>K. Stromberg, 1994. ''Probability Theory for Analysts''. Chapman and Hall.</ref> It is used as a method for proving joint convergence results. The theorem is named after [[Harald Cramér]] and [[Herman Ole Andreas Wold]].

==See also==
* [[Jacobi operator]]

==References==
{{reflist}}

==Further reading==
* [[Gaussian measure]], a finite-dimensional Borel measure
* {{Citation | last1=Feller | first1=William | author1-link=William Feller | title=An introduction to probability theory and its applications. Vol. II. | publisher=[[John Wiley & Sons]] | location=New York | series=Second edition | mr=0270403  | year=1971}}.
* {{cite book | author=J. D. Pryce | title=Basic methods of functional analysis | series=Hutchinson University Library | publisher=[[Hutchinson (publisher)|Hutchinson]] | year=1973 | isbn=0-09-113411-0 | page=217 }}
* {{cite book | last=Ransford | first=Thomas | title=Potential theory in the complex plane | series=London Mathematical Society Student Texts | volume=28 | location=Cambridge | publisher=[[Cambridge University Press]] | year=1995 | isbn=0-521-46654-7 | zbl=0828.31001 | pages=[https://archive.org/details/potentialtheoryi0000rans/page/209 209–218] | url=https://archive.org/details/potentialtheoryi0000rans/page/209 }}
* {{citation | last = Teschl| first = Gerald| authorlink = Gerald Teschl| title = Topics in Real Analysis| url = https://www.mat.univie.ac.at/~gerald/ftp/book-ra/index.html|publisher = (lecture notes)}}
* [[Wiener's lemma]]   related

==External links==
* [https://www.encyclopediaofmath.org/index.php/Borel_measure Borel measure] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

{{Measure theory}}

{{DEFAULTSORT:Borel Measure}}
[[Category:Measures (measure theory)]]