Borel measure – Wikipedia

Measure defined on all open sets of a topological space

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[edit]

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

${displaystyle mu (C)$

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[edit]

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[edit]

If X and Y are second-countable, Hausdorff topological spaces, then the set of Borel subsets

${displaystyle B(Xtimes Y)}$

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[edit]

Lebesgue–Stieltjes integral[edit]

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[edit]

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

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

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[edit]

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[edit]

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.

References[edit]

Further reading[edit]

• 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[edit]