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
[edit]Let be a locally compact Hausdorff space, and let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\mathfrak {B}}(X)} be the smallest σ-algebra that contains the open sets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} ; this is known as the σ-algebra of Borel sets. A Borel measure is any measure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} defined on the σ-algebra of Borel sets.[2] A few authors require in addition that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} is locally finite, meaning that every point has an open neighborhood with finite measure. For Hausdorff spaces, this implies that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu(C)<\infty} for every compact set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} ; and for locally compact Hausdorff spaces, the two conditions are equivalent. If a Borel measure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} is both inner regular and outer regular, it is called a regular Borel measure. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} is both inner regular, outer regular, and locally finite, it is called a Radon measure. Alternatively, if a regular Borel measure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} is tight, it is a Radon measure.
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a separable complete metric space, then every Borel measure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is a Radon measure.[3]
On the real line
[edit]The real line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{B}(\mathbb R)} is the smallest σ-algebra that contains the open intervals of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb R} . While there are many Borel measures μ, the choice of Borel measure that assigns Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu((a,b])=b-a} for every half-open interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b]} is sometimes called "the" Borel measure on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb R} . This measure turns out to be the restriction to the Borel σ-algebra of the Lebesgue measure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda(E)=\mu(E)} for every Borel measurable set, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} is the Borel measure described above). This idea extends to finite-dimensional spaces Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb R^n} (the Cramér–Wold theorem, below) but does not hold, in general, for infinite-dimensional spaces. Infinite-dimensional Lebesgue measures do not exist.
Product spaces
[edit]If X and Y are second-countable, Hausdorff topological spaces, then the set of Borel subsets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(X\times Y)} of their product coincides with the product of the sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(X)\times B(Y)} of Borel subsets of X and Y.[4] That is, the Borel functor
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.[5]
Laplace transform
[edit]One can define the Laplace transform of a finite Borel measure μ on the real line by the Lebesgue integral[6]
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathcal{L}f)(s) = \int_{0^-}^\infty e^{-st}f(t)\,dt}
where the lower limit of 0− is shorthand notation for
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.
Moment problem
[edit]One can define the moments of a finite Borel measure μ on the real line by the integral
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_n = \int_a^b x^n\,d\mu(x).}
For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b)=(-\infty,\infty),\;(0,\infty),\;(0,1)} these correspond to the Hamburger moment problem, the Stieltjes moment problem and the Hausdorff moment problem, respectively. The question or problem to be solved is, given a collection of such moments, is there a corresponding measure? For the Hausdorff moment problem, the corresponding measure is unique. For the other variants, in general, there are an infinite number of distinct measures that give the same moments.
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 the Frostman lemma:[7]
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
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb R^k} is uniquely determined by the totality of its one-dimensional projections.[8] It is used as a method for proving joint convergence results. The theorem is named after Harald Cramér and Herman Ole Andreas Wold.
See also
[edit]References
[edit]- ↑ 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.
- ↑ Bogachev, Vladimir I. (2007), "Measures on topological spaces", Measure Theory, Berlin, Heidelberg: Springer Berlin Heidelberg, p. 70, doi:10.1007/978-3-540-34514-5_7, ISBN 978-3-540-34513-8
- ↑ 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.
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. 28. Cambridge: Cambridge University Press. pp. 209–218. ISBN 0-521-46654-7. Zbl 0828.31001.
- Teschl, Gerald, Topics in Real Analysis, (lecture notes)
- Wiener's lemma related