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 intervalsof. 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 can be equipped with a complete measure. Also, the Borel measure and the Lebesgue measure coincide on the Borel sets (i.e., for every Borel measurable set, where is the Borel measure described above). This idea extends to finite-dimensional spaces (the Cramér–Wold theorem, below) but does not hold, in general, for infinite-dimensional spaces. Infinite-dimensional Lebesgue measures do not exist.
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.[4]
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 functionf. 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.
One can define the moments of a finite Borel measure μ on the real line by the integral
For 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.
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:[6]
Lemma: Let A be a Borel subset of Rn, and let s > 0. Then the following are equivalent:
^Rogers, C. A. (1998). Hausdorff measures. Cambridge Mathematical Library (Third ed.). Cambridge: Cambridge University Press. pp. xxx+195. ISBN0-521-62491-6.
^K. Stromberg, 1994. Probability Theory for Analysts. Chapman and Hall.