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).^{[1]} Some authors require additional restrictions on the measure, as described below.

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.^{[2]} A few authors require in addition that is locally finite, 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, outer regular, and locally finite, it is called a Radon measure.

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 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).

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*.^{[3]} That is, the Borel functor

from the category of second-countable Hausdorff spaces to the category of measurable spaces preserves finite products.

Main article: Lebesgue–Stieltjes integration |

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]}

Main article: Laplace transform |

One can define the Laplace transform of a finite Borel measure *μ* on the real line by the Lebesgue integral^{[5]}

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.

Main articles: Hausdorff dimension and Frostman 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

**Lemma:** Let *A* be a Borel subset of **R**^{n}, and let *s* > 0. Then the following are equivalent:

*H*^{s}(*A*) > 0, where*H*^{s}denotes the*s*-dimensional Hausdorff measure.- There is an (unsigned) Borel measure
*μ*satisfying*μ*(*A*) > 0, and such that

- holds for all
*x*∈**R**^{n}and*r*> 0.

Main article: Cramér–Wold theorem |

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.^{[7]} It is used as a method for proving joint convergence results. The theorem is named after Harald Cramér and Herman Ole Andreas Wold.