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 $X$ 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.^[2] 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). This idea extends to finite-dimensional spaces $\mathbb {R} ^{n}$ (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 $B(X\times Y)$ of their product coincides with the product of the sets $B(X)\times B(Y)$ of Borel subsets of X and Y.^[3] 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

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]

({\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

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

\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 $\mathbb {R} ^{k}$ 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

↑ D. H. Fremlin, 2000. Measure Theory Archived 2010-11-01 at the Wayback Machine. Torres Fremlin.
↑ Alan J. Weir (1974). General integration and measure. Cambridge University Press. pp. 158–184. ISBN 0-521-29715-X.
↑ Vladimir I. Bogachev. Measure Theory, Volume 1. Springer Science & Business Media, Jan 15, 2007
↑ Halmos, Paul R. (1974), Measure Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90088-9
↑ Feller 1971, §XIII.1
↑ Rogers, C. A. (1998). Hausdorff measures. Cambridge Mathematical Library (Third ed.). Cambridge: Cambridge University Press. pp. xxx+195. ISBN 0-521-62491-6.
↑ K. Stromberg, 1994. Probability Theory for Analysts. Chapman and Hall.

External links

Borel measure at Encyclopedia of Mathematics

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] D. H. Fremlin, 2000. Measure Theory Archived 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.

[1]

[2]

[3]

[4]

[5]

[6]

[7]