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] A few authors require in addition that ${\displaystyle \mu }$ is locally finite, 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. If ${\displaystyle \mu }$ is both inner regular, outer 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 that 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 to 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 that contains all the Borel sets and can be equipped with a complete measure. 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

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

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)) ≤ 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 the Frostman lemma:^[6]

Lemma: Let A be a Borel subset of Rⁿ, and let s > 0. Then the following are equivalent:

• H^s(A) > 0, where H^s 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 ${\displaystyle \mathbb {R} ^{k))$ 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.