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. 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 can be equipped with a complete measure. 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

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

from the category of second-countable Hausdorff spaces to the category of measurable spaces preserves finite products.

Applications

Lebesgue–Stieltjes integral

Main article: Lebesgue–Stieltjes integration

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

Main article: 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

Main articles: Hausdorff dimension and Frostman 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:

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

$\mu (B(x,r))\leq r^{s}$

holds for all x ∈ Rⁿ and r > 0.

Cramér–Wold theorem

Main article: 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.

Further reading

• Gaussian measure, a finite-dimensional Borel measure
• Feller, William (1971), An introduction to probability theory and its applications. Vol. II., Second edition, New York: John Wiley & Sons, MR 0270403.
• J. D. Pryce (1973). Basic methods of functional analysis. Hutchinson University Library. Hutchinson. p. 217. ISBN 0-09-113411-0.
• Ransford, Thomas (1995). Potential theory in the complex plane. London Mathematical Society Student Texts. Vol. 28. Cambridge: Cambridge University Press. pp. 209–218. ISBN 0-521-46654-7. Zbl 0828.31001.
• Teschl, Gerald, Topics in Real and Functional Analysis, (lecture notes)
• Wiener's lemma related

External links

• Borel measure at Encyclopedia of Mathematics