Borel measure

Formal definition

Let X be a locally compact Hausdorff space, and let B ( X ) 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 means μ(C) < ∞ for every compact set C.

On the real line

The real line R with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it. In this case, B ( R ) is the smallest σ-algebra that contains the open intervals of R . While there are many Borel measures μ, the choice of Borel measure which assigns μ ( ( a , b ] ) = b − a for every half-open interval ( a , b ] is sometimes called "the" Borel measure on R . This measure turns out to be the restriction on the Borel σ-algebra of the Lebesgue measure λ , 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., λ ( E ) = μ ( E ) for every Borel measurable set, where μ is the Borel measure described above).

Product spaces

If X and Y are second-countable, Hausdorff topological spaces, then the set of Borel subsets B ( X × Y ) of their product coincides with the product of the sets B ( X ) × B ( Y ) 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.

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(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:

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 R k 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.