“Borel measure”的意思、由来-开放百科全书

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

be a locally compact Hausdorff space, and let

be the smallest σ-algebra that contains the open sets of

; this is known as the σ-algebra of Borel sets. A Borel measure is any measure

defined on the σ-algebra of Borel sets.^[2] Some authors require in addition that

is locally compact, meaning that

for every compact set

. If a Borel measure

is both inner regular and outer regular, it is called a regular Borel measure (some authors also require it to be tight). If

is both inner regular and locally finite, it is called a Radon measure.

On the real line

The real line

with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it. In this case,

is the smallest σ-algebra that contains the open intervals of

. While there are many Borel measures μ, the choice of Borel measure which assigns

for every half-open interval

is sometimes called "the" Borel measure on

. This measure turns out to be the restriction on the Borel σ-algebra of the Lebesgue measure

, 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 which contains all the Borel sets and has a complete measure on it. Also, the Borel measure and the Lebesgue measure coincide on the Borel sets (i.e.,

for every Borel measurable set, where

is the Borel measure described above).

Product spaces

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

of their product coincides with the product of the sets

of Borel subsets of X and Y.^[3] That is, the Borel functor

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]

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

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 Frostman's lemma:^[6]

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

Cramér–Wold theorem

The Cramér–Wold theorem in measure theory states that a Borel probability measure on

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 {{Webarchive|url=https://web.archive.org/web/20101101220236/http://www.essex.ac.uk/maths/people/fremlin/mt.htm# |date=2010-11-01 }}. Torres Fremlin.
2. ^{{cite book | author=Alan J. Weir | title=General integration and measure | publisher=Cambridge University Press | year=1974 | isbn=0-521-29715-X | pages=158–184 }}
3. ^Vladimir I. Bogachev. Measure Theory, Volume 1. Springer Science & Business Media, Jan 15, 2007
4. ^{{Citation|last1=Halmos|first1=Paul R.|author1-link=Paul R. Halmos|title=Measure Theory | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-90088-9 | year=1974}}
5. ^{{harvnb|Feller|1971|loc=§XIII.1}}
6. ^{{cite book| author = Rogers, C. A.| title = Hausdorff measures| edition = Third| series = Cambridge Mathematical Library| publisher = Cambridge University Press| location = Cambridge| year = 1998| pages = xxx+195| isbn = 0-521-62491-6}}
7. ^K. Stromberg, 1994. Probability Theory for Analysts. Chapman and Hall.