In 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.
Formal definition
Let X be a locally compact Hausdorff space, and let \mathfrak{B}(X) be the smallest σalgebra that contains the open sets of X; this is known as the σalgebra of Borel sets. A Borel measure is any measure μ defined on the σalgebra of Borel sets.^{[2]} Some authors require in addition that μ(C) < ∞ for every compact set C. If a Borel measure μ is both inner regular and outer regular, it is called a regular Borel measure (some authors also require it to be tight). If μ is both inner regular and locally finite, it is called a Radon measure. Note that a locally finite Borel measure automatically satisfies μ(C) < ∞ for every compact set C.
On the real line
The real line \mathbb R with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it. In this case, \mathfrak{B}(\mathbb R) is the smallest σalgebra that contains the open intervals of \mathbb R. While there are many Borel measures μ, the choice of Borel measure which assigns \mu([a,b])=ba for every interval [a,b] is sometimes called "the" Borel measure on \mathbb R. In practice, even "the" Borel measure is not the most useful measure defined on the σalgebra of Borel sets; indeed, the Lebesgue measure \lambda 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 \lambda is an extension of the Borel measure \mu, it means that every Borelmeasurable set E is also a Lebesguemeasurable set, and the Borel measure and the Lebesgue measure coincide on the Borel sets (i.e., \lambda(E)=\mu(E) for every Borel measurable set).
Applications
Lebesgue–Stieltjes integral
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 regular Borel measure, and conversely every regular Borel measure on the real line is of this kind.^{[3]}
Laplace transform
One can define the Laplace transform of a finite Borel measure μ on the real line by the Lebesgue integral^{[4]}

(\mathcal{L}\mu)(s) = \int_{[0,\infty)} e^{st}\,d\mu(t).
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 distribution function f. In that case, to avoid potential confusion, one often writes

(\mathcal{L}f)(s) = \int_{0^}^\infty e^{st}f(t)\,dt
where the lower limit of 0^{−} is shorthand notation for

\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–Stieltjes transform.
Hausdorff dimension and Frostman's lemma
Given a Borel measure μ on a metric space X such that μ(X) > 0 and μ(B(x, r)) ≤ r^{s} holds for some constant s > 0 and for every ball B(x, r) in X, then the Hausdorff dimension dim_{Haus}(X) ≥ s. A partial converse is provided by Frostman's lemma:^{[5]}
Lemma: Let A be a Borel subset of R^{n}, and let s > 0. Then the following are equivalent:

H^{s}(A) > 0, where H^{s} denotes the sdimensional Hausdorff measure.

There is an (unsigned) Borel measure μ satisfying μ(A) > 0, and such that


\mu(B(x,r))\le r^s

holds for all x ∈ R^{n} and r > 0.
Cramér–Wold theorem
The Cramér–Wold theorem in measure theory states that a Borel probability measure on R^k is uniquely determined by the totality of its onedimensional projections.^{[6]} It is used as a method for proving joint convergence results. The theorem is named after Harald Cramér and Herman Ole Andreas Wold.
References

^ D. H. Fremlin, 2000. Measure Theory. Torres Fremlin.

^

^

^ Feller 1971, §XIII.1

^

^ K. Stromberg, 1994. Probability Theory for Analysts. Chapman and Hall.
Further reading
External links

Borel measure at Encyclopedia of Mathematics
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