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).^[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). This idea extends to finite-dimensional spaces ${\displaystyle \mathbb {R} ^{n}}$  (the Cramér–Wold theorem, below) but does not hold, in general, for infinite-dimensional spaces.

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

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.^[4]

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^[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

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:^[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

Main article: 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.

References

1. D. H. Fremlin, 2000. Measure Theory 2010-11-01 at the Wayback Machine. 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.

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