Borel measure

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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 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. Any measure μ defined on the σ-algebra of Borel sets is called a Borel measure.[2] Some authors require in addition that μ(C) < ∞ for every compact set C. If a Borel measure μ is both inner regular and outer regular, it is called a regular Borel measure (some authors also require it to be tight). If μ is both inner regular and locally finite, it is called a Radon measure. Note that a locally finite Borel measure automatically satisfies μ(C) < ∞ for every compact set C.

On the real line[edit]

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 which assigns \mu([a,b])=b-a for every interval [a,b] is sometimes called "the" Borel measure on \mathbb R. In practice, even "the" Borel measure is not the most useful measure defined on the σ-algebra of Borel sets; indeed, the Lebesgue measure \lambda is an extension of "the" Borel measure which possesses the crucial property that it is a complete measure (unlike the Borel measure). To clarify, when one says that the Lebesgue measure \lambda is an extension of the Borel measure \mu, it means that every Borel-measurable set E is also a Lebesgue-measurable set, and the Borel measure and the Lebesgue measure coincide on the Borel sets (i.e., \lambda(E)=\mu(E) for every Borel measurable set).

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.[3]

Laplace transform[edit]

Main article: Laplace transform

One can define the Laplace transform of a finite Borel measure μ on the real line by the Lebesgue integral[4]

(\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[edit]

Given a Borel measure μ on a metric space X such that μ(X) > 0 and μ(B(x, r)) ≤ rs holds for some constant s > 0 and for every ball B(x, r) in X, then the Hausdorff dimension dimHaus(X) ≥ s. A partial converse is provided by Frostman's lemma:[5]

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

  • Hs(A) > 0, where Hs denotes the s-dimensional Hausdorff measure.
  • There is an (unsigned) Borel measure μ satisfying μ(A) > 0, and such that
\mu(B(x,r))\le 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 R^k is uniquely determined by the totality of its one-dimensional projections.[6] 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]

  1. ^ D. H. Fremlin, 2000. Measure Theory. Torres Fremlin.
  2. ^ Alan J. Weir (1974). General integration and measure. Cambridge University Press. pp. 158–184. ISBN 0-521-29715-X. 
  3. ^ Halmos, Paul R. (1974), Measure Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90088-9 
  4. ^ Feller 1971, §XIII.1
  5. ^ Rogers, C. A. (1998). Hausdorff measures. Cambridge Mathematical Library (Third ed.). Cambridge: Cambridge University Press. pp. xxx+195. ISBN 0-521-62491-6. 
  6. ^ K. Stromberg, 1994. Probability Theory for Analysts. Chapman and Hall.

Further reading[edit]

Feller, William (1971), An introduction to probability theory and its applications. Vol. II., Second edition, New York: John Wiley & Sons, MR 0270403 .

External links[edit]