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). Some authors require additional restrictions on the measure, as described below.
Let X be a locally compact Hausdorff space, and let 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 μ defined on the σ-algebra of Borel sets. 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
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 . In practice, even "the" Borel measure is not the most useful measure defined on the σ-algebra of Borel sets; indeed, the Lebesgue measure 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 is an extension of the Borel measure , 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., for every Borel measurable set).
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. That is, the Borel functor
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.
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
where the lower limit of 0− is shorthand notation for
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)) ≤ 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:
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
- holds for all x ∈ Rn and r > 0.
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. It is used as a method for proving joint convergence results. The theorem is named after Harald Cramér and Herman Ole Andreas Wold.
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