# Borel measure

(Redirected from Borel probability 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).[1] Some authors require additional restrictions on the measure, as described below.

## Formal definition

Let ${\displaystyle X}$ be a locally compact Hausdorff space, and let ${\displaystyle {\mathfrak {B}}(X)}$ be the smallest σ-algebra that contains the open sets of ${\displaystyle X}$; this is known as the σ-algebra of Borel sets. A Borel measure is any measure ${\displaystyle \mu }$ defined on the σ-algebra of Borel sets.[2] Some authors require in addition that ${\displaystyle \mu }$ is locally compact, meaning that ${\displaystyle \mu (C)<\infty }$ for every compact set ${\displaystyle C}$. If a Borel measure ${\displaystyle \mu }$ is both inner regular and outer regular, it is called a regular Borel measure (some authors also require it to be tight). If ${\displaystyle \mu }$ is both inner regular and locally finite, it is called a Radon measure.

## On the real line

The real line ${\displaystyle \mathbb {R} }$ with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it. In this case, ${\displaystyle {\mathfrak {B}}(\mathbb {R} )}$ is the smallest σ-algebra that contains the open intervals of ${\displaystyle \mathbb {R} }$. While there are many Borel measures μ, the choice of Borel measure which assigns ${\displaystyle \mu ((a,b])=b-a}$ for every half-open interval ${\displaystyle (a,b]}$ is sometimes called "the" Borel measure on ${\displaystyle \mathbb {R} }$. This measure turns out to be the restriction on the Borel σ-algebra of the Lebesgue measure ${\displaystyle \lambda }$, 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 which 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., ${\displaystyle \lambda (E)=\mu (E)}$ for every Borel measurable set, where ${\displaystyle \mu }$ is the Borel measure described above).

## Product spaces

If X and Y are second-countable, Hausdorff topological spaces, then the set of Borel subsets ${\displaystyle B(X\times Y)}$ of their product coincides with the product of the sets ${\displaystyle B(X)\times B(Y)}$ of Borel subsets of X and Y.[3] That is, the Borel functor

${\displaystyle \mathbf {Bor} \colon \mathbf {Top} _{2CHaus}\to \mathbf {Meas} }$

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.[4]

### Laplace transform

One can define the Laplace transform of a finite Borel measure μ on the real line by the Lebesgue integral[5]

${\displaystyle ({\mathcal {L}}\mu )(s)=\int _{[0,\infty )}e^{-st}\,d\mu (t).}$

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

${\displaystyle ({\mathcal {L}}f)(s)=\int _{0^{-}}^{\infty }e^{-st}f(t)\,dt}$

where the lower limit of 0 is shorthand notation for

${\displaystyle \lim _{\varepsilon \downarrow 0}\int _{-\varepsilon }^{\infty }.}$

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:[6]

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
${\displaystyle \mu (B(x,r))\leq r^{s}}$
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 ${\displaystyle R^{k}}$ is uniquely determined by the totality of its one-dimensional projections.[7] It is used as a method for proving joint convergence results. The theorem is named after Harald Cramér and Herman Ole Andreas Wold.

## References

1. ^ D. H. Fremlin, 2000. Measure Theory. Torres Fremlin.
2. ^ Alan J. Weir (1974). General integration and measure. Cambridge University Press. pp. 158–184. ISBN 0-521-29715-X.
3. ^ Vladimir I. Bogachev. Measure Theory, Volume 1. Springer Science & Business Media, Jan 15, 2007
4. ^ Halmos, Paul R. (1974), Measure Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90088-9
5. ^ Feller 1971, §XIII.1
6. ^ Rogers, C. A. (1998). Hausdorff measures. Cambridge Mathematical Library (Third ed.). Cambridge: Cambridge University Press. pp. xxx+195. ISBN 0-521-62491-6.
7. ^ K. Stromberg, 1994. Probability Theory for Analysts. Chapman and Hall.