Editing Borel measure (section)

==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]].