[1] Some authors require additional restrictions on the measure, as described below.
defined on the σ-algebra of Borel sets.
; and for locally compact Hausdorff spaces, the two conditions are equivalent.
Alternatively, if a regular Borel measure
is a separable complete metric space, then every Borel measure
with its usual topology is a locally compact Hausdorff space; hence we can define a Borel measure on it.
, which is a complete measure and is defined on the Lebesgue σ-algebra.
(the Cramér–Wold theorem, below) but does not hold, in general, for infinite-dimensional spaces.
Infinite-dimensional Lebesgue measures do not exist.
If X and Y are second-countable, Hausdorff topological spaces, then the set of Borel subsets
[4] That is, the Borel functor from the category of second-countable Hausdorff spaces to the category of measurable spaces preserves finite products.
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.
[5] One can define the Laplace transform of a finite Borel measure μ on the real line by the Lebesgue integral[6] 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.
One can define the moments of a finite Borel measure μ on the real line by the integral For
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:[7] Lemma: Let A be a Borel subset of Rn, and let s > 0.
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.