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. 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. 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. 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 that assigns for every half-open interval is sometimes called "the" Borel measure on. This measure turns out to be the restriction to 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 that 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.

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. 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.

Laplace transform

One can define the Laplace transform of a finite Borel measure μ on the real line by the Lebesgue integral
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 > 0 and μ ≤ rs holds for some constant s > 0 and for every ball B in X, then the Hausdorff dimension dimHaus ≥ 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:
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.