# Borel measure

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). 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. Some authors require in addition that $\mu$  is locally compact, 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 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. 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
$\mu (B(x,r))\leq r^{s}$
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. It is used as a method for proving joint convergence results. The theorem is named after Harald Cramér and Herman Ole Andreas Wold.