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).D. H. Fremlin, 2000.
Measure Theory .
Torres Fremlin.
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 σ-algebra that contains the open sets of X; this is known as the σ-algebra of Borel sets.
A Borel measure is any measure \mu defined on the σ-algebra of Borel sets.
A few authors require in addition that \mu is locally finite, meaning that \mu(C)<\infty for every compact set C.
If a Borel measure \mu is both inner regular and outer regular, it is called a regular Borel measure.
If \mu is both inner regular, outer regular, and locally finite, it is called a Radon measure.
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 σ-algebra that contains the open intervals of \mathbb R.
While there are many Borel measures μ, the choice of Borel measure that assigns \mu((a,b])=b-a for every half-open interval (a,b] is sometimes called "the" Borel measure on \mathbb R.
This measure turns out to be the restriction to the Borel σ-algebra of the Lebesgue measure \lambda, 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 has a complete measure on it.
Also, the Borel measure and the Lebesgue measure coincide on the Borel sets (i.e., \lambda(E)=\mu(E) for every Borel measurable set, where \mu is the Borel measure described above).
Product spaces
If X and Y are second-countable, Hausdorff topological spaces, then the set of Borel subsets B(X\times Y) of their product coincides with the product of the sets B(X)\times B(Y) of Borel subsets of X and Y.Vladimir I. Bogachev.
Measure Theory, Volume 1.
Springer Science & Business Media, Jan 15, 2007 That is, the Borel functor
\mathbf{Bor}\colon\mathbf{Top}_{2CHaus}\to\mathbf{Meas}
from the category of second-countable Hausdorff spaces to the category of measurable spaces preserves finite products.
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.
Laplace transform
One can define the Laplace transform of a finite Borel measure μ on the real line by the Lebesgue integral
(\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)) ≤ rs holds for some constant s > 0 and for every ball B(x, r) in X, then the Hausdorff dimension dimHaus(X) ≥ s.
A partial converse is provided by Frostman's lemma:
Lemma: Let A be a Borel subset of Rn, and let s > 0.
Then the following are equivalent:
Hs(A) > 0, where Hs denotes the s-dimensional 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 ∈ Rn and r > 0.
Cramér–Wold theorem
The Cramér–Wold theorem in measure theory states that a Borel probability measure on \mathbb R^k is uniquely determined by the totality of its one-dimensional projections.K. Stromberg, 1994.
Probability Theory for Analysts.
Chapman and Hall.
It is used as a method for proving joint convergence results.
The theorem is named after Harald Cramér and Herman Ole Andreas Wold.
References
Further reading
Gaussian measure, a finite-dimensional Borel measure
.
Wiener's lemma   related
External links
Borel measure at Encyclopedia of Mathematics
